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(the)g(w)m(ork)118 4643 y(of)c(Sinai)f([Si)o(])h(for)g(Anoso)m(v)h (di\013eomorphisms,)f(later)f(extended)k(b)m(y)e(Ruelle)e(and)h(Bo)m(w) m(en)i([BR,)e(Ru])118 4763 y(for)33 b(Axiom)g(A)h(di\013eomorphisms)e (and)i(\015o)m(ws.)48 b(In)34 b(the)h(con)m(text)g(of)e(systems)i(with) f(no)g(uniformly)p 118 4831 1465 4 v 229 4892 a Fm(\003)268 4922 y Fl(W)-7 b(ork)26 b(partially)h(supp)r(orted)h(b)n(y)f(F)n(CT)g (through)g Fk(Centr)l(o)j(de)g(Matem\023)-42 b(atic)l(a)31 b(da)g(Universidade)h(do)e(Porto.)1924 5251 y Ft(1)p eop %%Page: 2 2 2 1 bop 118 548 a Ft(h)m(yp)s(erb)s(olic)38 b(structure)i(Jak)m(obson)f ([Ja])g(pro)m(v)m(ed)h(the)f(existence)h(of)e(suc)m(h)i(measures)f(for) f(certain)118 668 y(quadratic)i(transformations)f(of)h(the)h(in)m(terv) -5 b(al)39 b(exhibiting)g(c)m(haotic)h(b)s(eha)m(viour.)67 b(Another)41 b(im-)118 789 y(p)s(ortan)m(t)27 b(con)m(tribution)f(on)i (this)f(sub)5 b(ject)29 b(w)m(as)f(giv)m(en)f(b)m(y)i(Benedic)m(ks)g (and)e(Y)-8 b(oung)27 b([BY)q(],)h(based)h(on)118 909 y(the)i(previous)g(w)m(ork)g(of)f(Benedic)m(ks)i(and)e(Carleson)h ([BC1,)g(BC2],)g(where)h(this)e(kind)g(of)g(measures)118 1029 y(w)m(ere)36 b(constructed)g(for)e(H)m(\023)-46 b(enon)35 b(t)m(w)m(o)g(dimensional)d(maps)i(exhibiting)f(strange)i (attractors.)49 b(The)118 1150 y(recen)m(t)27 b(w)m(ork)g(of)f(Alv)m (es,)i(Bonatti)c(and)i(Viana)f([ABV])h(sho)m(ws)i(that)d(suc)m(h)j (measures)e(exist)h(in)e(great)118 1270 y(generalit)m(y)32 b(for)g(systems)i(exhibiting)d(some)h(non-uniformly)e(expanding)j(b)s (eha)m(viour.)264 1391 y(The)43 b(notion)d(of)g(stabilit)m(y)g(that)h (most)f(concerns)j(us)f(in)e(this)h(w)m(ork)h(can)f(b)s(e)h(form)m (ulated)d(in)118 1511 y(the)31 b(follo)m(wing)d(w)m(a)m(y)-8 b(.)44 b(Assume)31 b(that,)g(instead)f(of)g(time)f(a)m(v)m(erages)j(of) e(Dirac)f(measures)j(supp)s(orted)118 1631 y(on)j(the)g(iterates)g(of)g Fs(x)945 1646 y Fp(0)1016 1631 y Fo(2)e Fs(M)10 b Ft(,)36 b(w)m(e)g(consider)f(time)f(a)m(v)m(erages)i(of)f(Dirac)f(measures)h Fs(\016)3269 1646 y Fr(x)3309 1656 y Fn(j)3346 1631 y Ft(,)h(where)g(at)118 1752 y(eac)m(h)e(iteration)c(w)m(e)k(tak)m(e)f Fs(x)1143 1767 y Fr(j)t Fp(+1)1303 1752 y Ft(close)f(to)g Fs(f)11 b Ft(\()p Fs(x)1807 1767 y Fr(j)1844 1752 y Ft(\))32 b(with)h(a)f(con)m(trolled)f(error.)44 b(One)33 b(is)f(in)m(terested)h (in)118 1872 y(studying)j(the)g(existence)i(of)d(limit)d(measures)k (for)g(these)h(time)d(a)m(v)m(erages)j(and)f(their)f(relation)f(to)118 1993 y(the)f(analogous)e(ones)j(for)e(unp)s(erturb)s(ed)h(orbits,)f (that)h(is,)f(their)g(sto)s(c)m(hastic)h(stabilit)m(y)-8 b(.)264 2113 y(Systems)40 b(with)e(some)g(uniformly)e(h)m(yp)s(erb)s (olic)i(structure)h(are)g(quite)f(w)m(ell)g(understo)s(o)s(d)g(and)118 2233 y(stabilit)m(y)32 b(results)j(ha)m(v)m(e)g(b)s(een)g(established)f (in)f(general,)h(see)h([Ki1)o(,)f(Ki2)o(])g(and)g([Y)-8 b(o].)48 b(The)35 b(kno)m(wl-)118 2354 y(edge)j(of)f(the)h(sto)s(c)m (hastic)g(b)s(eha)m(viour)g(of)f(systems)i(that)e(do)h(not)f(exhibit)g (suc)m(h)i(uniform)d(expan-)118 2474 y(sion/con)m(traction)j(is)g (still)f(v)m(ery)j(incomplete.)64 b(Imp)s(ortan)m(t)39 b(results)h(on)g(this)f(sub)5 b(ject)42 b(w)m(ere)f(ob-)118 2594 y(tained)32 b(b)m(y)h(Benedic)m(ks,)h(Y)-8 b(oung)32 b([BY],)g(Baladi)e(and)j(Viana)e([BaV])h(for)f(certain)h(quadratic)g (maps)118 2715 y(of)37 b(the)i(in)m(terv)-5 b(al.)58 b(Another)38 b(imp)s(ortan)m(t)e(con)m(tribution)h(is)g(the)h (announced)i(w)m(ork)e(of)g(Benedic)m(ks)118 2835 y(and)k(Viana)f(for)g (H)m(\023)-46 b(enon-lik)m(e)42 b(strange)g(attractors.)71 b(As)43 b(far)e(as)h(w)m(e)h(kno)m(w)g(these)h(are)e(the)g(only)118 2956 y(results)33 b(of)f(this)g(t)m(yp)s(e)i(for)e(systems)i(with)e(no) h(uniform)d(expanding)j(b)s(eha)m(viour.)264 3076 y(Although)41 b(w)m(e)h(are)f(still)e(not)i(able)g(to)g(understand)i(whether,)i(in)40 b(general,)j(non-uniformly)118 3196 y(expanding)38 b(systems)g(admit)e (limit)d(measures)38 b(that)f(are)h(stable)f(under)h(small)c(random)j (p)s(ertur-)118 3317 y(bations,)29 b(in)f(this)h(w)m(ork)g(w)m(e)h (presen)m(t)h(b)s(oth)d(su\016cien)m(t)i(conditions)e(and)h(necessary)i (conditions)d(for)118 3437 y(their)g(sto)s(c)m(hastic)g(stabilit)m(y)-8 b(.)40 b(As)28 b(an)g(application)e(of)h(these)i(results)g(w)m(e)g(pro) m(v)m(e)g(that)f(the)g(classes)h(of)118 3557 y(non-uniformly)23 b(expanding)j(maps)f(in)m(tro)s(duced)h(in)e([Vi1])h(and)h([ABV])g(are) f(sto)s(c)m(hastically)g(stable.)118 3846 y Fj(1.1)135 b(Statemen)l(t)47 b(of)e(results)118 4031 y Ft(Let)33 b Fs(f)39 b Ft(:)28 b Fs(M)39 b Fo(!)28 b Fs(M)44 b Ft(b)s(e)33 b(a)f(smo)s(oth)g(map)g(de\014ned)i(on)f(a)g(compact)f(riemannian)f (manifold)f Fs(M)10 b Ft(.)45 b(W)-8 b(e)118 4151 y(\014x)33 b(some)g(normalized)d(riemannian)h(v)m(olume)h(form)f Fs(m)i Ft(on)f Fs(M)44 b Ft(that)32 b(w)m(e)i(call)d Fi(L)-5 b(eb)g(esgue)34 b(me)-5 b(asur)g(e)p Ft(.)264 4272 y(Giv)m(en)35 b Fs(\026)e Ft(an)i Fs(f)11 b Ft(-in)m(v)-5 b(arian)m(t)32 b(Borel)h(probabilit)m(y)f(measure)j(on)f Fs(M)10 b Ft(,)36 b(w)m(e)f(sa)m(y)g(that)f Fs(\026)g Ft(is)g(an)g Fi(SRB)118 4392 y(me)-5 b(asur)g(e)34 b Ft(if,)27 b(for)f(a)h(p)s(ositiv)m(e)f(Leb)s(esgue)i(measure)f(set)h (of)e(p)s(oin)m(ts)g Fs(x)j Fo(2)f Fs(M)10 b Ft(,)28 b(the)g(a)m(v)m(eraged)g(sequence)118 4513 y(of)d(Dirac)f(measures)i (along)e(the)h(orbit)g(\()p Fs(f)1632 4476 y Fr(n)1678 4513 y Ft(\()p Fs(x)p Ft(\)\))1847 4528 y Fr(n)p Fq(\025)p Fp(0)2010 4513 y Ft(con)m(v)m(erges)i(in)e(the)h(w)m(eak)2917 4476 y Fq(\003)2983 4513 y Ft(top)s(ology)d(to)i Fs(\026)p Ft(,)i(that)1924 5251 y(2)p eop %%Page: 3 3 3 2 bop 118 548 a Ft(is,)1330 853 y(lim)1279 912 y Fr(n)p Fq(!)p Fp(+)p Fq(1)1549 785 y Ft(1)p 1544 830 59 4 v 1544 921 a Fs(n)1634 728 y Fr(n)p Fq(\000)p Fp(1)1629 758 y Fh(X)1640 968 y Fr(j)t Fp(=0)1789 853 y Fs(')1853 772 y Fh(\000)1899 853 y Fs(f)1958 811 y Fr(n)2005 853 y Ft(\()p Fs(x)p Ft(\))2136 772 y Fh(\001)2209 853 y Ft(=)2313 717 y Fh(Z)2429 853 y Fs(')17 b(d\026)1035 b Ft(\(1\))118 1168 y(for)39 b(ev)m(ery)i(con)m(tin)m(uous)f(map)f Fs(')g Ft(:)g Fs(M)50 b Fo(!)39 b Fg(R)5 b Ft(.)69 b(W)-8 b(e)40 b(de\014ne)h(the)e Fi(b)-5 b(asin)39 b Ft(of)g Fs(\026)g Ft(as)g(the)h(set)g(of)f(those)118 1288 y(p)s(oin)m(ts)g Fs(x)i Ft(in)e Fs(M)50 b Ft(for)39 b(whic)m(h)h(\(1\))g(holds)f(for)g (all)f(con)m(tin)m(uous)i Fs(')p Ft(.)65 b(The)41 b(maps)e(to)h(b)s(e)g (considered)118 1409 y(in)e(this)f(w)m(ork)i(will)d(only)i(ha)m(v)m(e)h (a)f(\014nite)g(n)m(um)m(b)s(er)h(of)f(SRB)g(measures)h(whose)g(basins) f(co)m(v)m(er)i(the)118 1529 y(whole)33 b(manifold)c Fs(M)10 b Ft(,)34 b(up)f(to)f(a)g(set)h(of)f(zero)h(Leb)s(esgue)h (measure.)264 1649 y(W)-8 b(e)40 b(are)g(in)m(terested)h(in)d(studying) i(random)f(p)s(erturbations)g(of)g(the)h(map)f Fs(f)11 b Ft(.)64 b(F)-8 b(or)39 b(that,)i(w)m(e)118 1770 y(tak)m(e)33 b(a)g(con)m(tin)m(uous)g(map)1426 1981 y(\010)28 b(:)83 b Fs(T)97 b Fo(\000)-16 b(!)83 b Fs(C)2109 1945 y Fp(2)2148 1981 y Ft(\()p Fs(M)5 b(;)17 b(M)10 b Ft(\))1652 2102 y Fs(t)101 b Fo(7\000)-16 b(!)83 b Fs(f)2080 2117 y Fr(t)118 2325 y Ft(from)30 b(a)g(metric)g(space)i Fs(T)45 b Ft(in)m(to)30 b(the)h(space)h(of)f Fs(C)1897 2289 y Fp(2)1967 2325 y Ft(maps)f(from)g Fs(M)42 b Ft(to)30 b Fs(M)10 b Ft(,)32 b(with)f Fs(f)38 b Ft(=)28 b Fs(f)3324 2340 y Fr(t)3349 2321 y Ff(\003)3420 2325 y Ft(for)j(some)118 2445 y(\014xed)f Fs(t)385 2409 y Fq(\003)452 2445 y Fo(2)e Fs(T)14 b Ft(.)42 b(Giv)m(en)28 b Fs(x)h Fo(2)f Fs(M)39 b Ft(w)m(e)29 b(call)e(the)i (sequence)2153 2364 y Fh(\000)2199 2445 y Fs(f)2258 2409 y Fr(n)2247 2470 y(t)p 2247 2482 30 3 v 2305 2445 a Ft(\()p Fs(x)p Ft(\))2436 2364 y Fh(\001)2482 2484 y Fr(n)p Fq(\025)p Fp(1)2647 2445 y Ft(a)f Fi(r)-5 b(andom)31 b(orbit)37 b Ft(of)28 b Fs(x)p Ft(,)i(where)118 2586 y Fs(t)p 118 2601 36 4 v 33 w Ft(denotes)k(an)e(elemen)m(t)h(\()p Fs(t)1108 2601 y Fp(1)1147 2586 y Fs(;)17 b(t)1226 2601 y Fp(2)1265 2586 y Fs(;)g(t)1344 2601 y Fp(3)1384 2586 y Fs(;)g(:)g(:)g(:)f Ft(\))32 b(in)g(the)h(pro)s(duct)g(space)g Fs(T)2608 2549 y Fe(N)2693 2586 y Ft(and)1273 2806 y Fs(f)1332 2764 y Fr(n)1321 2830 y(t)p 1321 2842 30 3 v 1406 2806 a Ft(=)28 b Fs(f)1558 2821 y Fr(t)1583 2829 y Fn(n)1652 2806 y Fo(\016)22 b(\001)17 b(\001)g(\001)j(\016)i Fs(f)1982 2821 y Fr(t)2007 2830 y Fd(1)2144 2806 y Ft(for)97 b Fs(n)28 b Fo(\025)g Ft(1)p Fs(:)118 3026 y Ft(W)-8 b(e)34 b(also)e(tak)m(e)i(a)e(family)f(\()p Fs(\022)1158 3041 y Fr(\017)1191 3026 y Ft(\))1229 3041 y Fr(\017>)p Fp(0)1385 3026 y Ft(of)i(probabilit)m(y)e(measures)j(on)f Fs(T)46 b Ft(suc)m(h)35 b(that)e Fo(f)p Ft(supp)17 b(\()p Fs(\022)3437 3041 y Fr(\017)3470 3026 y Ft(\))p Fo(g)3558 3041 y Fr(\017>)p Fp(0)3714 3026 y Ft(is)118 3146 y(a)33 b(nested)i(family)c(of)i(connected)j(compact)d(sets)h(and)g(supp)18 b(\()p Fs(\022)2447 3161 y Fr(\017)2480 3146 y Ft(\))29 b Fo(!)g(f)p Fs(t)2761 3110 y Fq(\003)2800 3146 y Fo(g)34 b Ft(when)g Fs(\017)c Fo(!)f Ft(0.)46 b(W)-8 b(e)34 b(will)118 3266 y(also)41 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3740 y Fn(n)2755 3783 y(t)p 2755 3793 28 3 v 2805 3763 a Fp(\()p Fr(x)p Fp(\))2930 3748 y Ft(along)e(random)g(orbits)118 3807 y Fh(\000)164 3888 y Fs(f)223 3852 y Fr(n)212 3913 y(t)p 212 3925 30 3 v 269 3888 a Ft(\()p Fs(x)p Ft(\))400 3807 y Fh(\001)446 3927 y Fr(n)p Fq(\025)p Fp(0)614 3888 y Ft(con)m(v)m(erges)33 b(in)d(the)i(w)m(eak)1538 3852 y Fq(\003)1609 3888 y Ft(top)s(ology)d(to)i Fs(\026)2185 3852 y Fr(\017)2248 3888 y Ft(for)f Fs(\022)2443 3852 y Fe(N)2440 3913 y Fr(\017)2526 3888 y Ft(almost)f(ev)m(ery)k Fs(t)p 3095 3904 36 4 v 28 w Fo(2)28 b Fs(T)3323 3852 y Fe(N)3375 3888 y Ft(.)43 b(That)31 b(is,)645 4224 y(lim)594 4283 y Fr(n)p Fq(!)p Fp(+)p Fq(1)864 4156 y Ft(1)p 859 4201 59 4 v 859 4292 a Fs(n)949 4099 y Fr(n)p Fq(\000)p Fp(1)944 4129 y Fh(X)955 4339 y Fr(j)t Fp(=0)1104 4224 y Fs(')1168 4143 y Fh(\000)1214 4224 y Fs(f)1273 4182 y Fr(n)1262 4248 y(t)p 1262 4260 30 3 v 1320 4224 a Ft(\()p Fs(x)p Ft(\))1451 4143 y Fh(\001)1524 4224 y Ft(=)1628 4088 y Fh(Z)1744 4224 y Fs(')17 b(d\026)1935 4182 y Fr(\017)2064 4224 y Ft(for)32 b(all)f(con)m(tin)m(uous)i Fs(')11 b Ft(:)33 b Fs(M)39 b Fo(!)27 b Fg(R)362 b Ft(\(2\))118 4552 y(and)30 b Fs(\022)353 4515 y Fe(N)350 4576 y Fr(\017)436 4552 y Ft(almost)e(ev)m(ery)k Fs(t)p 1003 4568 36 4 v 28 w Fo(2)c Fs(T)1231 4515 y Fe(N)1283 4552 y Fs(:)j Ft(W)-8 b(e)30 b(denote)h(the)g(set)g(of)e(p)s(oin)m(ts)h Fs(x)e Fo(2)g Fs(M)41 b Ft(for)30 b(whic)m(h)h(\(2\))f(holds)g(b)m(y) 118 4672 y Fs(B)5 b Ft(\()p Fs(\026)294 4636 y Fr(\017)327 4672 y Ft(\))25 b(and)h(call)f(it)g(the)h Fi(b)-5 b(asin)28 b(of)h Fs(\026)1409 4636 y Fr(\017)1441 4672 y Ft(.)41 b(The)27 b(map)e Fs(f)d Ft(:)33 b Fs(M)39 b Fo(!)27 b Fs(M)37 b Ft(is)25 b(said)h(to)g(b)s(e)g Fi(sto)-5 b(chastic)g(al)5 b(ly)28 b(stable)118 4792 y Ft(if)j(the)g(w)m(eak)583 4756 y Fq(\003)656 4792 y Ft(accum)m(ulation)f(p)s(oin)m(ts)h(\(when)h Fs(\017)c(>)g Ft(0)j(go)s(es)h(to)f(zero\))h(of)f(the)h(ph)m(ysical)f (probabilit)m(y)118 4913 y(measures)c(of)e Fs(f)36 b Ft(are)25 b(con)m(v)m(ex)j(linear)c(com)m(binations)g(of)h(the)h (\(\014nitely)f(man)m(y\))h(SRB)f(measures)i(of)e Fs(f)11 b Ft(.)1924 5251 y(3)p eop %%Page: 4 4 4 3 bop 118 548 a Fc(1.1.1)113 b(Lo)s(cal)37 b(di\013eomorphisms)118 733 y Ft(Let)g Fs(f)46 b Ft(:)36 b Fs(M)46 b Fo(!)35 b Fs(M)47 b Ft(b)s(e)38 b(a)e Fs(C)1170 697 y Fp(2)1247 733 y Ft(lo)s(cal)f(di\013eomorphism)f(of)j(the)g(manifold)d Fs(M)10 b Ft(.)58 b(W)-8 b(e)37 b(sa)m(y)h(that)f Fs(f)48 b Ft(is)118 853 y Fi(non-uniformly)34 b(exp)-5 b(anding)31 b Ft(if)g(there)j(is)e(some)g(constan)m(t)i Fs(c)27 b(>)h Ft(0)k(for)g(whic)m(h)1057 1113 y(lim)17 b(sup)1087 1192 y Fr(n)p Fq(!)p Fp(+)p Fq(1)1387 1046 y Ft(1)p 1382 1090 59 4 v 1382 1182 a Fs(n)1472 989 y Fr(n)p Fq(\000)p Fp(1)1467 1019 y Fh(X)1478 1229 y Fr(j)t Fp(=0)1627 1113 y Ft(log)g Fo(k)p Fs(D)s(f)11 b Ft(\()p Fs(f)2060 1072 y Fr(j)2095 1113 y Ft(\()p Fs(x)p Ft(\)\))2264 1072 y Fq(\000)p Fp(1)2359 1113 y Fo(k)27 b(\024)i(\000)p Fs(c)f(<)f Ft(0)814 b(\(3\))118 1390 y(for)37 b(Leb)s(esgue)j(almost)c(ev)m(ery)j Fs(x)e Fo(2)g Fs(M)10 b Ft(.)60 b(It)38 b(w)m(as)h(pro)m(v)m(ed)g(in)e([ABV)q (])h(that)f(for)g(a)h(non-uniformly)118 1510 y(expanding)33 b(lo)s(cal)d(di\013eomorphism)g Fs(f)44 b Ft(the)33 b(follo)m(wing)c (prop)s(ert)m(y)34 b(holds:)171 1675 y(\(P\))49 b Fi(Ther)-5 b(e)26 b(is)h(a)f(\014nite)h(numb)-5 b(er)26 b(of)g(er)-5 b(go)g(dic)26 b(absolutely)h(c)-5 b(ontinuous)27 b(\()16 b Ft(SRB)p Fi(\))26 b Fs(f)11 b Fi(-invariant)26 b(pr)-5 b(ob-)362 1796 y(ability)37 b(me)-5 b(asur)g(es)36 b Fs(\026)1136 1811 y Fp(1)1175 1796 y Fs(;)17 b(:)g(:)g(:)f(;)h(\026) 1453 1811 y Fr(p)1529 1796 y Fi(whose)36 b(b)-5 b(asins)37 b(c)-5 b(over)36 b(a)h(ful)5 b(l)37 b(L)-5 b(eb)g(esgue)36 b(me)-5 b(asur)g(e)37 b(subset)g(of)362 1916 y Fs(M)10 b Fi(.)44 b(Mor)-5 b(e)g(over,)29 b(every)g(absolutely)g(c)-5 b(ontinuous)29 b Fs(f)11 b Fi(-invariant)28 b(pr)-5 b(ob)g(ability)29 b(me)-5 b(asur)g(e)28 b Fs(\026)h Fi(may)362 2036 y(b)-5 b(e)39 b(written)h(as)g(a)f(c)-5 b(onvex)39 b(line)-5 b(ar)39 b(c)-5 b(ombination)38 b(of)i Fs(\026)2378 2051 y Fp(1)2417 2036 y Fs(;)17 b(:)g(:)g(:)f(;)h(\026)2695 2051 y Fr(p)2734 2036 y Fi(:)54 b(ther)-5 b(e)40 b(ar)-5 b(e)40 b(r)-5 b(e)g(al)39 b(numb)-5 b(ers)362 2157 y Fs(w)432 2172 y Fp(1)471 2157 y Fs(;)17 b(:)g(:)g(:)f(;)h(w)760 2172 y Fr(p)827 2157 y Fo(\025)28 b Ft(0)35 b Fi(with)f Fs(w)1297 2172 y Fp(1)1359 2157 y Ft(+)22 b Fo(\001)17 b(\001)g(\001)j Ft(+)i Fs(w)1763 2172 y Fr(p)1830 2157 y Ft(=)28 b(1)34 b Fi(for)h(which)f Fs(\026)27 b Ft(=)h Fs(w)2707 2172 y Fp(1)2746 2157 y Fs(\026)2805 2172 y Fp(1)2866 2157 y Ft(+)22 b Fo(\001)17 b(\001)g(\001)k Ft(+)h Fs(w)3271 2172 y Fr(p)3310 2157 y Fs(\026)3369 2172 y Fr(p)3408 2157 y Ft(.)118 2322 y(The)36 b(pro)s(of)e(of)g(the)h (previous)g(result)g(w)m(as)g(based)h(on)e(the)i(existence)g(of)e Fs(\013)q Ft(-h)m(yp)s(erb)s(olic)f(times)h(for)118 2442 y(the)i(p)s(oin)m(ts)f(in)g Fs(M)10 b Ft(:)50 b(giv)m(en)36 b(0)d Fs(<)g(\013)g(<)g Ft(1,)j(w)m(e)h(sa)m(y)f(that)g Fs(n)d Fo(2)g Fg(Z)2440 2406 y Fp(+)2532 2442 y Ft(is)i(a)g Fs(\013)q Fi(-hyp)-5 b(erb)g(olic)37 b(time)e Ft(for)g(the)118 2563 y(p)s(oin)m(t)d Fs(x)c Fo(2)g Fs(M)43 b Ft(if)924 2698 y Fr(n)p Fq(\000)p Fp(1)927 2728 y Fh(Y)879 2941 y Fr(j)t Fp(=)p Fr(n)p Fq(\000)p Fr(k)1119 2823 y Fo(k)p Fs(D)s(f)11 b Ft(\()p Fs(f)1409 2782 y Fr(j)1444 2823 y Ft(\()p Fs(x)p Ft(\)\))1613 2782 y Fq(\000)p Fp(1)1708 2823 y Fo(k)27 b(\024)i Fs(\013)1954 2782 y Fr(k)2093 2823 y Ft(for)j(ev)m(ery)100 b(1)27 b Fo(\024)i Fs(k)h Fo(\024)e Fs(n:)637 b Ft(\(4\))118 3102 y(The)41 b(existence)g(of)f (\(a)f(p)s(ositiv)m(e)h(frequency)i(of)7 b(\))39 b Fs(\013)q Ft(-h)m(yp)s(erb)s(olic)g(times)g(for)g(p)s(oin)m(ts)g Fs(x)i Fo(2)g Fs(M)50 b Ft(is)40 b(a)118 3222 y(consequence)c(of)d(the) g(h)m(yp)s(othesis)h(of)f(non-uniform)e(expansion)j(of)e(the)i(map)e Fs(f)44 b Ft(and)33 b(p)s(ermits)f(us)118 3342 y(to)25 b(de\014ne)i(a)e(map)g Fs(h)i Ft(:)h Fs(M)39 b Fo(!)27 b Fg(Z)1256 3306 y Fp(+)1338 3342 y Ft(giving)c(the)j(\014rst)g(h)m(yp) s(erb)s(olic)f(time)f(for)h Fs(m)g Ft(almost)f(ev)m(ery)j Fs(x)i Fo(2)f Fs(M)10 b Ft(.)264 3463 y(In)34 b(the)h(con)m(text)g(of)e (random)g(p)s(erturbations)g(of)g(a)h(non-uniformly)d(expanding)j(map)f (w)m(e)h(are)118 3583 y(also)e(able)g(to)g(pro)m(v)m(e)i(a)e(result)h (on)f(the)h(\014nitness)g(of)g(ph)m(ysical)f(measures.)118 3748 y Fc(Theorem)37 b(A.)49 b Fi(L)-5 b(et)32 b Fs(f)22 b Ft(:)33 b Fs(M)39 b Fo(!)27 b Fs(M)43 b Fi(b)-5 b(e)32 b(a)g Fs(C)1726 3712 y Fp(2)1798 3748 y Fi(non-uniformly)f(exp)-5 b(anding)30 b(lo)-5 b(c)g(al)32 b(di\013e)-5 b(omorphism.)118 3869 y(If)41 b Fs(\017)g(>)e Ft(0)j Fi(is)f(su\016ciently)h(smal)5 b(l,)42 b(then)f(ther)-5 b(e)42 b(ar)-5 b(e)41 b(physic)-5 b(al)41 b(me)-5 b(asur)g(es)41 b Fs(\026)2921 3833 y Fr(\017)2921 3893 y Fp(1)2960 3869 y Fs(;)17 b(:)g(:)g(:)f(;)h(\026) 3238 3833 y Fr(\017)3238 3895 y(l)3311 3869 y Fi(\(with)42 b Fs(l)i Fi(not)118 3989 y(dep)-5 b(ending)33 b(on)i Fs(\017)p Fi(\))g(such)g(that:)234 4154 y(1.)48 b(for)31 b(e)-5 b(ach)30 b Fs(x)e Fo(2)g Fs(M)42 b Fi(and)30 b Fs(\022)1275 4118 y Fe(N)1272 4179 y Fr(\017)1358 4154 y Fi(almost)h(every)g Fs(t)p 1920 4170 36 4 v 27 w Fo(2)e Fs(T)2148 4118 y Fe(N)2199 4154 y Fi(,)j(the)f(aver)-5 b(age)30 b(of)g(Dir)-5 b(ac)31 b(me)-5 b(asur)g(es)30 b Fs(\016)3597 4170 y Fr(f)3638 4147 y Fn(n)3631 4190 y(t)p 3631 4200 28 3 v 3681 4170 a Fp(\()p Fr(x)p Fp(\))362 4275 y Fi(c)-5 b(onver)g(ges)34 b(in)g(the)h(we)-5 b(ak)1278 4239 y Fq(\003)1352 4275 y Fi(top)g(olo)g(gy)34 b(to)h(some)f Fs(\026)2150 4239 y Fr(\017)2150 4299 y(i)2218 4275 y Fi(with)g Ft(1)28 b Fo(\024)g Fs(i)g Fo(\024)g Fs(l)r Fi(;)234 4466 y(2.)48 b(for)35 b(e)-5 b(ach)34 b Ft(1)27 b Fo(\024)h Fs(i)g Fo(\024)h Fs(l)37 b Fi(we)d(have)1065 4726 y Fs(\026)1124 4685 y Fr(\017)1124 4750 y(i)1184 4726 y Ft(=)27 b Fs(w)1360 4690 y Fq(\003)1399 4726 y Fi(-)41 b Ft(lim)1450 4785 y Fr(n)p Fq(!1)1682 4658 y Ft(1)p 1678 4703 59 4 v 1678 4794 a Fs(n)1768 4601 y Fr(n)p Fq(\000)p Fp(1)1762 4631 y Fh(X)1773 4841 y Fr(j)t Fp(=0)1923 4590 y Fh(Z)2039 4645 y(\000)2085 4726 y Fs(f)2144 4678 y Fr(j)2133 4748 y(t)p 2133 4760 30 3 v 2180 4645 a Fh(\001)2226 4765 y Fq(\003)2265 4645 y Fh(\000)2311 4726 y Fs(m)28 b Fo(j)f Fs(B)5 b Ft(\()p Fs(\026)2655 4685 y Fr(\017)2655 4750 y(i)2688 4726 y Ft(\))2726 4645 y Fh(\001)2788 4726 y Fs(d\022)2887 4685 y Fe(N)2884 4750 y Fr(\017)2939 4726 y Ft(\()p Fs(t)p 2977 4742 36 4 v Ft(\))p Fs(;)362 5002 y Fi(wher)-5 b(e)28 b Fs(m)g Fo(j)f Fs(B)5 b Ft(\()p Fs(\026)975 4966 y Fr(\017)975 5027 y(i)1008 5002 y Ft(\))28 b Fi(is)g(the)h(normalization)e(of)h(the) g(L)-5 b(eb)g(esgue)28 b(me)-5 b(asur)g(e)28 b(r)-5 b(estricte)g(d)29 b(to)f Fs(B)5 b Ft(\()p Fs(\026)3679 4966 y Fr(\017)3679 5027 y(i)3712 5002 y Ft(\))p Fi(;)1924 5251 y Ft(4)p eop %%Page: 5 5 5 4 bop 234 548 a Fi(3.)48 b(if)35 b Fs(f)45 b Fi(is)35 b(tr)-5 b(ansitive,)34 b(then)h Fs(l)30 b Ft(=)e(1)p Fi(.)264 743 y Ft(W)-8 b(e)42 b(sa)m(y)g(that)f(the)g(map)f Fs(f)52 b Ft(is)41 b Fi(non-uniformly)g(exp)-5 b(anding)41 b(for)i(r)-5 b(andom)41 b(orbits)g Ft(if)f(there)i(is)118 863 y(some)33 b(constan)m(t)g Fs(c)28 b(>)f Ft(0)32 b(suc)m(h)j(that)d (for)g Fs(\017)c(>)g Ft(0)k(small)e(enough)1035 1157 y(lim)17 b(sup)1065 1236 y Fr(n)p Fq(!)p Fp(+)p Fq(1)1365 1090 y Ft(1)p 1360 1134 59 4 v 1360 1226 a Fs(n)1450 1033 y Fr(n)p Fq(\000)p Fp(1)1445 1063 y Fh(X)1456 1273 y Fr(j)t Fp(=0)1606 1157 y Ft(log)f Fo(k)p Fs(D)s(f)11 b Ft(\()p Fs(f)2038 1110 y Fr(j)2027 1180 y(t)p 2027 1192 30 3 v 2074 1157 a Ft(\()p Fs(x)p Ft(\)\))2243 1116 y Fq(\000)p Fp(1)2337 1157 y Fo(k)28 b(\024)g(\000)p Fs(c)g(<)f Ft(0)p Fs(;)809 b Ft(\(5\))118 1475 y(for)41 b Fs(\022)324 1439 y Fe(N)321 1500 y Fr(\017)405 1475 y Fo(\002)29 b Fs(m)42 b Ft(almost)e(ev)m(ery)k(\()p Fs(t)p 1267 1491 36 4 v(;)17 b(x)p Ft(\))44 b Fo(2)g Fs(T)1664 1439 y Fe(N)1744 1475 y Fo(\002)29 b Fs(M)10 b Ft(.)72 b(Similarly)38 b(to)k(the)g(deterministic)e(situation,)118 1595 y(condition)46 b(\(5\))h(p)s(ermits)f(us)i(to)f(in)m(tro)s(duce)g (a)g(notion)f(of)h Fs(\013)q Ft(-h)m(yp)s(erb)s(olic)e(times)i(for)f(p) s(oin)m(ts)h(in)118 1716 y Fs(T)189 1680 y Fe(N)263 1716 y Fo(\002)23 b Fs(M)43 b Ft(and)33 b(de\014ne)h(a)e(map)1554 1836 y Fs(h)1610 1851 y Fr(\017)1654 1836 y Ft(:)h Fs(T)1785 1795 y Fe(N)1859 1836 y Fo(\002)23 b Fs(M)38 b Fo(!)27 b Fg(Z)2287 1795 y Fp(+)118 2006 y Ft(b)m(y)38 b(taking)e Fs(h)619 2021 y Fr(\017)652 2006 y Ft(\()p Fs(t)p 690 2022 V(;)17 b(x)p Ft(\))37 b(the)h(\014rst)g Fs(\013)q Ft(-h)m(yp)s(erb)s(olic)d(time)h(for)g(the)i(p)s(oin)m(t)e(\()p Fs(t)p 2704 2022 V(;)17 b(x)p Ft(\))36 b Fo(2)f Fs(T)3084 1970 y Fe(N)3161 2006 y Fo(\002)26 b Fs(M)48 b Ft(\(see)38 b(Sec-)118 2127 y(tion)32 b(2\).)43 b(Assuming)32 b(that)g Fs(h)1191 2142 y Fr(\017)1257 2127 y Ft(is)g(in)m(tegrable)f(with)h (resp)s(ect)i(to)f Fs(\022)2531 2090 y Fe(N)2528 2151 y Fr(\017)2605 2127 y Fo(\002)23 b Fs(m)p Ft(,)33 b(then)846 2412 y Fo(k)p Fs(h)952 2427 y Fr(\017)985 2412 y Fo(k)1035 2427 y Fp(1)1102 2412 y Ft(=)1242 2288 y Fq(1)1205 2318 y Fh(X)1213 2530 y Fr(k)r Fp(=0)1366 2412 y Fs(k)20 b Ft(\()p Fs(\022)1523 2371 y Fe(N)1520 2437 y Fr(\017)1597 2412 y Fo(\002)j Fs(m)p Ft(\))1820 2331 y Fh(\000)1866 2412 y Fo(f)p Ft(\()p Fs(t)p 1954 2428 V(;)17 b(x)p Ft(\))11 b(:)33 b Fs(h)2253 2427 y Fr(\017)2286 2412 y Ft(\()p Fs(t)p 2324 2428 V(;)17 b(x)p Ft(\))28 b(=)f Fs(k)s Fo(g)2748 2331 y Fh(\001)2821 2412 y Fs(<)h Fo(1)p Fs(:)603 b Ft(\(6\))118 2716 y(W)-8 b(e)36 b(sa)m(y)h(that)f(the)g(family)d(\()p Fs(h)1242 2731 y Fr(\017)1275 2716 y Ft(\))1313 2731 y Fr(\017>)p Fp(0)1472 2716 y Ft(has)j Fi(uniform)h Fs(L)2090 2680 y Fp(1)2130 2716 y Fi(-tail)p Ft(,)g(if)d(the)j(series)f(in)f (\(6\))g(con)m(v)m(erges)k(uni-)118 2837 y(formly)31 b(to)h Fo(k)p Fs(h)652 2852 y Fr(\017)685 2837 y Fo(k)735 2852 y Fp(1)807 2837 y Ft(\(as)g(a)h(series)g(of)f(functions)h(on)f (the)h(v)-5 b(ariable)31 b Fs(\017)p Ft(\).)118 3031 y Fc(Theorem)37 b(B.)49 b Fi(L)-5 b(et)33 b Fs(f)22 b Ft(:)33 b Fs(M)38 b Fo(!)28 b Fs(M)43 b Fi(b)-5 b(e)33 b(a)f(non-uniformly)g(exp)-5 b(anding)32 b Fs(C)2812 2995 y Fp(2)2884 3031 y Fi(lo)-5 b(c)g(al)32 b(di\013e)-5 b(omorphism.)234 3226 y(1.)48 b(If)29 b Fs(f)41 b Fi(is)29 b(sto)-5 b(chastic)g(al)5 b(ly)29 b(stable,)i(then)e Fs(f)41 b Fi(is)29 b(non-uniformly)g(exp)-5 b(anding)28 b(for)i(r)-5 b(andom)29 b(orbits.)234 3426 y(2.)48 b(If)31 b Fs(f)43 b Fi(is)32 b(non-uniformly)f(exp)-5 b(anding)30 b(for)i(r)-5 b(andom)31 b(orbits)h(and)g Ft(\()p Fs(h)2796 3441 y Fr(\017)2828 3426 y Ft(\))2866 3441 y Fr(\017)2931 3426 y Fi(has)f(uniform)h Fs(L)3538 3390 y Fp(1)3578 3426 y Fi(-tail,)362 3547 y(then)j Fs(f)45 b Fi(is)35 b(sto)-5 b(chastic)g(al)5 b(ly)34 b(stable.)118 3805 y Fc(1.1.2)113 b(Maps)38 b(with)f(critical)d(sets)118 3990 y Ft(Similar)g(results)39 b(to)e(those)i(presen)m(ted)h(for)e (random)f(p)s(erturbations)g(of)h(lo)s(cal)d(di\013eomorphisms)118 4110 y(will)i(also)h(b)s(e)h(obtained)g(for)f(maps)h(with)g(critical)e (sets)j(in)e(the)i(sense)h(of)d([ABV)q(].)63 b(W)-8 b(e)39 b(start)h(b)m(y)118 4230 y(describing)k(the)i(class)e(of)h(maps)f(that) g(w)m(e)i(are)f(going)e(to)h(consider.)80 b(Let)45 b Fs(f)22 b Ft(:)38 b Fs(M)59 b Fo(!)47 b Fs(M)56 b Ft(b)s(e)45 b(a)118 4351 y(con)m(tin)m(uous)31 b(map)e(of)h(the)g(compact)g (manifold)d Fs(M)41 b Ft(that)30 b(fails)e(to)i(b)s(e)g(a)g Fs(C)2829 4315 y Fp(2)2898 4351 y Ft(lo)s(cal)e(di\013eomorphism)118 4471 y(on)k(a)f(critical)e(set)j Fo(C)i(\032)28 b Fs(M)43 b Ft(with)31 b(zero)h(Leb)s(esgue)h(measure.)43 b(W)-8 b(e)32 b(assume)g(that)f Fs(f)43 b Fi(b)-5 b(ehaves)32 b(like)i(a)118 4592 y(p)-5 b(ower)36 b(of)g(the)g(distanc)-5 b(e)34 b Ft(close)g(to)g(the)h(critical)d(set)j Fo(C)6 b Ft(:)47 b(there)35 b(are)f(constan)m(ts)i Fs(B)f(>)c Ft(1)j(and)g Fs(\014)i(>)31 b Ft(0)118 4712 y(for)h(whic)m(h)134 4959 y(\(S1\))411 4891 y(1)p 396 4936 80 4 v 396 5027 a Fs(B)485 4959 y Ft(dist)o(\()p Fs(x;)17 b Fo(C)6 b Ft(\))875 4918 y Fr(\014)950 4959 y Fo(\024)1066 4891 y(k)p Fs(D)s(f)11 b Ft(\()p Fs(x)p Ft(\))p Fs(v)t Fo(k)p 1066 4936 425 4 v 1202 5027 a(k)p Fs(v)t Fo(k)1527 4959 y(\024)28 b Fs(B)5 b Ft(dist\()p Fs(x;)17 b Fo(C)6 b Ft(\))2102 4918 y Fq(\000)p Fr(\014)2205 4959 y Ft(;)1924 5251 y(5)p eop %%Page: 6 6 6 5 bop 134 590 a Ft(\(S2\))386 506 y Fh(\014)386 566 y(\014)419 590 y Ft(log)16 b Fo(k)p Fs(D)s(f)11 b Ft(\()p Fs(x)p Ft(\))885 549 y Fq(\000)p Fp(1)979 590 y Fo(k)22 b(\000)h Ft(log)16 b Fo(k)p Fs(D)s(f)11 b Ft(\()p Fs(y)t Ft(\))1614 549 y Fq(\000)p Fp(1)1707 590 y Fo(k)1779 506 y Fh(\014)1779 566 y(\014)1840 590 y Fo(\024)28 b Fs(B)2061 523 y Ft(dist\()p Fs(x;)17 b(y)t Ft(\))p 2034 568 439 4 v 2034 659 a(dist\()p Fs(x;)g Fo(C)6 b Ft(\))2425 630 y Fr(\014)2482 590 y Ft(;)134 904 y(\(S3\))386 819 y Fh(\014)386 879 y(\014)419 904 y Ft(log)16 b Fo(j)h Ft(det)g Fs(D)s(f)11 b Ft(\()p Fs(x)p Ft(\))1032 863 y Fq(\000)p Fp(1)1126 904 y Fo(j)22 b(\000)g Ft(log)17 b Fo(j)g Ft(det)f Fs(D)s(f)11 b Ft(\()p Fs(y)t Ft(\))1885 863 y Fq(\000)p Fp(1)1978 904 y Fo(j)2028 819 y Fh(\014)2028 879 y(\014)2089 904 y Fo(\024)28 b Fs(B)2310 836 y Ft(dist\()p Fs(x;)17 b(y)t Ft(\))p 2283 881 V 2283 972 a(dist\()p Fs(x;)g Fo(C)6 b Ft(\))2674 943 y Fr(\014)2731 904 y Ft(;)118 1135 y(for)38 b(ev)m(ery)i Fs(x;)17 b(y)40 b Fo(2)d Fs(M)g Fo(n)26 b(C)44 b Ft(with)38 b(dist\()p Fs(x;)17 b(y)t Ft(\))36 b Fs(<)h Ft(dist)o(\()p Fs(x;)17 b Fo(C)6 b Ft(\))p Fs(=)p Ft(2)38 b(and)h Fs(v)h Fo(2)e Fs(T)2861 1150 y Fr(x)2905 1135 y Fs(M)10 b Ft(.)61 b(Giv)m(en)38 b Fs(\016)j(>)c Ft(0)g(w)m(e)118 1256 y(de\014ne)d(the)f Fs(\016)t Ft(-)p Fi(trunc)-5 b(ate)g(d)35 b(distanc)-5 b(e)31 b Ft(from)h Fs(x)c Fo(2)g Fs(M)43 b Ft(to)32 b Fo(C)1007 1466 y Ft(dist)1165 1481 y Fr(\016)1203 1466 y Ft(\()p Fs(x;)17 b Fo(C)6 b Ft(\))28 b(=)1567 1326 y Fh(\032)1683 1405 y Ft(1)425 b(if)32 b(dist)o(\()p Fs(x;)17 b Fo(C)6 b Ft(\))28 b Fo(\025)h Fs(\016)n(;)1683 1526 y Ft(dist\()p Fs(x;)17 b Fo(C)6 b Ft(\))83 b(otherwise.)264 1693 y(Assume)35 b(that)e Fs(f)44 b Ft(is)32 b(a)h(non-uniformly)e (expanding)i(map,)g(in)g(the)g(sense)i(that)e(there)h(is)f Fs(c)c(>)g Ft(0)118 1813 y(suc)m(h)38 b(that)d(the)i(limit)32 b(in)j(\(3\))h(holds)f(for)h(Leb)s(esgue)h(almost)d(ev)m(ery)k Fs(x)c Fo(2)g Fs(M)46 b Ft(\(recall)35 b(that)g(w)m(e)i(are)118 1933 y(taking)26 b Fo(C)33 b Ft(with)26 b(zero)h(Leb)s(esgue)h (measure\))f(and,)h(moreo)m(v)m(er,)h(supp)s(ose)f(that)e(the)h(orbits) g(of)f Fs(f)37 b Ft(ha)m(v)m(e)118 2054 y Fi(slow)d(appr)-5 b(oximation)34 b(to)h(the)g(critic)-5 b(al)34 b(set)p Ft(:)44 b(giv)m(en)32 b(small)f Fs(\015)h(>)c Ft(0)k(there)h(is)g Fs(\016)e(>)d Ft(0)k(suc)m(h)i(that)1151 2312 y(lim)17 b(sup)1181 2391 y Fr(n)p Fq(!)p Fp(+)p Fq(1)1481 2245 y Ft(1)p 1476 2289 59 4 v 1476 2380 a Fs(n)1566 2188 y Fr(n)p Fq(\000)p Fp(1)1561 2217 y Fh(X)1572 2427 y Fr(j)t Fp(=0)1721 2312 y Fo(\000)g Ft(log)g(dist)2116 2327 y Fr(\016)2154 2312 y Ft(\()p Fs(f)2251 2271 y Fr(j)2287 2312 y Ft(\()p Fs(x)p Ft(\))p Fs(;)g Fo(C)6 b Ft(\))28 b Fo(\024)g Fs(\015)913 b Ft(\(7\))118 2587 y(for)38 b(Leb)s(esgue)i(almost)c(ev)m(ery)41 b Fs(x)d Fo(2)g Fs(M)10 b Ft(.)62 b(The)39 b(results)g(in)f([ABV])h(sho)m(w)g(that)f (in)g(this)g(situation)118 2707 y(w)m(e)g(obtain)d(the)i(same)f (conclusion)g(on)h(the)f(\014niteness)i(of)e(SRB)h(measures)g(for)f (suc)m(h)i(an)e Fs(f)11 b Ft(,)37 b(also)118 2827 y(holding)31 b(prop)s(ert)m(y)i(\(P\).)264 2948 y(In)47 b(order)g(to)f(pro)m(v)m(e)i (the)f(sto)s(c)m(hastic)g(stabilit)m(y)e(of)h(maps)g(with)g(critical)e (sets)k(w)m(e)f(need)h(to)118 3068 y(restrict)34 b(the)h(class)f(of)g (p)s(erturbations)g(w)m(e)h(are)f(going)f(to)g(consider:)47 b(w)m(e)36 b(tak)m(e)f(maps)e Fs(f)3356 3083 y Fr(t)3420 3068 y Ft(with)h(the)118 3189 y(same)f(critical)d(set)j Fo(C)39 b Ft(and)32 b(imp)s(ose)g(that)892 3362 y Fs(D)s(f)1024 3377 y Fr(t)1054 3362 y Ft(\()p Fs(x)p Ft(\))c(=)f Fs(D)s(f)11 b Ft(\()p Fs(x)p Ft(\))98 b(for)32 b(ev)m(ery)i Fs(x)28 b Fo(2)g Fs(M)33 b Fo(n)22 b(C)39 b Ft(and)33 b Fs(t)27 b Fo(2)i Fs(T)13 b(:)650 b Ft(\(8\))118 3536 y(This)37 b(ma)m(y)g(b)s(e)g(implemen)m(ted,)g(for)f(instance,)j(in)d (parallelizable)d(manifolds)i(\(with)h(an)h(additiv)m(e)118 3656 y(group)c(structure,)g(e.g.)44 b(tori)31 b Fg(T)1276 3620 y Fr(d)1353 3656 y Ft(or)h(cylinders)h Fg(T)1945 3620 y Fr(d)p Fq(\000)p Fr(k)2104 3656 y Fo(\002)23 b Fg(R)2270 3620 y Fr(k)2318 3656 y Ft(\))33 b(b)m(y)g(considering)1454 3830 y Fs(T)42 b Ft(=)27 b Fo(f)p Fs(t)h Fo(2)g Fg(R)1929 3789 y Fr(d)1987 3830 y Ft(:)33 b Fo(k)p Fs(t)p Fo(k)27 b(\024)i Fs(\017)2354 3845 y Fp(0)2393 3830 y Fo(g)118 4004 y Ft(for)j(some)h Fs(\017)551 4019 y Fp(0)619 4004 y Fs(>)28 b Ft(0,)33 b Fs(\022)877 4019 y Fr(\017)943 4004 y Ft(the)g(normalized)e(Leb)s(esgue)j(measure)f(on)g(the)g(ball)e (of)h(radius)h Fs(\017)28 b Fo(\024)h Fs(\017)3523 4019 y Fp(0)3562 4004 y Ft(,)k(and)118 4124 y(taking)41 b Fs(f)476 4139 y Fr(t)550 4124 y Ft(=)i Fs(f)d Ft(+)28 b Fs(t)p Ft(;)47 b(that)42 b(is,)i(adding)d(at)h(eac)m(h)h(step)g(a)e (random)g(noise)h(to)g(the)g(unp)s(erturb)s(ed)118 4244 y(dynamics.)264 4365 y(F)-8 b(or)39 b(the)h(case)h(of)e(maps)g(with)g (critical)f(sets)i(w)m(e)h(also)e(need)h(to)g(imp)s(ose)e(an)i(analog)e (of)h(con-)118 4485 y(dition)f(\(7\))h(for)g(random)f(orbits;)k(w)m(e)f (assume)f Fi(slow)g(appr)-5 b(oximation)40 b(of)h(r)-5 b(andom)40 b(orbits)h(to)h(the)118 4606 y(critic)-5 b(al)35 b(set)p Ft(:)43 b(giv)m(en)33 b(an)m(y)g(small)d Fs(\015)j(>)28 b Ft(0)k(there)h(is)f Fs(\016)g(>)27 b Ft(0)33 b(suc)m(h)h(that)1151 4864 y(lim)17 b(sup)1181 4943 y Fr(n)p Fq(!)p Fp(+)p Fq(1)1481 4796 y Ft(1)p 1476 4841 V 1476 4932 a Fs(n)1566 4739 y Fr(n)p Fq(\000)p Fp(1)1561 4769 y Fh(X)1572 4979 y Fr(j)t Fp(=0)1721 4864 y Fo(\000)g Ft(log)g(dist)2116 4879 y Fr(\016)2154 4864 y Ft(\()p Fs(f)2251 4817 y Fr(j)2240 4886 y(t)p 2240 4898 30 3 v 2287 4864 a Ft(\()p Fs(x)p Ft(\))p Fs(;)g Fo(C)6 b Ft(\))28 b Fo(\024)g Fs(\015)913 b Ft(\(9\))1924 5251 y(6)p eop %%Page: 7 7 7 6 bop 118 548 a Ft(for)39 b Fs(\022)322 512 y Fe(N)319 573 y Fr(\017)400 548 y Fo(\002)27 b Fs(m)40 b Ft(almost)d(ev)m(ery)k (\()p Fs(t)p 1252 564 36 4 v(;)17 b(x)p Ft(\))39 b Fo(2)g Fs(T)1639 512 y Fe(N)1717 548 y Fo(\002)27 b Fs(M)50 b Ft(and)39 b(small)e Fs(\017)i(>)f Ft(0.)63 b(Results)39 b(similar)d(to)j(those)118 668 y(presen)m(ted)d(for)d(lo)s(cal)f (di\013eomorphisms)g(on)h(the)i(\014niteness)g(of)e(ph)m(ysical)h (measures)g(can)g(also)f(b)s(e)118 789 y(obtained)f(in)g(this)g(case.) 118 955 y Fc(Theorem)37 b(C.)49 b Fi(L)-5 b(et)38 b Fs(f)22 b Ft(:)35 b Fs(M)44 b Fo(!)33 b Fs(M)48 b Fi(b)-5 b(e)38 b(a)g Fs(C)1758 919 y Fp(2)1835 955 y Fi(non-uniformly)f(exp)-5 b(anding)36 b(map)i(b)-5 b(ehaving)36 b(like)i(a)118 1076 y(p)-5 b(ower)31 b(of)g(the)h(distanc)-5 b(e)30 b(close)h(to)g(the)h(critic)-5 b(al)31 b(set)h Fo(C)6 b Fi(,)32 b(and)f(whose)f(orbits)i(have)e(slow)h(appr)-5 b(oxima-)118 1196 y(tion)37 b(to)f Fo(C)6 b Fi(.)50 b(If)36 b Fs(f)48 b Fi(is)36 b(non-uniformly)g(exp)-5 b(anding)35 b(for)h(r)-5 b(andom)36 b(orbits)g(and)g(r)-5 b(andom)36 b(orbits)g(have)118 1316 y(slow)e(appr)-5 b(oximation)34 b(to)h Fo(C)6 b Fi(,)35 b(then)f(we)h(arrive)f(at)h(the)g(same)f(c)-5 b(onclusions)34 b(of)g(The)-5 b(or)g(em)34 b(A.)264 1483 y Ft(The)j(prop)s(ert)m(y)e(of)g(non-uniform)e(expansion)j(for)e (random)h(orbits,)g(together)g(with)g(the)g(slo)m(w)118 1603 y(appro)m(ximation)e(of)j(random)e(orbits)h(to)g(the)i(critical)c (set)j(p)s(ermit)e(us)i(to)g(in)m(tro)s(duce)f(a)g(notion)g(of)118 1723 y(\()p Fs(\013)q(;)17 b(\016)t Ft(\)-h)m(yp)s(erb)s(olic)31 b(times)g(for)h(p)s(oin)m(ts)h(in)e(\()p Fs(t)p 1711 1739 V 1 w(;)17 b(x)p Ft(\))27 b Fo(2)h Fs(T)2076 1687 y Fe(N)2150 1723 y Fo(\002)23 b Fs(M)43 b Ft(and)33 b(de\014ne)h(a)e (map)1540 1900 y Fs(h)1596 1915 y Fr(\017)1640 1900 y Ft(:)h Fs(T)1771 1859 y Fe(N)1845 1900 y Fo(\002)23 b Fs(M)39 b Fo(!)27 b Fg(Z)2274 1859 y Fp(+)2330 1900 y Fs(;)118 2077 y Ft(b)m(y)41 b(taking)f Fs(h)626 2092 y Fr(\017)659 2077 y Ft(\()p Fs(t)p 697 2093 V(;)17 b(x)p Ft(\))41 b(the)f(\014rst)h(\()p Fs(\013)q(;)17 b(\016)t Ft(\)-h)m(yp)s(erb)s(olic)39 b(time)g(for)h(the)h(p)s(oin)m(t)e(\()p Fs(t)p 2904 2093 V(;)17 b(x)p Ft(\))41 b Fo(2)h Fs(T)3296 2041 y Fe(N)3375 2077 y Fo(\002)28 b Fs(M)10 b Ft(,)43 b(see)118 2198 y(Section)37 b(2.)56 b(Assuming)37 b(that)f Fs(h)1320 2213 y Fr(\017)1390 2198 y Ft(is)h(in)m(tegrable)f(with)g (resp)s(ect)i(to)f Fs(\022)2683 2213 y Fr(\017)2741 2198 y Fo(\002)26 b Fs(m)p Ft(,)38 b(then)g(w)m(e)g(obtain)e(an)118 2318 y(analog)k(to)h(\(6\),)j(whic)m(h)e(enables)g(us)h(to)e(de\014ne)i (a)e(notion)g(of)g Fi(uniform)i Fs(L)2923 2282 y Fp(1)2963 2318 y Fi(-tail)e Ft(exactly)h(in)f(the)118 2438 y(same)33 b(w)m(a)m(y)g(as)g(b)s(efore.)264 2559 y(Due)c(to)g(the)h(fact)e(that)h (log)17 b Fo(k)p Fs(D)s(f)1478 2523 y Fq(\000)p Fp(1)1571 2559 y Fo(k)29 b Ft(is)g(not)g(a)f(con)m(tin)m(uous)i(map)e(\(it)g(is)h (not)g(ev)m(en)h(ev)m(erywhere)118 2679 y(de\014ned\))49 b(w)m(e)g(are)f(not)f(able)g(to)g(presen)m(t)j(in)d(this)g(con)m(text)i (a)e(similar)e(to)i(Theorem)h(B)g(in)f(all)118 2800 y(its)40 b(strength.)67 b(Ho)m(w)m(ev)m(er,)45 b(w)m(e)c(obtain)e(the)i(same)f (kind)g(of)g(conclusion)g(of)g(the)g(second)i(item)d(of)118 2920 y(Theorem)33 b(B.)118 3086 y Fc(Theorem)k(D.)49 b Fi(L)-5 b(et)45 b Fs(f)21 b Ft(:)37 b Fs(M)56 b Fo(!)45 b Fs(M)55 b Fi(b)-5 b(e)44 b(non-uniformly)f(exp)-5 b(anding)43 b Fs(C)2831 3050 y Fp(2)2914 3086 y Fi(map)h(b)-5 b(ehaving)43 b(like)h(a)118 3207 y(p)-5 b(ower)26 b(of)g(the)g(distanc)-5 b(e)26 b(close)f(to)i(its)f(critic)-5 b(al)26 b(set)h Fo(C)32 b Fi(and)26 b(whose)g(orbits)g(have)f(slow)h(appr)-5 b(oximation)118 3327 y(to)35 b Fo(C)6 b Fi(.)44 b(Assume)35 b(that)f Fs(f)45 b Fi(is)34 b(non-uniformly)f(exp)-5 b(anding)33 b(for)h(r)-5 b(andom)34 b(orbits)g(and)f(r)-5 b(andom)34 b(orbits)118 3448 y(have)43 b(slow)h(appr)-5 b(oximation)42 b(to)i Fo(C)6 b Fi(.)72 b(If)43 b Ft(\()p Fs(h)1716 3463 y Fr(\017)1749 3448 y Ft(\))1787 3463 y Fr(\017)1864 3448 y Fi(has)g(uniform)g Fs(L)2494 3411 y Fp(1)2534 3448 y Fi(-tail,)j(then)e Fs(f)54 b Fi(is)44 b(sto)-5 b(chastic)g(al)5 b(ly)118 3568 y(stable.)264 3734 y Ft(As)34 b(a)f(ma)5 b(jor)32 b(application)e(of)i(the)i (previous)f(theorem)g(w)m(e)h(are)f(thinking)f(of)g(a)h(class)g(of)g (maps)118 3855 y(on)38 b(the)h(cylinder)e Fs(S)875 3819 y Fp(1)941 3855 y Fo(\002)26 b Fg(R)49 b Ft(in)m(tro)s(duced)38 b(in)g([Vi1)o(].)60 b(Subsequen)m(t)41 b(w)m(orks)e([Al)o(])g(and)f([A) -11 b(V])38 b(sho)m(w)m(ed)118 3975 y(that)h(suc)m(h)h(systems)h(are)d (top)s(ologically)d(mixing)i(\(th)m(us)j(transitiv)m(e\))e(and)h(ha)m (v)m(e)h(a)f(unique)g(SRB)118 4095 y(measure.)k(In)29 b(Section)f(6)g(w)m(e)i(pro)m(v)m(e)g(that)e(Viana)g(maps)g(satisfy)h (the)g(h)m(yp)s(otheses)i(of)d(Theorem)h(D,)118 4216 y(hence)38 b(b)s(eing)f(sto)s(c)m(hastically)e(stable.)57 b(An)37 b(application)d(of)i(Theorem)i(B)f(will)d(also)i(b)s(e)h(giv)m (en)g(in)118 4336 y(Section)28 b(6)g(for)g(an)g(op)s(en)h(class)f(of)g (lo)s(cal)e(di\013eomorphisms)h(in)m(tro)s(duced)h(in)g([ABV,)h(App)s (endix)g(A].)118 4663 y Fu(2)161 b(Distortion)54 b(b)t(ounds)118 4882 y Ft(In)46 b(this)g(section)g(w)m(e)h(generalize)e(some)h(of)f (the)h(results)g(in)g([Al)o(])g(and)g([ABV])g(for)f(the)i(setting)118 5002 y(of)37 b(sto)s(c)m(hastic)h(p)s(erturbations)f(of)g(a)g (non-uniformly)e(expanding)i(map.)57 b(These)39 b(results)f(will)d(b)s (e)1924 5251 y(7)p eop %%Page: 8 8 8 7 bop 118 548 a Ft(pro)m(v)m(ed)37 b(in)e(the)i(setting)e(of)g(maps)h (with)f(critical)f(sets.)54 b(Then)37 b(ev)m(erything)g(follo)m(ws)d (in)h(the)i(same)118 668 y(w)m(a)m(y)h(for)f(lo)s(cal)e (di\013eomorphisms)g(if)h(w)m(e)i(think)f(of)f Fo(C)44 b Ft(as)37 b(b)s(eing)f(equal)h(to)g(the)g(empt)m(y)h(set,)h(with)118 789 y(the)f(only)e(exception)i(of)f(a)g(particular)f(p)s(oin)m(t)g (that)h(w)m(e)h(clarify)e(in)g(Remark)h(2.4)g(b)s(elo)m(w)g(\(due)h(to) 118 909 y(the)29 b(fact)f(that)h(w)m(e)g(are)f(not)h(assuming)e (condition)h(\(8\))g(for)f(maps)i(with)f(no)g(critical)e(sets\).)43 b(F)-8 b(or)28 b(the)118 1029 y(next)34 b(de\014nition)d(w)m(e)j(tak)m (e)f(0)28 b Fs(<)f(b)h(<)g Ft(min)n Fo(f)p Ft(1)p Fs(=)p Ft(2)p Fs(;)17 b Ft(1)p Fs(=)p Ft(\(2)p Fs(\014)6 b Ft(\))p Fo(g)p Ft(.)118 1208 y Fc(De\014nition)36 b(2.1.)49 b Fi(Given)31 b Ft(0)d Fs(<)f(\013)i(<)e Ft(1)k Fi(and)g Fs(\016)h(>)27 b Ft(0)p Fi(,)32 b(we)f(say)g(that)h Fs(n)c Fo(2)g Fg(Z)2828 1171 y Fp(+)2916 1208 y Fi(is)j(a)g Ft(\()p Fs(\013)q(;)17 b(\016)t Ft(\))p Fi(-hyp)-5 b(erb)g(olic)118 1328 y(time)35 b(for)f Ft(\()p Fs(t)p 533 1344 36 4 v 1 w(;)17 b(x)p Ft(\))27 b Fo(2)h Fs(T)898 1292 y Fe(N)972 1328 y Fo(\002)23 b Fs(M)46 b Fi(if)711 1484 y Fr(n)p Fq(\000)p Fp(1)714 1514 y Fh(Y)666 1726 y Fr(j)t Fp(=)p Fr(n)p Fq(\000)p Fr(k)906 1608 y Fo(k)p Fs(D)s(f)1088 1623 y Fr(t)1113 1633 y Fn(j)s Fd(+1)1227 1608 y Ft(\()p Fs(f)1324 1561 y Fr(j)1313 1631 y(t)p 1313 1643 30 3 v 1360 1608 a Ft(\()p Fs(x)p Ft(\)\))1529 1567 y Fq(\000)p Fp(1)1624 1608 y Fo(k)27 b(\024)h Fs(\013)1869 1567 y Fr(k)2011 1608 y Fi(and)99 b(dist)2417 1623 y Fr(\016)2455 1608 y Ft(\()p Fs(f)2552 1567 y Fr(n)p Fq(\000)p Fr(k)2541 1633 y(t)p 2541 1645 V 2692 1608 a Ft(\()p Fs(x)p Ft(\))p Fs(;)17 b Fo(C)6 b Ft(\))28 b Fo(\025)g Fs(\013)3159 1567 y Fr(bk)118 1896 y Fi(for)35 b(every)g Ft(1)27 b Fo(\024)h Fs(k)j Fo(\024)d Fs(n)p Fi(.)264 2074 y Ft(The)37 b(follo)m(wing)32 b(lemma,)i(due)i(to)f(Pliss)g([Pl)o(],)i(pro)m(vides) f(the)f(main)f(to)s(ol)f(in)i(the)h(pro)s(of)e(of)h(the)118 2195 y(existence)c(of)d(h)m(yp)s(erb)s(olic)g(times)g(for)h(p)s(oin)m (ts)f(with)h(non-uniform)e(expansion)i(on)g(random)f(orbits.)118 2373 y Fc(Lemma)37 b(2.2.)49 b Fi(L)-5 b(et)40 b Fs(H)j Fo(\025)36 b Fs(c)1190 2388 y Fp(2)1265 2373 y Fs(>)g(c)1419 2388 y Fp(1)1494 2373 y Fs(>)f Ft(0)k Fi(and)g Fs(\020)j Ft(=)36 b(\()p Fs(c)2164 2388 y Fp(2)2225 2373 y Fo(\000)23 b Fs(c)2367 2388 y Fp(1)2406 2373 y Ft(\))p Fs(=)p Ft(\()p Fs(H)30 b Fo(\000)22 b Fs(c)2783 2388 y Fp(1)2823 2373 y Ft(\))p Fi(.)57 b(Given)39 b(r)-5 b(e)g(al)39 b(numb)-5 b(ers)118 2493 y Fs(a)169 2508 y Fp(1)209 2493 y Fs(;)17 b(:)g(:)g(:)32 b(;)17 b(a)495 2508 y Fr(N)597 2493 y Fi(satisfying)962 2646 y Fr(N)921 2676 y Fh(X)932 2886 y Fr(j)t Fp(=1)1082 2771 y Fs(a)1133 2786 y Fr(j)1197 2771 y Fo(\025)28 b Fs(c)1344 2786 y Fp(2)1384 2771 y Fs(N)110 b Fi(and)99 b Fs(a)1877 2786 y Fr(j)1941 2771 y Fo(\024)28 b Fs(H)63 b Fi(for)35 b(al)5 b(l)55 b Ft(1)27 b Fo(\024)h Fs(j)34 b Fo(\024)28 b Fs(N)5 b(;)118 3057 y Fi(ther)-5 b(e)35 b(ar)-5 b(e)35 b Fs(l)30 b(>)d(\020)8 b(N)45 b Fi(and)34 b Ft(1)27 b Fs(<)h(n)1290 3072 y Fp(1)1357 3057 y Fs(<)g(:)17 b(:)g(:)27 b(<)g(n)1764 3072 y Fr(l)1818 3057 y Fo(\024)i Fs(N)45 b Fi(such)35 b(that)805 3193 y Fr(n)848 3203 y Fn(i)767 3224 y Fh(X)729 3434 y Fr(j)t Fp(=)p Fr(n)p Fp(+1)966 3319 y Fs(a)1017 3334 y Fr(j)1081 3319 y Fo(\025)29 b Fs(c)1229 3334 y Fp(1)1290 3319 y Fo(\001)22 b Ft(\()p Fs(n)1436 3334 y Fr(i)1486 3319 y Fo(\000)h Fs(n)p Ft(\))56 b Fi(for)34 b(e)-5 b(ach)55 b Ft(0)27 b Fo(\024)i Fs(n)e(<)h(n)2562 3334 y Fr(i)2590 3319 y Fs(;)45 b(i)28 b Ft(=)f(1)p Fs(;)17 b(:)g(:)g(:)32 b(;)17 b(l)r(:)118 3610 y Fi(Pr)-5 b(o)g(of.)49 b Ft(See)33 b([ABV,)g(Lemma)e(3.1].)p 3709 3610 4 66 v 3713 3547 59 4 v 3713 3610 V 3771 3610 4 66 v 118 3788 a Fc(Prop)s(osition)36 b(2.3.)49 b Fi(Ther)-5 b(e)35 b(ar)-5 b(e)35 b Fs(\013)29 b(>)g Ft(0)35 b Fi(and)g Fs(\016)e(>)c Ft(0)35 b Fi(for)g(which)g Fs(\022)2608 3752 y Fe(N)2605 3813 y Fr(\017)2683 3788 y Fo(\002)23 b Fs(m)36 b Fi(almost)f(every)g Ft(\()p Fs(t)p 3512 3804 36 4 v(;)17 b(x)p Ft(\))29 b Fo(2)118 3908 y Fs(T)189 3872 y Fe(N)263 3908 y Fo(\002)23 b Fs(M)45 b Fi(has)35 b(some)f Ft(\()p Fs(\013)q(;)17 b(\016)t Ft(\))p Fi(-hyp)-5 b(erb)g(olic)33 b(time.)118 4086 y(Pr)-5 b(o)g(of.)49 b Ft(Let)32 b(\()p Fs(t)p 636 4102 V(;)17 b(x)p Ft(\))28 b Fo(2)g Fs(T)1001 4050 y Fe(N)1075 4086 y Fo(\002)23 b Fs(M)43 b Ft(b)s(e)33 b(a)f(p)s(oin)m(t)g(satisfying)g (\(5\).)43 b(F)-8 b(or)31 b(large)h Fs(N)43 b Ft(w)m(e)34 b(ha)m(v)m(e)1172 4369 y Fo(\000)1266 4245 y Fr(N)7 b Fq(\000)p Fp(1)1271 4275 y Fh(X)1281 4485 y Fr(j)t Fp(=0)1436 4369 y Ft(log)1579 4285 y Fh(\015)1579 4345 y(\015)1634 4369 y Fs(D)s(f)k Ft(\()p Fs(f)1874 4322 y Fr(j)1863 4392 y(t)p 1863 4404 30 3 v 1910 4369 a Ft(\()p Fs(x)p Ft(\)\))2079 4328 y Fq(\000)p Fp(1)2173 4285 y Fh(\015)2173 4345 y(\015)2256 4369 y Fo(\025)2375 4302 y Fs(c)p 2371 4347 49 4 v 2371 4438 a Ft(2)2430 4369 y Fs(N)38 b(>)28 b Ft(0)p Fs(;)118 4655 y Ft(b)m(y)40 b(de\014nition)d(of)h(non-uniform) f(expansion)i(on)g(random)e(orbits.)61 b(Fixing)37 b Fs(\032)h(>)g(\014)44 b Ft(w)m(e)c(see)g(that)118 4776 y(condition)31 b(\(S1\))i(implies)1194 4881 y Fh(\014)1194 4941 y(\014)1227 4966 y Ft(log)1370 4881 y Fh(\015)1370 4941 y(\015)1425 4966 y Fs(D)s(f)11 b Ft(\()p Fs(x)p Ft(\))1699 4925 y Fq(\000)p Fp(1)1793 4881 y Fh(\015)1793 4941 y(\015)1849 4881 y(\014)1849 4941 y(\014)1909 4966 y Fo(\024)29 b Fs(\032)17 b Fo(j)o Ft(log)g(dist)f(\()p Fs(x;)h Fo(C)6 b Ft(\))p Fo(j)919 b Ft(\(10\))1924 5251 y(8)p eop %%Page: 9 9 9 8 bop 118 548 a Ft(for)32 b(ev)m(ery)j Fs(x)e Ft(in)f(a)h(neigh)m(b)s (orho)s(o)s(d)e Fs(V)55 b Ft(of)32 b Fo(C)6 b Ft(.)44 b(No)m(w)34 b(w)m(e)g(tak)m(e)f Fs(\015)2402 563 y Fp(1)2469 548 y Fs(>)28 b Ft(0)33 b(so)g(that)f Fs(\032\015)3087 563 y Fp(1)3155 548 y Fo(\024)c Fs(c=)p Ft(10)k(and)h(let)118 668 y Fs(\016)161 683 y Fp(1)228 668 y Fs(>)28 b Ft(0)k(b)s(e)h(small)d (enough)j(to)g(get)953 973 y Fo(\000)1047 848 y Fr(N)7 b Fq(\000)p Fp(1)1052 878 y Fh(X)1063 1088 y Fr(j)t Fp(=0)1217 973 y Ft(log)17 b(dist)1534 988 y Fr(\016)1565 997 y Fd(1)1604 973 y Ft(\()p Fs(f)1701 925 y Fr(j)1690 995 y(t)p 1690 1007 30 3 v 1737 973 a Ft(\()p Fs(x)p Ft(\))p Fs(;)g(S)6 b Ft(\))28 b Fo(\024)g Fs(\015)2200 988 y Fp(1)2239 973 y Fs(N)108 b Ft(for)32 b(large)f Fs(N)11 b(;)678 b Ft(\(11\))118 1291 y(whic)m(h)48 b(is)g(p)s(ossible)f(after)g (prop)s(ert)m(y)i(\(7\))e(of)h(slo)m(w)f(appro)m(ximation)f(to)h Fo(C)6 b Ft(.)90 b(Moreo)m(v)m(er,)53 b(\014xing)118 1411 y Fs(H)41 b Fo(\025)34 b Fs(\032)p Fo(j)17 b Ft(log)f Fs(\016)t Fo(j)36 b Ft(su\016cien)m(tly)g(large)f(in)g(order)h(that)g (it)f(b)s(e)h(also)f(an)h(upp)s(er)g(b)s(ound)h(for)e(for)g(the)i(set) 118 1532 y Fo(f\000)17 b Ft(log)g Fo(k)p Fs(D)s(f)598 1490 y Fq(\000)p Fp(1)587 1554 y Fr(t)p 587 1566 V 691 1532 a Fo(k)28 b Ft(:)f Fs(t)h Fo(2)g Fs(T)8 b(;)45 b(x)28 b Fo(2)g Fs(M)33 b Fo(n)22 b Fs(V)g Fo(g)p Ft(,)32 b(then)h(the)g(set) 936 1765 y Fs(E)h Ft(=)27 b Fo(f)p Ft(1)h Fo(\024)g Fs(j)33 b Fo(\024)28 b Fs(N)39 b Ft(:)27 b Fo(\000)17 b Ft(log)g Fo(k)p Fs(D)s(f)11 b Ft(\()p Fs(f)2253 1718 y Fr(j)t Fq(\000)p Fp(1)2242 1788 y Fr(t)p 2242 1800 V 2379 1765 a Ft(\()p Fs(x)p Ft(\)\))2548 1724 y Fq(\000)p Fp(1)2642 1765 y Fo(k)28 b Fs(>)f(H)8 b Fo(g)118 1999 y Ft(is)32 b(suc)m(h)i(that)f Fs(f)707 1952 y Fr(j)t Fq(\000)p Fp(1)696 2021 y Fr(t)p 696 2033 V 833 1999 a Ft(\()p Fs(x)p Ft(\))28 b Fo(2)g Fs(V)54 b Ft(for)32 b(all)f Fs(j)i Fo(2)28 b Fs(E)39 b Ft(and)590 2233 y Fs(\032)657 2148 y Fh(\014)657 2208 y(\014)690 2233 y Ft(log)17 b(dist)f(\()p Fs(f)1104 2185 y Fr(j)t Fq(\000)p Fp(1)1093 2255 y Fr(t)p 1093 2267 V 1231 2233 a Ft(\()p Fs(x)p Ft(\))p Fs(;)h Fo(C)6 b Ft(\))1502 2148 y Fh(\014)1502 2208 y(\014)1563 2233 y Fs(>)27 b Fo(\000)17 b Ft(log)1903 2148 y Fh(\015)1903 2208 y(\015)1958 2233 y Fs(D)s(f)11 b Ft(\()p Fs(f)2198 2185 y Fr(j)t Fq(\000)p Fp(1)2187 2255 y Fr(t)p 2187 2267 V 2324 2233 a Ft(\()p Fs(x)p Ft(\)\))2493 2192 y Fq(\000)p Fp(1)2588 2148 y Fh(\015)2588 2208 y(\015)2671 2233 y Fs(>)27 b(H)35 b Fo(\025)29 b Fs(\032)p Fo(j)17 b Ft(log)f Fs(\016)t Fo(j)118 2466 y Ft(i.e.,)41 b(dist)16 b(\()p Fs(f)582 2419 y Fr(j)t Fq(\000)p Fp(1)571 2489 y Fr(t)p 571 2501 V 708 2466 a Ft(\()p Fs(x)p Ft(\))p Fs(;)h Fo(C)6 b Ft(\))39 b Fs(<)g(\016)1176 2481 y Fp(1)1215 2466 y Ft(,)i(in)e(particular)e(dist)2018 2481 y Fr(\016)2049 2490 y Fd(1)2088 2466 y Ft(\()p Fs(f)2185 2419 y Fr(j)t Fq(\000)p Fp(1)2174 2489 y Fr(t)p 2174 2501 V 2311 2466 a Ft(\()p Fs(x)p Ft(\))p Fs(;)17 b Fo(C)6 b Ft(\))39 b(=)g(dist)o(\()p Fs(f)2990 2419 y Fr(j)t Fq(\000)p Fp(1)2979 2489 y Fr(t)p 2979 2501 V 3117 2466 a Ft(\()p Fs(x)p Ft(\))p Fs(;)17 b Fo(C)6 b Ft(\))39 b Fs(<)f(\016)3584 2481 y Fp(1)3663 2466 y Ft(for)118 2587 y(all)31 b Fs(j)i Fo(2)28 b Fs(E)6 b Ft(.)44 b(Hence,)34 b(de\014ning)1033 2861 y Fs(a)1084 2876 y Fr(j)1148 2861 y Ft(=)1252 2720 y Fh(\032)1368 2802 y Fo(\000)17 b Ft(log)1605 2718 y Fh(\015)1605 2777 y(\015)1660 2802 y Fs(D)s(f)11 b Ft(\()p Fs(f)1900 2755 y Fr(j)t Fq(\000)p Fp(1)1889 2825 y Fr(t)p 1889 2837 V 2026 2802 a Ft(\()p Fs(x)p Ft(\)\))2195 2766 y Fq(\000)p Fp(1)2289 2718 y Fh(\015)2289 2777 y(\015)2428 2802 y Ft(if)82 b Fs(j)33 b Fo(62)28 b Fs(E)1368 2924 y Ft(0)1011 b(if)82 b Fs(j)33 b Fo(2)28 b Fs(E)118 3140 y Ft(it)k(holds)g Fs(a)522 3155 y Fr(j)586 3140 y Fo(\024)c Fs(H)41 b Ft(for)32 b(1)27 b Fo(\024)h Fs(j)34 b Fo(\024)28 b Fs(N)10 b Ft(,)33 b(and)g(\(10\))f(and)g(\(11\))g(imply)578 3378 y Fo(\000)672 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b(w)m(e)i(sho)m(w)g(that)f (there)h(is)e(a)h(w)m(ell)118 1645 y(de\014ned)34 b(branc)m(h)f(of)f Fs(f)946 1609 y Fq(\000)p Fr(j)1070 1645 y Ft(on)g(a)h(ball)d(of)i (small)e(enough)j(radius)g(around)f Fs(f)2861 1598 y Fr(j)2850 1668 y(t)p 2850 1680 V 2897 1645 a Ft(\()p Fs(x)p Ft(\).)44 b(No)m(w)33 b(w)m(e)h(observ)m(e)118 1766 y(that)e(Lemma)g(2.5)g(giv)m(es)h(for)f Fs(j)h Ft(=)28 b(1)1001 1976 y Fo(k)p Fs(D)s(f)11 b Ft(\()p Fs(y)t Ft(\))1322 1935 y Fq(\000)p Fp(1)1414 1976 y Fo(k)28 b(\024)g Fs(\013)1660 1935 y Fq(\000)p Fp(1)p Fr(=)p Fp(2)1825 1976 y Fo(k)p Fs(D)s(f)11 b Ft(\()p Fs(f)2115 1935 y Fr(n)p Fq(\000)p Fp(1)2104 2001 y Fr(t)p 2104 2013 V 2251 1976 a Ft(\()p Fs(x)p Ft(\)\))2420 1935 y Fq(\000)p Fp(1)2514 1976 y Fo(k)28 b(\024)g Fs(\013)2760 1935 y Fp(1)p Fr(=)p Fp(2)2870 1976 y Fs(;)118 2200 y Ft(b)s(ecause)i Fs(n)e Ft(is)g(a)g(\()p Fs(\013)q(;)17 b(\016)t Ft(\)-h)m(yp)s(erb)s(olic)26 b(time)h(for)g(\()p Fs(t)p 1866 2216 36 4 v 1 w(;)17 b(x)p Ft(\).)42 b(This)28 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4900 y Fr(\017)711 4875 y Fo(\034)27 b Fs(m)33 b Ft(for)f(all)f Fs(x)d Fo(2)g Fs(M)43 b Ft(and)33 b Fs(n)28 b Fo(\025)g Fs(n)1994 4890 y Fp(0)2034 4875 y Ft(.)1900 5251 y(13)p eop %%Page: 14 14 14 13 bop 118 548 a Ft(Here)23 b(\()p Fs(f)435 512 y Fr(n)424 573 y(x)481 548 y Ft(\))519 563 y Fq(\003)559 548 y Fs(\022)607 512 y Fe(N)604 573 y Fr(\017)681 548 y Ft(is)e(the)h(push-forw)m(ard)g(of)g Fs(\022)1654 512 y Fe(N)1651 573 y Fr(\017)1728 548 y Ft(to)f Fs(M)33 b Ft(via)21 b Fs(f)2171 512 y Fr(n)2160 573 y(x)2245 548 y Ft(:)28 b Fs(T)2371 512 y Fe(N)2450 548 y Fo(!)g Fs(M)10 b Ft(,)24 b(de\014ned)f(as)f Fs(f)3226 512 y Fr(n)3215 573 y(x)3273 548 y Ft(\()p Fs(t)p 3311 564 36 4 v Ft(\))28 b(=)f Fs(f)3574 512 y Fr(n)3563 573 y(t)p 3563 585 30 3 v 3621 548 a Ft(\()p Fs(x)p Ft(\).)118 668 y(Condition)22 b(1)h(means)g(that)g(p)s(erturb)s(ed)g(iterates)g (co)m(v)m(er)i(a)d(full)g(neigh)m(b)s(orho)s(o)s(d)f(of)i(the)g(unp)s (erturb)s(ed)118 789 y(ones)41 b(after)g(a)f(threshold)g(for)g(all)f 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Ft(is)g(equal)f(to)h(the)118 1511 y(normalized)32 b(restriction)i(of)f(the)i(Leb)s(esgue)h(measure)e (to)g(the)h(ball)d(of)i(radius)g Fs(\017)g Ft(around)h(0,)f(for)g(a)118 1631 y(su\016cien)m(tly)h(big)e(n)m(um)m(b)s(er)i Fs(p)c Fo(2)f Fg(N)50 b Ft(of)33 b(parameters.)49 b(F)-8 b(or)33 b(parallelizable)e(manifolds)h(the)i(random)118 1752 y(p)s(erturbations)c(whic)m(h)h(consist)g(in)e(adding)h(at)g(eac)m(h)h (step)g(a)f(random)g(noise)g(to)g(the)g(unp)s(erturb)s(ed)118 1872 y(dynamics,)h(as)h(describ)s(ed)g(in)e(the)i(In)m(tro)s(duction,)f (clearly)f(satisfy)h(nondegeneracy)j(conditions)c(1)118 1993 y(and)j(2)f(for)g Fs(n)596 2008 y Fp(0)663 1993 y Ft(=)c(1.)264 2163 y(In)44 b(the)f(con)m(text)i(of)e(random)f(p)s (erturbations)h(of)f(a)h(map,)i(w)m(e)f(sa)m(y)h(that)d(a)h(set)h Fs(A)i Fo(\032)g Fs(M)54 b Ft(is)118 2283 y Fi(invariant)28 b Ft(if)f Fs(f)667 2298 y Fr(t)696 2283 y Ft(\()p Fs(A)p Ft(\))h Fo(\032)g Fs(A)p Ft(,)h(at)f(least)g(for)g Fs(t)f Fo(2)i Ft(supp)17 b(\()p Fs(\022)2048 2298 y Fr(\017)2081 2283 y Ft(\))28 b(with)g Fs(\017)g(>)g Ft(0)g(small.)39 b(The)30 b(usual)d(in)m(v)-5 b(ariance)118 2403 y(of)46 b(a)g(measure)g(with)g(resp)s(ect)i(to)d(a)h(transformation)e(is)i (replaced)g(b)m(y)i(the)e(follo)m(wing)e(one:)71 b(a)118 2524 y(probabilit)m(y)35 b(measure)j Fs(\026)f Ft(is)g(said)f(to)h(b)s (e)h Fi(stationary)p Ft(,)g(if)f(for)f(ev)m(ery)j(con)m(tin)m(uous)f Fs(')e Ft(:)g Fs(M)46 b Fo(!)35 b Fg(R)48 b Ft(it)118 2644 y(holds)1161 2752 y Fh(Z)1278 2888 y Fs(')17 b(d\026)26 b Ft(=)1599 2752 y Fh(Z)61 b(Z)1831 2888 y Fs(')1895 2807 y Fh(\000)1941 2888 y Fs(f)1989 2903 y Fr(t)2018 2888 y Ft(\()p Fs(x)p Ft(\))2149 2807 y Fh(\001)2212 2888 y Fs(d\026)p Ft(\()p Fs(x)p Ft(\))17 b Fs(d\022)2566 2903 y Fr(\017)2598 2888 y Ft(\()p Fs(t)p Ft(\))p Fs(:)870 b Ft(\(16\))118 3156 y Fc(Remark)37 b(3.1.)49 b Fi(If)38 b Ft(\()p Fs(\026)969 3120 y Fr(\017)1002 3156 y Ft(\))1040 3171 y Fr(\017>)p Fp(0)1201 3156 y Fi(is)h(a)g(family)f(of)h (stationary)g(me)-5 b(asur)g(es)38 b(having)g Fs(\026)3077 3171 y Fp(0)3155 3156 y Fi(as)h(a)f(we)-5 b(ak)3572 3120 y Fq(\003)3650 3156 y Fi(ac-)118 3277 y(cumulation)41 b(p)-5 b(oint)41 b(when)g Fs(\017)h Fi(go)-5 b(es)40 b(to)i Ft(0)p Fi(,)g(then)g(it)f(fol)5 b(lows)40 b(fr)-5 b(om)41 b(\(16\))g(and)g(the)g(c)-5 b(onver)g(genc)g(e)40 b(of)118 3397 y(supp)16 b Ft(\()p Fs(\022)409 3412 y Fr(\017)443 3397 y Ft(\))34 b Fi(to)h Fo(f)p Fs(t)717 3361 y Fq(\003)757 3397 y Fo(g)g Fi(that)g Fs(\026)1100 3412 y Fp(0)1174 3397 y Fi(must)g(b)-5 b(e)35 b(invariant)f(by)h Fs(f)j Ft(=)28 b Fs(f)2320 3412 y Fr(t)2345 3393 y Ff(\003)2385 3397 y Fi(.)264 3600 y Ft(It)33 b(is)f(not)h(di\016cult)e(to)i(see)g (\(cf.)g([Ar1]\))f(that)h(a)f(stationary)g(measure)h Fs(\026)f Ft(satis\014es)680 3820 y Fs(x)c Fo(2)g Ft(supp)18 b(\()p Fs(\026)p Ft(\))124 b Fo(\))h Fs(f)1607 3835 y Fr(t)1637 3820 y Ft(\()p Fs(x)p Ft(\))28 b Fo(2)g Ft(supp)18 b(\()p Fs(\026)p Ft(\))97 b(for)32 b(all)95 b Fs(t)28 b Fo(2)g Ft(supp)18 b(\()p Fs(\022)3147 3835 y Fr(\017)3180 3820 y Ft(\))118 4040 y(just)38 b(b)m(y)g(con)m(tin)m(uit)m(y)f(of)g (\010.)59 b(This)37 b(means)g(that)h(if)e Fs(\026)h Ft(is)g(a)g (stationary)f(measure,)j(then)f(supp)18 b(\()p Fs(\026)p Ft(\))118 4161 y(is)33 b(an)h(in)m(v)-5 b(arian)m(t)32 b(set.)47 b(Nondegeneracy)35 b(condition)d(1)h(ensures)j(that)d(the)h (in)m(terior)e(of)h(supp)18 b(\()p Fs(\026)p Ft(\))33 b(is)118 4281 y(nonempt)m(y)-8 b(.)264 4402 y(Let)23 b(us)f(write)g(supp)c(\()p Fs(\026)p Ft(\))j(as)h(a)g(disjoin)m(t)f (union)1933 4327 y Fh(S)2017 4431 y Fr(i)2061 4402 y Fs(C)2131 4417 y Fr(i)2181 4402 y Ft(of)h(connected)h(comp)s(onen)m(ts) g(and)f(consider)118 4522 y(only)27 b(those)h Fs(C)647 4537 y Fr(i)702 4522 y Ft(for)e(whic)m(h)i Fs(m)p Ft(\()p Fs(C)1312 4537 y Fr(i)1340 4522 y Ft(\))g Fs(>)f Ft(0)g({)g(this)g (collection)e(is)i(nonempt)m(y)h(since)f(supp)18 b(\()p Fs(\026)p Ft(\))26 b(con)m(tains)118 4642 y(op)s(en)j(sets.)43 b(Moreo)m(v)m(er)30 b(eac)m(h)g Fs(f)1265 4657 y Fr(t)1323 4642 y Ft(m)m(ust)f(p)s(erm)m(ute)g(these)h(comp)s(onen)m(ts)f(for)f Fs(t)g Fo(2)g Ft(supp)18 b(\()p Fs(\022)3324 4657 y Fr(\017)3357 4642 y Ft(\),)29 b(b)s(ecause)118 4763 y Fs(f)166 4778 y Fr(t)196 4763 y Ft(\()p Fs(C)304 4778 y Fr(i)332 4763 y Ft(\))34 b(is)g(connected)j(b)m(y)e(con)m(tin)m(uit)m(y)-8 b(,)36 b Fs(f)1630 4778 y Fr(t)1659 4763 y Ft(\()p Fs(C)1767 4778 y Fr(i)1795 4763 y Ft(\))31 b Fo(\032)h Ft(supp)18 b(\()p Fs(\026)p Ft(\))34 b(b)m(y)i(in)m(v)-5 b(ariance,)34 b(and)h Fs(m)p Ft(\()p Fs(f)3351 4778 y Fr(t)3381 4763 y Ft(\()p Fs(C)3489 4778 y Fr(i)3517 4763 y Ft(\)\))c Fs(>)g Ft(0)118 4883 y(since)i(w)m(e)h(ha)m(v)m(e)g(\()p Fs(f)812 4898 y Fr(t)841 4883 y Ft(\))879 4898 y Fq(\003)919 4883 y Fs(m)28 b Fo(\034)f Fs(m)p Ft(.)1900 5251 y(14)p eop %%Page: 15 15 15 14 bop 264 548 a Ft(The)49 b(connectedness)i(of)c Fs(C)1321 563 y Fr(i)1396 548 y Ft(and)h(con)m(tin)m(uit)m(y)f(of)g (\010)h(guaran)m(tee)g(that)g(the)g(ab)s(o)m(v)m(emen)m(tion)118 668 y(p)s(erturbation)39 b(of)g(the)i(comp)s(onen)m(ts)f Fs(C)1605 683 y Fr(i)1672 668 y Ft(induced)h(b)m(y)f Fs(f)2233 683 y Fr(t)2303 668 y Ft(do)s(es)g(not)g(dep)s(end)h(on)e Fs(t)h Fo(2)h Ft(supp)17 b(\()p Fs(\022)3681 683 y Fr(\017)3714 668 y Ft(\).)118 789 y(Indeed,)34 b(supp)s(osing)f(that)f Fs(t;)17 b(t)1241 753 y Fq(0)1292 789 y Fo(2)28 b Ft(supp)18 b(\()p Fs(\022)1687 804 y Fr(\017)1720 789 y Ft(\))33 b(are)f(suc)m(h)i(that)1246 1006 y Fs(f)1294 1021 y Fr(t)1324 1006 y Ft(\()p Fs(C)1432 1021 y Fr(i)1460 1006 y Ft(\))27 b Fo(\032)h Fs(C)1700 1021 y Fr(j)1834 1006 y Ft(and)98 b Fs(f)2137 1021 y Fr(t)2162 1002 y Ff(0)2189 1006 y Ft(\()p Fs(C)2297 1021 y Fr(i)2325 1006 y Ft(\))28 b Fo(\032)g Fs(C)2566 1021 y Fr(j)2599 1002 y Ff(0)2624 1006 y Fs(;)118 1224 y Ft(then)38 b(\014xing)f(some)g Fs(z)k Fo(2)36 b Fs(C)1125 1239 y Fr(i)1191 1224 y Ft(w)m(e)i(ha)m(v)m (e)h(that)e Fo(f)p Fs(f)1883 1239 y Fr(t)1912 1224 y Ft(\()p Fs(z)t Ft(\))11 b(:)36 b Fs(t)g Fo(2)g Ft(supp)18 b(\()p Fs(\022)2585 1239 y Fr(\017)2618 1224 y Ft(\))p Fo(g)37 b Ft(is)g(a)g(connected)i(set)f(in)m(ter-)118 1344 y(secting)33 b(b)s(oth)f Fs(C)744 1359 y Fr(j)813 1344 y Ft(and)h Fs(C)1073 1359 y Fr(j)1106 1340 y Ff(0)1164 1344 y Ft(inside)f(supp)17 b(\()p Fs(\026)p Ft(\),)33 b(and)f(so)h Fs(C)2232 1359 y Fr(j)2296 1344 y Ft(=)28 b Fs(C)2470 1359 y Fr(j)2503 1340 y Ff(0)2528 1344 y Ft(.)264 1464 y(W)-8 b(e)40 b(will)d(sho)m(w)j(that)f(these)i (connected)g(comp)s(onen)m(ts)f(are)f(p)s(erio)s(dic)f(under)i(the)f (action)g(in-)118 1585 y(duced)e(b)m(y)g Fs(f)590 1600 y Fr(t)656 1585 y Ft(with)e Fs(t)f Fo(2)g Ft(supp)17 b(\()p Fs(\022)1350 1600 y Fr(\017)1383 1585 y Ft(\).)54 b(After)36 b(this,)g(w)m(e)h(ma)m(y)f(use)h(nondegeneracy)h(condition)c (1)i(to)118 1705 y(conclude)41 b(that)g(eac)m(h)g(comp)s(onen)m(t)g (con)m(tains)f(a)h(ball)e(of)h(uniform)f(radius)h(and)g(th)m(us)i(that) f(eac)m(h)118 1826 y(comp)s(onen)m(t)36 b(satis\014es)h Fs(m)p Ft(\()p Fs(C)1176 1841 y Fr(i)1204 1826 y Ft(\))c Fs(>)g Ft(const)h Fs(>)f Ft(0.)53 b(Hence)37 b(there)g(existing)e(only) h(a)f(\014nite)h(n)m(um)m(b)s(er)g(of)118 1946 y(suc)m(h)e(comp)s(onen) m(ts.)264 2066 y(A)m(t)f(this)g(p)s(oin)m(t)e(it)h(is)g(useful)h(to)f (in)m(tro)s(duce)g(the)h(sk)m(ew-pro)s(duct)i(map)1232 2275 y Fs(F)41 b Ft(:)84 b Fs(T)1518 2239 y Fe(N)1592 2275 y Fo(\002)22 b Fs(M)94 b Fo(\000)-16 b(!)179 b Fs(T)2290 2239 y Fe(N)2364 2275 y Fo(\002)23 b Fs(M)1519 2396 y Ft(\()p Fs(t)p 1557 2412 36 4 v(;)17 b(z)t Ft(\))156 b Fo(7\000)-16 b(!)2122 2316 y Fh(\000)2168 2396 y Fs(\033)t Ft(\()p Fs(t)p 2265 2412 V Ft(\))p Fs(;)17 b(f)2430 2411 y Fr(t)2455 2420 y Fd(1)2494 2396 y Ft(\()p Fs(z)t Ft(\))2619 2316 y Fh(\001)118 2620 y Ft(where)33 b Fs(\033)i Ft(is)30 b(the)i(left)e(shift)h(on)g(sequences)k Fs(t)p 1716 2636 V 28 w Ft(=)27 b(\()p Fs(t)1955 2635 y Fp(1)1995 2620 y Fs(;)17 b(t)2074 2635 y Fp(2)2113 2620 y Fs(;)g(:)g(:)g(:)f Ft(\))27 b Fo(2)i Fs(T)2519 2583 y Fe(N)2570 2620 y Ft(.)43 b(It)32 b(is)e(easy)j(to)e(c)m(hec)m(k)i(that)e(the)118 2740 y(pro)s(duct)37 b(measure)g Fs(\022)920 2704 y Fe(N)917 2765 y Fr(\017)998 2740 y Fo(\002)25 b Fs(\026)37 b Ft(is)f Fs(F)14 b Ft(-in)m(v)-5 b(arian)m(t,)36 b(as)h(so)g(is)f(the)h(set)h (supp)17 b(\()p Fs(\022)2831 2704 y Fe(N)2828 2765 y Fr(\017)2909 2740 y Fo(\002)25 b Fs(\026)p Ft(\))34 b(=)h(supp)18 b(\()p Fs(\022)3554 2755 y Fr(\017)3587 2740 y Ft(\))3625 2704 y Fe(N)3702 2740 y Fo(\002)118 2860 y Ft(supp)g(\()p Fs(\026)p Ft(\).)118 3062 y Fc(Lemma)37 b(3.2.)49 b Fi(The)35 b(supp)-5 b(ort)35 b(of)f(a)h(stationary)g(me)-5 b(asur)g(e)35 b Fs(\026)f Fi(c)-5 b(ontains)35 b(a)f(\014nite)h(numb)-5 b(er)35 b(of)f(c)-5 b(on-)118 3182 y(ne)g(cte)g(d)35 b(c)-5 b(omp)g(onents)33 b(arr)-5 b(ange)g(d)34 b(in)h(cycles)f(p)-5 b(ermute)g(d)35 b(by)g(the)g(action)f(of)h Fs(f)2910 3197 y Fr(t)2974 3182 y Fi(for)g Fs(t)28 b Fo(2)g Fi(supp)16 b Ft(\()p Fs(\022)3578 3197 y Fr(\017)3611 3182 y Ft(\))p Fi(.)118 3383 y(Pr)-5 b(o)g(of.)49 b Ft(Is)32 b(is)g(enough)g(to)g (obtain)f(that)h(eac)m(h)h(connected)h(comp)s(onen)m(t)e Fs(C)2837 3398 y Fr(i)2897 3383 y Ft(is)g(p)s(erio)s(dic)e(under)j(the) 118 3504 y(action)39 b(of)g Fs(f)584 3519 y Fr(t)653 3504 y Ft(for)g Fs(t)h Fo(2)f Ft(supp)18 b(\()p Fs(\022)1290 3519 y Fr(\017)1323 3504 y Ft(\),)41 b(in)e(the)h(sense)h(that)f Fs(f)2260 3456 y Fr(p)2249 3526 y(t)p 2249 3538 30 3 v 2299 3504 a Ft(\()p Fs(C)2407 3519 y Fr(i)2435 3504 y Ft(\))f Fo(\032)h Fs(C)2699 3519 y Fr(i)2767 3504 y Ft(for)f(some)g Fs(p)h Fo(2)g Fg(N)54 b Ft(and)40 b(all)118 3624 y Fs(t)p 118 3640 36 4 v 33 w Fo(2)33 b Ft(supp)18 b(\()p Fs(\022)589 3588 y Fe(N)586 3649 y Fr(\017)641 3624 y Ft(\).)53 b(There)36 b(are)g(comp)s(onen)m(ts)g Fs(C)1819 3639 y Fr(i)1882 3624 y Ft(with)g(nonempt)m(y)g(in)m(terior,) f(since)h(the)g(in)m(terior)e(of)118 3744 y(supp)18 b(\()p Fs(\026)p Ft(\))41 b(is)g(nonempt)m(y)-8 b(.)72 b(So)42 b(w)m(e)g(ma)m(y)g(tak)m(e)h(a)e(comp)s(onen)m(t)h Fs(C)2539 3759 y Fr(i)2608 3744 y Ft(that)g(con)m(tains)g(some)f(ball)f Fs(B)5 b Ft(.)118 3865 y(Then)40 b(w)m(e)g(ha)m(v)m(e)h Fs(m)p Ft(\()p Fs(B)5 b Ft(\))39 b Fs(>)f Ft(0)h(and)g(so)g(\()p Fs(\022)1650 3829 y Fe(N)1647 3889 y Fr(\017)1729 3865 y Fo(\002)27 b Fs(\026)p Ft(\)\(supp)17 b(\()p Fs(\022)2271 3829 y Fe(N)2268 3889 y Fr(\017)2323 3865 y Ft(\))27 b Fo(\002)g Fs(B)5 b Ft(\))39 b Fs(>)f Ft(0.)63 b(P)m(oincar)m(\023)-46 b(e)39 b(Recurrence)118 3985 y(Theorem)32 b(no)m(w)g(guaran)m(tees)h (there)f(is)f(\()p Fs(t)p 1599 4001 V 1 w(;)17 b(x)p Ft(\))27 b Fo(2)h Ft(supp)18 b(\()p Fs(\022)2197 3949 y Fe(N)2194 4010 y Fr(\017)2249 3985 y Ft(\))i Fo(\002)h Fs(B)37 b Ft(suc)m(h)c(that)e(the)h Fs(F)14 b Ft(-orbit)30 b(of)h(\()p Fs(t)p 3607 4001 V(;)17 b(x)p Ft(\))118 4106 y(has)29 b(the)g(same)f(\()p Fs(t)p 730 4121 V(;)17 b(x)p Ft(\))28 b(as)h(an)f(accum)m(ulation)f(p)s(oin)m(t.)41 b(W)-8 b(e)29 b(see)g(that)f(there)i(m)m(ust)e(exist)h(some)f Fs(p)f Fo(2)i Fg(N)118 4226 y Ft(suc)m(h)42 b(that)e Fs(f)624 4179 y Fr(p)613 4248 y(t)p 613 4260 30 3 v 663 4226 a Ft(\()p Fs(x)p Ft(\))h Fo(2)g Fs(B)46 b Fo(\032)c Fs(C)1251 4241 y Fr(i)1279 4226 y Ft(.)66 b(In)41 b(view)f(of)g(the)h (indep)s(endence)h(of)d(the)i(p)s(erm)m(utation)e(on)h(the)118 4346 y(c)m(hoice)33 b(of)f Fs(t)p 519 4362 36 4 v Ft(,)h(w)m(e)h (conclude)f(that)f Fs(C)1440 4361 y Fr(i)1500 4346 y Ft(is)h(sen)m(t)g(inside)f(itself)g(b)m(y)h Fs(f)2510 4299 y Fr(p)2499 4369 y(t)p 2499 4381 30 3 v 2582 4346 a Ft(for)f(all)e Fs(t)p 2866 4362 36 4 v 28 w Fo(2)e Ft(supp)18 b(\()p Fs(\022)3327 4310 y Fe(N)3324 4371 y Fr(\017)3379 4346 y Ft(\).)p 3709 4346 4 66 v 3713 4284 59 4 v 3713 4346 V 3771 4346 4 66 v 264 4541 a(It)36 b(is)f(clear)h(that)f(the)h(cycles)h(obtained)e(ab)s(o)m(v)m(e)h(are)g (in)m(v)-5 b(arian)m(t)34 b(sets.)54 b(W)-8 b(e)36 b(are)g(no)m(w)g (ready)h(to)118 4661 y(decomp)s(ose)c Fs(\026)f Ft(in)m(to)g(some)h (simpler)e(measures.)44 b(F)-8 b(or)32 b(that)g(w)m(e)i(need)f(the)g (follo)m(wing)d(result.)118 4862 y Fc(Lemma)37 b(3.3.)49 b Fi(The)39 b(normalize)-5 b(d)37 b(r)-5 b(estriction)39 b(of)g(a)g(stationary)g(me)-5 b(asur)g(e)39 b(to)g(an)g(invariant)f (set)118 4983 y(is)d(a)f(stationary)h(me)-5 b(asur)g(e.)1900 5251 y Ft(15)p eop %%Page: 16 16 16 15 bop 118 548 a Fi(Pr)-5 b(o)g(of.)49 b Ft(See)33 b([Ar1,)f(Lemma)g(8.2].)p 3709 548 4 66 v 3713 485 59 4 v 3713 548 V 3771 548 4 66 v 264 740 a(W)-8 b(e)29 b(de\014ne)g(an)f Fi(invariant)h(domain)e Ft(in)g Fs(M)39 b Ft(as)28 b(a)f(\014nite)h(collection)e(\()p Fs(U)2806 755 y Fp(0)2845 740 y Fs(;)17 b(:)g(:)g(:)f(;)h(U)3130 755 y Fr(p)p Fq(\000)p Fp(1)3260 740 y Ft(\))28 b(of)f(pairwise)118 861 y(separated)33 b(op)s(en)f(sets,)h(that)f(is,)p 1344 781 77 4 v 31 w Fs(U)1421 876 y Fr(i)1470 861 y Fo(\\)p 1557 781 V 21 w Fs(U)1633 876 y Fr(j)1698 861 y Ft(=)27 b Fo(;)32 b Ft(if)e Fs(i)e Fo(6)p Ft(=)g Fs(j)6 b Ft(,)32 b(suc)m(h)h(that)f Fs(f)2730 825 y Fr(k)2719 885 y(t)p 2719 897 30 3 v 2772 861 a Ft(\()p Fs(U)2876 876 y Fr(i)2904 861 y Ft(\))c Fo(\032)g Fs(U)3141 876 y Fp(\()p Fr(k)r Fp(+)p Fr(i)p Fp(\))t(mo)r(d)t Fr(p)3528 861 y Ft(for)k(all)118 991 y Fs(k)f Fo(\025)d Ft(1,)k Fs(i)c Ft(=)g(0)p Fs(;)17 b(:)g(:)g(:)e(;)i(p)22 b Fo(\000)h Ft(1)32 b(and)h Fs(t)p 1287 1006 36 4 v 28 w Fo(2)28 b Ft(supp)18 b(\()p Fs(\022)1748 954 y Fe(N)1745 1015 y Fr(\017)1800 991 y Ft(\).)264 1111 y(In)27 b(order)g(to)f(get)h(the)f(separation)g(of)g(the)h (connected)i(comp)s(onen)m(ts)e(in)e(a)i(cycle,)h(w)m(e)f(ma)m(y)g (unite)118 1231 y(those)33 b(comp)s(onen)m(ts)g Fs(C)980 1246 y Fr(i)1040 1231 y Ft(and)f Fs(C)1299 1246 y Fr(j)1367 1231 y Ft(suc)m(h)i(that)p 1798 1151 77 4 v 32 w Fs(C)1875 1246 y Fr(i)1925 1231 y Fo(\\)p 2012 1151 V 21 w Fs(C)2089 1246 y Fr(j)2153 1231 y Fo(6)p Ft(=)28 b Fo(;)k Ft(and)g(observ)m(e)i (that)e(the)h(p)s(erm)m(utation)118 1352 y(no)m(w)g(induced)f(in)f(the) h(new)h(sets)g(b)m(y)g Fs(f)1536 1367 y Fr(t)1597 1352 y Ft(also)e(do)s(es)i(not)e(dep)s(end)i(on)f(the)g(c)m(hoice)g(of)g Fs(t)c Fo(2)g Ft(supp)17 b(\()p Fs(\022)3681 1367 y Fr(\017)3714 1352 y Ft(\).)118 1472 y(In)41 b(this)g(manner)f(w)m(e)i(construct)f (in)m(v)-5 b(arian)m(t)40 b(domains)f(inside)h(the)h(supp)s(ort)h(of)e (an)m(y)h(stationary)118 1592 y(probabilit)m(y)31 b(measure.)264 1713 y(The)g(next)f(step)h(is)e(to)g(lo)s(ok)g(for)g Fi(minimal)h(invariant)i(domains)k Ft(with)30 b(resp)s(ect)g(to)g(the)g (natural)118 1833 y(order)38 b(relation)f(of)g(inclusion)g(of)g(sets.) 62 b(Let)38 b Fs(D)i Ft(=)d(\()p Fs(U)2151 1848 y Fp(0)2190 1833 y Fs(;)17 b(:)g(:)g(:)f(;)h(U)2475 1848 y Fr(p)p Fq(\000)p Fp(1)2605 1833 y Ft(\))38 b(and)g Fs(D)2960 1797 y Fq(0)3020 1833 y Ft(=)f(\()p Fs(W)3263 1848 y Fp(0)3303 1833 y Fs(;)17 b(:)g(:)g(:)f(;)h(W)3614 1848 y Fr(q)r Fq(\000)p Fp(1)3742 1833 y Ft(\))118 1954 y(b)s(e)47 b(in)m(v)-5 b(arian)m(t)45 b(domains.)85 b(On)47 b(the)g(one)g(hand,)k Fs(D)j Ft(=)e Fs(D)2342 1917 y Fq(0)2411 1954 y Ft(if)46 b(there)h(are)g Fs(i;)17 b(j)58 b Fo(2)52 b Fg(N)62 b Ft(suc)m(h)48 b(that)118 2074 y Fs(U)184 2089 y Fp(\()p Fr(i)p Fp(+)p Fr(k)r Fp(\))t(mo)r(d)t Fr(p)570 2074 y Ft(=)31 b Fs(W)769 2089 y Fp(\()p Fr(j)t Fp(+)p Fr(k)r Fp(\))t(mo)r(d)s Fr(q)1165 2074 y Ft(for)j(all)e Fs(k)i Fo(\025)d Ft(1,)k(whic)m(h)g(implies)d Fs(p)e Ft(=)h Fs(q)38 b Ft(b)s(ecause)e(the)f(op)s(en)f(sets)i(that)118 2194 y(form)d(eac)m(h)j(in)m(v)-5 b(arian)m(t)33 b(domain)g(are)h (pairwise)g(disjoin)m(t.)48 b(On)35 b(the)f(other)h(hand,)g(w)m(e)h(sa) m(y)f Fs(D)f Fo(\036)e Fs(D)3757 2158 y Fq(0)118 2315 y Ft(if)39 b(there)i(are)f Fs(i;)17 b(j)47 b Fo(2)41 b Fg(N)55 b Ft(suc)m(h)41 b(that)f Fs(U)1537 2330 y Fr(i)t Fp(mo)r(d)t Fr(p)1785 2315 y Fg(\()h Fs(W)1995 2330 y Fr(j)8 b Fp(mo)r(d)s Fr(q)2249 2315 y Ft(and)40 b Fs(U)2512 2330 y Fp(\()p Fr(i)p Fp(+)p Fr(k)r Fp(\))t(mo)r(d)t Fr(p)2908 2315 y Fo(\032)h Fs(W)3118 2330 y Fp(\()p Fr(j)t Fp(+)p Fr(k)r Fp(\))t(mo)r(d)s Fr(q)3520 2315 y Ft(for)f(all)118 2435 y Fs(k)31 b Fo(\025)d Ft(1.)118 2630 y Fc(Lemma)37 b(3.4.)49 b Fi(In)39 b(the)f(p)-5 b(artial)5 b(ly)39 b(or)-5 b(der)g(e)g(d)38 b(family)h(of)f(al)5 b(l)39 b(invariant)f(domains)f(in)i Fs(M)10 b Fi(,)40 b(with)f(r)-5 b(e-)118 2751 y(sp)g(e)g(ct)39 b(to)g(the)f(r)-5 b(elation)39 b Fo(\036)p Fi(,)h(the)f(numb)-5 b(er)38 b(of)g Fo(\036)p Fi(-minimal)g(domains)g(is)g(\014nite.)56 b(Mor)-5 b(e)g(over,)40 b(every)118 2871 y(invariant)34 b(domain)g(c)-5 b(ontains)34 b(at)h(le)-5 b(ast)35 b(one)f(minimal)g(domain.)118 3066 y(Pr)-5 b(o)g(of.)49 b Ft(The)32 b(pro)s(of)e(relies)h(in)g(sho)m(wing) g(that)g(Zorn's)h(Lemma)e(can)h(b)s(e)h(applied)e(to)h(this)g (partially)118 3187 y(ordered)c(set)h(and)e(that)h(minimal)22 b(domains)j(are)i(pairwise)f(separated.)42 b(See)28 b([Ar1,)g(Section)e (3].)p 3709 3187 4 66 v 3713 3124 59 4 v 3713 3187 V 3771 3187 4 66 v 264 3379 a(Let)33 b(us)g(no)m(w)h(\014x)f Fs(x)28 b Fo(2)g Fs(M)43 b Ft(and)33 b(consider)1476 3659 y Fs(\026)1535 3674 y Fr(n)1582 3659 y Ft(\()p Fs(x)p Ft(\))28 b(=)1859 3591 y(1)p 1855 3636 59 4 v 1855 3727 a Fs(n)1945 3534 y Fr(n)p Fq(\000)p Fp(1)1939 3564 y Fh(X)1950 3774 y Fr(j)t Fp(=0)2083 3659 y Ft(\()p Fs(f)2180 3618 y Fr(j)2169 3684 y(x)2216 3659 y Ft(\))2254 3674 y Fq(\003)2294 3659 y Fs(\022)2342 3618 y Fe(N)2339 3684 y Fr(\017)2394 3659 y Fs(:)1185 b Ft(\(17\))118 3965 y(Since)37 b(this)f(is)g(a)g(sequence)j(of)d(probabilit)m(y)f(measures) i(on)f(the)h(compact)f(manifold)e Fs(M)10 b Ft(,)38 b(then)f(it)118 4085 y(has)c(w)m(eak)502 4049 y Fq(\003)575 4085 y Ft(accum)m(ulation)e (p)s(oin)m(ts.)118 4280 y Fc(Lemma)37 b(3.5.)49 b Fi(Every)30 b(we)-5 b(ak)1212 4244 y Fq(\003)1281 4280 y Fi(ac)g(cumulation)29 b(p)-5 b(oint)30 b(of)2222 4200 y Fh(\000)2268 4280 y Fs(\026)2327 4295 y Fr(n)2374 4280 y Ft(\()p Fs(x)p Ft(\))2505 4200 y Fh(\001)2551 4319 y Fr(n)2628 4280 y Fi(is)f(stationary)h(and)g (absolutely)118 4401 y(c)-5 b(ontinuous)35 b(with)f(r)-5 b(esp)g(e)g(ct)35 b(to)g(the)g(L)-5 b(eb)g(esgue)34 b(me)-5 b(asur)g(e.)118 4596 y(Pr)g(o)g(of.)49 b Ft(Let)32 b Fs(\026)g Ft(b)s(e)h(a)g(w)m(eak)1114 4560 y Fq(\003)1187 4596 y Ft(accum)m(ulation)e(p)s(oin)m(t)g(of)2152 4515 y Fh(\000)2197 4596 y Fs(\026)2256 4611 y Fr(n)2303 4596 y Ft(\()p Fs(x)p Ft(\))2434 4515 y Fh(\001)2480 4635 y Fr(n)2527 4596 y Ft(.)43 b(W)-8 b(e)33 b(ma)m(y)g(write)364 4778 y Fh(Z)61 b(Z)596 4914 y Fs(')660 4833 y Fh(\000)706 4914 y Fs(f)754 4929 y Fr(t)784 4914 y Ft(\()p Fs(x)p Ft(\))915 4833 y Fh(\001)977 4914 y Fs(d\026)p Ft(\()p Fs(x)p Ft(\))17 b Fs(d\022)1331 4929 y Fr(\017)1363 4914 y Ft(\()p Fs(t)p Ft(\))28 b(=)1606 4778 y Fh(Z)1771 4914 y Ft(lim)1722 4976 y Fr(k)r Fq(!)p Fp(+)p Fq(1)2009 4846 y Ft(1)p 1983 4891 101 4 v 1983 4982 a Fs(n)2041 4997 y Fr(k)2111 4786 y(n)2154 4798 y Fn(k)2191 4786 y Fq(\000)p Fp(1)2124 4819 y Fh(X)2135 5029 y Fr(j)t Fp(=0)2298 4778 y Fh(Z)2414 4914 y Fs(')2495 4833 y Fh(\000)2541 4914 y Fs(f)2589 4929 y Fr(t)2618 4833 y Fh(\000)2664 4914 y Fs(f)2723 4866 y Fr(j)2712 4936 y(t)p 2712 4948 30 3 v 2759 4914 a Ft(\()p Fs(x)p Ft(\))2890 4833 y Fh(\001\001)3015 4914 y Fs(d\022)3114 4873 y Fe(N)3111 4938 y Fr(\017)3166 4914 y Ft(\()p Fs(t)p 3204 4930 36 4 v Ft(\))17 b Fs(d\022)3390 4929 y Fr(\017)3423 4914 y Ft(\()p Fs(t)p Ft(\))1900 5251 y(16)p eop %%Page: 17 17 17 16 bop 118 548 a Ft(for)40 b(eac)m(h)i(con)m(tin)m(uous)f Fs(')g Ft(:)g Fs(M)52 b Fo(!)41 b Fg(R)5 b Ft(.)73 b(Moreo)m(v)m(er)42 b(dominated)d(con)m(v)m(ergence)44 b(ensures)e(that)e(w)m(e)118 668 y(ma)m(y)32 b(exc)m(hange)j(the)e(limit)c(and)j(the)h(outer)g(in)m (tegral)e(sign)h(and,)h(b)m(y)g(de\014nition)f(of)g Fs(f)3272 621 y Fr(j)3261 691 y(t)p 3261 703 30 3 v 3308 668 a Ft(\()p Fs(x)p Ft(\),)h(w)m(e)h(get)1045 994 y(lim)1022 1057 y Fr(k)r Fq(!1)1255 926 y Ft(1)p 1229 971 101 4 v 1229 1062 a Fs(n)1287 1077 y Fr(k)1356 866 y(n)1399 878 y Fn(k)1437 866 y Fq(\000)p Fp(1)1370 899 y Fh(X)1380 1109 y Fr(j)t Fp(=0)1544 858 y Fh(Z)1660 994 y Fs(')1724 913 y Fh(\000)1770 994 y Fs(f)1829 947 y Fr(j)t Fp(+1)1818 1016 y Fr(t)p 1818 1028 30 3 v 1955 994 a Ft(\()p Fs(x)p Ft(\))2086 913 y Fh(\001)2149 994 y Fs(d\022)2248 953 y Fe(N)2245 1018 y Fr(\017)2300 994 y Ft(\()p Fs(t)p 2338 1010 36 4 v Ft(\))27 b(=)2542 858 y Fh(Z)2658 994 y Fs(')17 b(d\026;)118 1315 y Ft(according)32 b(to)g(the)h (de\014nition)f(of)g Fs(\026)p Ft(.)43 b(Th)m(us)34 b(\(16\))e(m)m(ust) h(hold)f(and)g Fs(\026)h Ft(is)f(stationary)-8 b(.)264 1435 y(Noting)42 b(that)h Fs(C)896 1399 y Fp(0)936 1435 y Ft(\()p Fs(M)5 b(;)17 b Fg(R)5 b Ft(\))49 b(is)43 b(dense)h(in)f Fs(L)1846 1399 y Fp(1)1885 1435 y Ft(\()p Fs(M)5 b(;)17 b(\026)p Ft(\))43 b(with)g(the)h Fs(L)2684 1399 y Fp(1)2767 1435 y Ft(norm,)h(w)m(e)f(see)g(that)f(\(16\))118 1555 y(holds)35 b(for)h(all)d Fs(\026)p Ft(-in)m(tegrable)h(functions)i Fs(')c Ft(:)h Fs(M)44 b Fo(!)33 b Fg(R)5 b Ft(.)58 b(In)36 b(particular,)f(if)g Fs(E)k Fo(\032)33 b Fs(M)46 b Ft(is)36 b(suc)m(h)h(that)118 1676 y Fs(m)p Ft(\()p Fs(E)6 b Ft(\))28 b(=)g(0,)k(then)803 1809 y Fh(Z)919 1944 y Ft(1)968 1959 y Fr(E)1044 1944 y Fs(d\026)83 b Ft(=)1395 1809 y Fh(Z)62 b(Z)1628 1944 y Ft(1)1677 1959 y Fr(E)1736 1864 y Fh(\000)1782 1944 y Fs(f)1830 1959 y Fr(t)1860 1944 y Ft(\()p Fs(x)p Ft(\))1991 1864 y Fh(\001)2053 1944 y Fs(d\026)p Ft(\()p Fs(x)p Ft(\))17 b Fs(d\022)2407 1959 y Fr(\017)2439 1944 y Ft(\()p Fs(t)p Ft(\))1237 2199 y(=)1395 2063 y Fh(Z)62 b(Z)1628 2199 y Ft(1)1677 2214 y Fr(E)1736 2118 y Fh(\000)1782 2199 y Fs(f)1830 2214 y Fr(t)1860 2199 y Ft(\()p Fs(x)p Ft(\))1991 2118 y Fh(\001)2053 2199 y Fs(d\022)2149 2214 y Fr(\017)2182 2199 y Ft(\()p Fs(t)p Ft(\))17 b Fs(d\026)p Ft(\()p Fs(x)p Ft(\))1237 2453 y(=)1395 2318 y Fh(Z)62 b(Z)f(Z)1744 2453 y Ft(1)1793 2468 y Fr(E)1853 2373 y Fh(\000)1898 2453 y Fs(f)1946 2468 y Fr(t)1976 2453 y Ft(\()p Fs(f)2062 2468 y Fr(s)2099 2453 y Ft(\()p Fs(x)p Ft(\)\))2268 2373 y Fh(\001)2330 2453 y Fs(d\022)2426 2468 y Fr(\017)2459 2453 y Ft(\()p Fs(t)p Ft(\))17 b Fs(d\026)p Ft(\()p Fs(x)p Ft(\))g Fs(d\022)2941 2468 y Fr(\017)2973 2453 y Ft(\()p Fs(s)p Ft(\))1237 2708 y(=)1395 2572 y Fh(Z)62 b(Z)1628 2708 y Ft(1)1677 2723 y Fr(E)1736 2627 y Fh(\000)1782 2708 y Fs(f)1841 2667 y Fp(2)1830 2733 y Fr(t)p 1830 2745 30 3 v 1880 2708 a Ft(\()p Fs(x)p Ft(\))2011 2627 y Fh(\001)2074 2708 y Fs(d\022)2173 2667 y Fe(N)2170 2733 y Fr(\017)2225 2708 y Ft(\()p Fs(t)p 2263 2724 36 4 v Ft(\))17 b Fs(d\026)p Ft(\()p Fs(x)p Ft(\))1237 2963 y(=)1395 2827 y Fh(Z)1495 2963 y Ft(\()p Fs(f)1592 2922 y Fp(2)1581 2987 y Fr(x)1631 2963 y Ft(\))1669 2978 y Fq(\003)1709 2963 y Fs(\022)1757 2922 y Fe(N)1754 2987 y Fr(\017)1809 2963 y Ft(\()p Fs(E)6 b Ft(\))17 b Fs(d\026)p Ft(\()p Fs(x)p Ft(\))p Fs(:)118 3226 y Ft(This)33 b(pro)s(cess)h(ma)m(y)e(b)s(e)h(iterated)f(to)g (yield)1355 3488 y Fs(\026)p Ft(\()p Fs(E)6 b Ft(\))27 b(=)1698 3353 y Fh(Z)1798 3488 y Ft(\()p Fs(f)1895 3447 y Fr(n)1938 3456 y Fd(0)1884 3513 y Fr(x)1976 3488 y Ft(\))2014 3503 y Fq(\003)2054 3488 y Fs(\022)2099 3503 y Fr(\017)2132 3488 y Ft(\()p Fs(E)6 b Ft(\))17 b Fs(d\026)p Ft(\()p Fs(x)p Ft(\))118 3757 y(and,)33 b(since)g(\()p Fs(f)671 3721 y Fr(n)714 3730 y Fd(0)660 3782 y Fr(x)752 3757 y Ft(\))790 3772 y Fq(\003)829 3757 y Fs(\022)874 3772 y Fr(\017)935 3757 y Fo(\034)28 b Fs(m)k Ft(b)m(y)i(nondegeneracy) g(condition)e(2,)g(w)m(e)i(m)m(ust)e(ha)m(v)m(e)i Fs(\026)p Ft(\()p Fs(E)6 b Ft(\))27 b(=)h(0.)p 3709 3757 4 66 v 3713 3694 59 4 v 3713 3757 V 3771 3757 4 66 v 264 3952 a(Clearly)36 b(if)f Fs(x)f Fo(2)h Fs(M)47 b Ft(b)s(elongs)36 b(to)g(some)g(set)h(of)f(an)g(in)m(v)-5 b(arian)m(t)35 b(domain)f(\()p Fs(U)3033 3967 y Fp(0)3073 3952 y Fs(;)17 b(:)g(:)g(:)f(;)h(U)3358 3967 y Fr(p)p Fq(\000)p Fp(1)3488 3952 y Ft(\),)37 b(then)118 4072 y Fs(\026)177 4087 y Fr(n)224 4072 y Ft(\()p Fs(x)p Ft(\))22 b(ha)m(v)m(e)i(supp)s(orts)f (con)m(tained)f(in)p 1515 3992 77 4 v 21 w Fs(U)1592 4087 y Fp(0)1632 4072 y Fo([)q(\001)17 b(\001)g(\001[)p 1884 3992 V 1 w Fs(U)1961 4087 y Fr(p)p Fq(\000)p Fp(1)2113 4072 y Ft(for)22 b(all)e Fs(n)28 b Fo(\025)g Ft(1)22 b(and)g(an)m(y)h(w)m(eak)3202 4036 y Fq(\003)3265 4072 y Ft(accumlation)118 4193 y(p)s(oin)m(t)33 b Fs(\026)h Ft(of)g(\()p Fs(\026)677 4208 y Fr(n)724 4193 y Ft(\()p Fs(x)p Ft(\)\))893 4208 y Fr(n)974 4193 y Ft(is)g(a)g(stationary)f (measure)i(with)f(supp)17 b(\()p Fs(\026)p Ft(\))30 b Fo(\032)p 2718 4113 V 31 w Fs(U)2795 4208 y Fp(0)2858 4193 y Fo([)23 b(\001)17 b(\001)g(\001)22 b([)p 3176 4113 V 23 w Fs(U)3253 4208 y Fr(p)p Fq(\000)p Fp(1)3383 4193 y Ft(.)48 b(W)-8 b(e)35 b(will)118 4313 y(no)m(w)e(see)h(these)g (measures)f(are)g(ph)m(ysical.)118 4517 y Fc(Lemma)k(3.6.)49 b Fi(If)35 b Ft(\()p Fs(U)945 4532 y Fp(0)985 4517 y Fs(;)17 b(:)g(:)g(:)f(;)h(U)1270 4532 y Fr(p)p Fq(\000)p Fp(1)1399 4517 y Ft(\))36 b Fi(is)f(a)g(minimal)f(invariant)h(domain,)f (then)i(ther)-5 b(e)35 b(is)g(a)h(unique)118 4637 y(absolutely)30 b(c)-5 b(ontinuous)29 b(stationary)h(me)-5 b(asur)g(e)29 b Fs(\027)36 b Fi(such)30 b(that)g(supp)16 b Ft(\()p Fs(\027)6 b Ft(\))28 b Fo(\032)p 2839 4557 V 28 w Fs(U)2916 4652 y Fp(0)2966 4637 y Fo([)11 b(\001)17 b(\001)g(\001)9 b([)p 3247 4557 V 11 w Fs(U)3324 4652 y Fr(p)p Fq(\000)p Fp(1)3454 4637 y Fi(.)43 b(Mor)-5 b(e-)118 4757 y(over,)34 b(this)h Fs(\027)42 b Fi(is)34 b(a)h(physic)-5 b(al)34 b(me)-5 b(asur)g(e)34 b(and)h(supp)16 b Ft(\()p Fs(\027)6 b Ft(\))28 b(=)p 2232 4677 V 28 w Fs(U)2308 4772 y Fp(0)2370 4757 y Fo([)22 b(\001)17 b(\001)g(\001)k([)p 2685 4677 V 22 w Fs(U)2762 4772 y Fr(p)p Fq(\000)p Fp(1)2892 4757 y Fi(.)1900 5251 y Ft(17)p eop %%Page: 18 18 18 17 bop 118 548 a Fi(Pr)-5 b(o)g(of.)49 b Ft(Let)36 b(us)g(assume)h Fs(n)1129 563 y Fp(0)1202 548 y Ft(=)d(1)h(for)h (simplicit)m(y)d(\(see)38 b([Ar1)o(,)f(Section)f(7])g(for)g(the)g (general)g(case\))118 668 y(and)49 b(let)e(us)i(consider)g(a)f (stationary)f(absolutely)h(con)m(tin)m(uous)h(probabilit)m(y)d(measure) j Fs(\027)55 b Ft(with)118 789 y(supp)18 b(\()p Fs(\027)6 b Ft(\))28 b Fo(\032)p 599 709 77 4 v 28 w Fs(U)676 804 y Fp(0)735 789 y Fo([)20 b(\001)d(\001)g(\001)h([)p 1043 709 V 20 w Fs(U)1120 804 y Fr(p)p Fq(\000)p Fp(1)1250 789 y Ft(.)43 b(W)-8 b(e)32 b(\014rst)g(sho)m(w)g(the)g(ergo)s(dicit)m (y)f(of)f Fs(\027)6 b Ft(,)33 b(in)d(the)i(sense)h(that)f Fs(\022)3556 753 y Fe(N)3553 813 y Fr(\017)3628 789 y Fo(\002)20 b Fs(\027)118 909 y Ft(is)43 b Fs(F)14 b Ft(-ergo)s(dic.)74 b(It)43 b(turns)h(out)g(that)f(to)g(b)s(e)g Fs(F)14 b Ft(-ergo)s(dic)42 b(it)g(su\016ces)j(that)f(either)f Fs(\027)6 b Ft(\()p Fs(G)p Ft(\))46 b(=)g(0)d(or)118 1029 y Fs(\027)6 b Ft(\()p Fs(G)p Ft(\))28 b(=)g(1)k(for)g(ev)m(ery)i (Borel)e(set)h Fs(G)28 b Fo(\032)g Fs(M)44 b Ft(satisfying)1364 1298 y(1)1413 1313 y Fr(G)1472 1298 y Ft(\()p Fs(x)p Ft(\))28 b(=)1735 1162 y Fh(Z)1851 1298 y Ft(1)1900 1313 y Fr(G)1975 1298 y Ft(\()p Fs(f)2061 1313 y Fr(t)2091 1298 y Ft(\()p Fs(x)p Ft(\)\))34 b Fs(d\022)2390 1313 y Fr(\017)2422 1298 y Ft(\()p Fs(t)p Ft(\))1073 b(\(18\))118 1566 y(for)37 b Fs(\027)44 b Ft(almost)37 b(ev)m(ery)i Fs(x)f Ft(\(cf.)g([Ar1])g(and)g([Vi2)o(]\).)59 b(So)37 b(let)h(us)g(tak)m(e)g Fs(G)g Ft(suc)m(h)h(that)f Fs(\027)6 b Ft(\()p Fs(G)p Ft(\))37 b Fs(>)f Ft(0)h(and)118 1687 y Fs(G)d Ft(satis\014es)h(the)g(left)e(hand)h(side)g(of)g(\(18\).)47 b(Then)36 b(it)d(m)m(ust)h(b)s(e)g Fs(m)p Ft(\()p Fs(G)p Ft(\))d Fs(>)f Ft(0)k(b)s(ecause)h Fs(\027)i Fo(\034)30 b Fs(m)k Ft(and)118 1807 y(there)40 b(is)g(a)f(closed)h(set)g Fs(J)49 b Fo(\032)40 b Fs(G)g Ft(suc)m(h)h(that)e Fs(m)p Ft(\()p Fs(G)27 b Fo(n)g Fs(J)9 b Ft(\))40 b(=)f(0)h(and)g(also)e Fs(\027)6 b Ft(\()p Fs(G)28 b Fo(n)e Fs(J)9 b Ft(\))40 b(=)g(0.)64 b(Hence)118 1927 y Fs(J)45 b Ft(also)35 b(satis\014es)i (the)f(left)g(hand)g(side)g(of)f(\(18\))h(b)s(ecause)h(of)f (nondegeneracy)i(condition)c(2)i(\(with)118 2048 y Fs(n)176 2063 y Fp(0)243 2048 y Ft(=)28 b(1\),)k(since)1256 2081 y Fh(Z)1372 2217 y Ft(1)1421 2232 y Fr(E)1481 2217 y Ft(\()p Fs(f)1567 2232 y Fr(t)1596 2217 y Ft(\()p Fs(x)p Ft(\)\))17 b Fs(d\022)1878 2232 y Fr(\017)1911 2217 y Ft(\()p Fs(t)p Ft(\))28 b(=)f(\()p Fs(f)2239 2232 y Fr(x)2283 2217 y Ft(\))2321 2232 y Fq(\003)2360 2217 y Fs(\022)2408 2176 y Fe(N)2405 2241 y Fr(\017)2461 2217 y Ft(\()p Fs(E)6 b Ft(\))p Fs(:)118 2439 y Ft(This)48 b(means)g(that)g(when)h Fs(x)54 b Fo(2)g Fs(J)j Ft(w)m(e)49 b(ha)m(v)m(e)g Fs(f)1954 2454 y Fr(t)1984 2439 y Ft(\()p Fs(x)p Ft(\))54 b Fo(2)g Fs(J)j Ft(for)48 b Fs(\022)2610 2454 y Fr(\017)2691 2439 y Ft(almost)e(all)g Fs(t)54 b Fo(2)g Ft(supp)17 b(\()p Fs(\022)3681 2454 y Fr(\017)3714 2439 y Ft(\).)118 2560 y(Since)40 b(a)f(set)h(of)g Fs(\022)791 2575 y Fr(\017)863 2560 y Ft(measure)g(1)f(is)h(dense)h(in)d(supp)18 b(\()p Fs(\022)2138 2575 y Fr(\017)2171 2560 y Ft(\))40 b(\(w)m(e)g(are)g (supp)s(osing)f Fs(\022)3112 2575 y Fr(\017)3185 2560 y Ft(to)g(b)s(e)h(p)s(ositiv)m(e)118 2680 y(on)49 b(op)s(en)f(sets\))i (and)e Fs(f)1019 2695 y Fr(t)1049 2680 y Ft(\()p Fs(x)p Ft(\))h(v)-5 b(aries)48 b(con)m(tin)m(uously)h(with)f Fs(t)p Ft(,)53 b(w)m(e)c(see)h(that)e Fs(f)3063 2695 y Fr(t)3093 2680 y Ft(\()p Fs(x)p Ft(\))55 b Fo(2)g Fs(J)j Ft(for)48 b(all)118 2801 y Fs(t)30 b Fo(2)f Ft(supp)18 b(\()p Fs(\022)579 2816 y Fr(\017)612 2801 y Ft(\))33 b(b)s(ecause)i Fs(J)43 b Ft(is)33 b(closed.)46 b(W)-8 b(e)34 b(then)g(ha)m(v)m(e)h(that)e(the)h(in)m(terior)e(of)h Fs(J)42 b Ft(is)33 b(nonempt)m(y)i(b)m(y)118 2921 y(condition)27 b(1)i(on)f(random)g(p)s(erturbations)g(and)h(w)m(e)h(ma)m(y)e(apply)h (the)g(metho)s(ds)f(of)g(decomp)s(osition)118 3041 y(in)m(to)39 b(connected)i(comp)s(onen)m(ts)g(as)f(b)s(efore)f(\(Lemma)g(3.2\).)64 b(In)40 b(this)g(manner)f(w)m(e)i(construct)g(an)118 3162 y(in)m(v)-5 b(arian)m(t)34 b(domain)g(inside)i Fs(J)44 b Ft(whic)m(h,)38 b(in)d(turn,)h(is)g(inside)f(a)g(minimal)d(in)m(v)-5 b(arian)m(t)34 b(domain.)51 b(This)118 3282 y(con)m(tradicts)32 b(minimalit)m(y)26 b(and)32 b(so)f(w)m(e)h(conclude)g(that)f Fs(J)40 b Ft(m)m(ust)31 b(con)m(tain)p 2846 3202 V 31 w Fs(U)2923 3297 y Fp(0)2981 3282 y Fo([)20 b(\001)d(\001)g(\001)h([)p 3288 3202 V 19 w Fs(U)3365 3297 y Fr(p)p Fq(\000)p Fp(1)3495 3282 y Ft(.)43 b(Th)m(us)118 3403 y(w)m(e)34 b(ha)m(v)m(e)g Fs(\027)6 b Ft(\()p Fs(G)p Ft(\))28 b(=)f Fs(\027)6 b Ft(\()p Fs(J)j Ft(\))28 b(=)g(1)k(pro)m(ving)g Fs(\022)1628 3418 y Fr(\017)1684 3403 y Fo(\002)22 b Fs(\027)39 b Ft(to)33 b(b)s(e)f Fs(F)14 b Ft(-ergo)s(dic.)264 3523 y(No)m(w,)38 b(giv)m(en)e Fs(')d Ft(:)g Fs(M)45 b Fo(!)32 b Fg(R)47 b Ft(con)m(tin)m(uous)37 b(w)m(e)g(consider)f(the)g(map)f Fs( )j Ft(=)33 b Fs(')24 b Fo(\016)g Fs(\031)40 b Ft(from)35 b Fs(T)3497 3487 y Fe(N)3573 3523 y Fo(\002)25 b Fs(M)118 3643 y Ft(to)36 b Fg(R)5 b Ft(,)43 b(where)37 b Fs(\031)g Ft(:)d Fs(T)886 3607 y Fe(N)962 3643 y Fo(\002)25 b Fs(M)44 b Fo(!)33 b Fs(M)47 b Ft(is)36 b(the)g(natural)f(pro)5 b(jection.)53 b(The)37 b(Ergo)s(dic)f(Theorem)g(then)118 3764 y(ensures)1145 3949 y(lim)1093 4009 y Fr(n)p Fq(!)p Fp(+)p Fq(1)1363 3882 y Ft(1)p 1359 3926 59 4 v 1359 4018 a Fs(n)1449 3825 y Fr(n)p Fq(\000)p Fp(1)1443 3855 y Fh(X)1454 4065 y Fr(j)t Fp(=0)1604 3949 y Fs( )t Ft(\()p Fs(F)1786 3908 y Fr(j)1822 3949 y Ft(\()p Fs(t)p 1860 3965 36 4 v(;)17 b(x)p Ft(\)\))28 b(=)2201 3814 y Fh(Z)2318 3949 y Fs( )20 b(d)p Ft(\()p Fs(\022)2538 3908 y Fe(N)2535 3974 y Fr(\017)2612 3949 y Fo(\002)j Fs(\027)6 b Ft(\))118 4232 y(for)32 b Fs(\022)315 4195 y Fe(N)312 4256 y Fr(\017)390 4232 y Fo(\002)22 b Fs(\027)39 b Ft(almost)31 b(all)g(\()p Fs(t)p 1065 4248 V(;)17 b(x)p Ft(\),)32 b(whic)m(h)i(is)e(just)h(the)g (same)f(as)1345 4542 y(lim)1294 4602 y Fr(n)p Fq(!)p Fp(+)p Fq(1)1564 4474 y Ft(1)p 1559 4519 59 4 v 1559 4610 a Fs(n)1649 4417 y Fr(n)p Fq(\000)p Fp(1)1644 4447 y Fh(X)1655 4657 y Fr(j)t Fp(=0)1804 4542 y Fs(')p Ft(\()p Fs(f)1965 4495 y Fr(j)1954 4564 y(t)p 1954 4576 30 3 v 2002 4542 a Ft(\()p Fs(x)p Ft(\)\))27 b(=)2302 4406 y Fh(Z)2418 4542 y Fs(')17 b(d\027)1008 b Ft(\(19\))118 4870 y(for)33 b Fs(\022)316 4834 y Fe(N)313 4894 y Fr(\017)390 4870 y Fo(\002)23 b Fs(\027)40 b Ft(almost)31 b(all)g(\()p Fs(t)p 1067 4886 36 4 v(;)17 b(x)p Ft(\).)45 b(Finally)31 b(considering)h(the)i(ergo)s(dic)e(basin)h Fs(B)5 b Ft(\()p Fs(\027)h Ft(\),)33 b(de\014ned)i(as)e(the)1900 5251 y(18)p eop %%Page: 19 19 19 18 bop 118 548 a Ft(set)33 b(of)f(p)s(oin)m(ts)h Fs(x)28 b Fo(2)g Fs(M)43 b Ft(for)32 b(whic)m(h)1345 853 y(lim)1294 912 y Fr(n)p Fq(!)p Fp(+)p Fq(1)1564 785 y Ft(1)p 1559 830 59 4 v 1559 921 a Fs(n)1649 728 y Fr(n)p Fq(\000)p Fp(1)1644 758 y Fh(X)1655 968 y Fr(j)t Fp(=0)1804 853 y Fs(')p Ft(\()p Fs(f)1965 805 y Fr(j)1954 875 y(t)p 1954 887 30 3 v 2002 853 a Ft(\()p Fs(x)p Ft(\)\))27 b(=)2302 717 y Fh(Z)2418 853 y Fs(')17 b(d\027)118 1181 y Ft(for)25 b(all)e Fs(')k Fo(2)h Fs(C)650 1144 y Fp(0)690 1181 y Ft(\()p Fs(M)5 b(;)17 b Fg(R)5 b Ft(\))31 b(and)25 b Fs(\022)1236 1144 y Fe(N)1233 1205 y Fr(\017)1313 1181 y Ft(almost)f(ev)m(ery)j Fs(t)p 1871 1197 36 4 v 27 w Fo(2)i Fs(T)2099 1144 y Fe(N)2150 1181 y Ft(,)e(it)d(is)h(easy)h(to)f (see)h(that)f Fs(B)5 b Ft(\()p Fs(\027)h Ft(\))25 b(satis\014es)h(the) 118 1301 y(left)21 b(hand)h(side)g(of)f(\(18\))h(in)f(the)h(place)g(of) f Fs(G)h Ft(and)g(w)m(e)h(m)m(ust)e(ha)m(v)m(e)j(as)e(b)s(efore)g Fs(B)5 b Ft(\()p Fs(\027)h Ft(\))28 b Fo(\033)p 3180 1221 77 4 v 28 w Fs(U)3257 1316 y Fp(0)3296 1301 y Fo([\001)17 b(\001)g(\001)o([)p 3545 1221 V Fs(U)3623 1316 y Fr(p)p Fq(\000)p Fp(1)3752 1301 y Ft(.)264 1421 y(This)34 b(sho)m(ws)g(that)f (if)f(another)h(stationary)g(absolutely)f(con)m(tin)m(uous)i (probabilit)m(y)d(measure)39 b(~)-55 b Fs(\027)118 1542 y Ft(is)27 b(suc)m(h)h(that)f(supp)18 b(\()6 b(~)-55 b Fs(\027)6 b Ft(\))28 b Fo(\032)p 1112 1462 V 28 w Fs(U)1188 1557 y Fp(0)1239 1542 y Fo([)11 b(\001)17 b(\001)g(\001)9 b([)p 1520 1462 V 11 w Fs(U)1597 1557 y Fr(p)p Fq(\000)p Fp(1)1726 1542 y Ft(,)29 b(then)e(the)h(basins)f(of)f Fs(\027)34 b Ft(and)e(~)-54 b Fs(\027)33 b Ft(m)m(ust)27 b(ha)m(v)m(e)i(nonempt)m(y)118 1662 y(in)m(tersection.)56 b(Th)m(us)39 b(these)f(measures)f(m)m(ust)g(b)s(e)g(equal.)57 b(Moreo)m(v)m(er)38 b Fs(\027)2798 1581 y Fh(\000)2844 1662 y Fs(B)5 b Ft(\()p Fs(\027)h Ft(\))3053 1581 y Fh(\001)3134 1662 y Ft(=)35 b(1)h(and)h(so,)i(b)m(y)118 1782 y(absolute)32 b(con)m(tin)m(uit)m(y)-8 b(,)33 b Fs(m)1068 1702 y Fh(\000)1114 1782 y Fs(B)5 b Ft(\()p Fs(\027)h Ft(\))1323 1702 y Fh(\001)1397 1782 y Fs(>)27 b Ft(0)33 b(and)f(th)m(us)i Fs(\027)39 b Ft(is)32 b(a)g(ph)m(ysical)h(probabilit)m(y)-8 b(.)p 3709 1782 4 66 v 3713 1720 59 4 v 3713 1782 V 3771 1782 4 66 v 118 2115 a Fu(4)161 b(The)54 b(n)l(um)l(b)t(er)d(of)j(ph)l (ysical)g(measures)118 2334 y Ft(In)31 b(this)f(section)g(w)m(e)h(will) d(pro)m(v)m(e)k(that)e(the)h(n)m(um)m(b)s(er)f Fs(l)j Ft(of)d(ph)m(ysical)g(measures)h(is)f(b)s(ounded)h(b)m(y)g(the)118 2455 y(n)m(um)m(b)s(er)i Fs(p)g Ft(of)g(SRB)g(measures.)45 b(Moreo)m(v)m(er)34 b(w)m(e)g(will)d(presen)m(t)k(examples)d(of)h (dynamical)e(systems)118 2575 y(for)h(whic)m(h)h Fs(l)d Ft(=)e Fs(p)k Ft(and)h Fs(l)d(<)d(p)p Ft(.)264 2695 y(Let)c Fs(\026)488 2710 y Fp(1)528 2695 y Fs(;)17 b(:)g(:)g(:)32 b(;)17 b(\026)822 2710 y Fr(l)870 2695 y Ft(b)s(e)23 b(the)g(ph)m(ysical)g(measures)h(supp)s(orted)g(on)e(the)i(minimal)18 b(in)m(v)-5 b(arian)m(t)22 b(domains)118 2816 y(in)28 b Fs(M)10 b Ft(,)30 b(whic)m(h)g(exist)f(b)m(y)g(Lemmas)f(3.2)g(and)h (3.4)f(through)h(3.6.)42 b(If)29 b Fs(\026)f Ft(is)g(an)h(absolutely)f (con)m(tin)m(uous)118 2936 y(stationary)f(measure,)i(its)e (restrictions)g(to)g(the)h(minimal)c(in)m(v)-5 b(arian)m(t)26 b(domains)g(of)h Fs(M)10 b Ft(,)29 b(normalized)118 3057 y(when)46 b(not)e(equal)h(to)f(the)h(constan)m(t)h(zero)f(measure,)j (are)d(absolutely)f(con)m(tin)m(uous)h(stationary)118 3177 y(measures)g(b)m(y)h(Lemma)c(3.3.)79 b(After)44 b(Lemma)f(3.6)h(these)i(restrictions)e(m)m(ust)g(b)s(e)h(the)g(ph)m (ysical)118 3297 y(measures)39 b Fs(\026)601 3312 y Fp(1)640 3297 y Fs(;)17 b(:)g(:)g(:)33 b(;)17 b(\026)935 3312 y Fr(l)998 3297 y Ft(of)38 b(the)h(minimal)34 b(domains.)59 b(Hence)40 b Fs(\026)e Ft(m)m(ust)g(decomp)s(ose)h(in)m(to)e(a)h (linear)118 3418 y(com)m(bination)22 b(of)h(ph)m(ysical)h(measures.)41 b(Moreo)m(v)m(er,)27 b(the)e(union)e(of)g(supp)18 b(\()p Fs(\026)2873 3433 y Fp(1)2912 3418 y Ft(\))p Fs(;)f(:)g(:)g(:)f(;)h Ft(supp)g(\()p Fs(\026)3483 3433 y Fr(l)3509 3418 y Ft(\))23 b(m)m(ust)118 3538 y(con)m(tain)32 b(supp)18 b(\()p Fs(\026)p Ft(\),)32 b(except)i(p)s(ossibly)e(for)g(a)h Fs(\026)f Ft(n)m(ull)f(set.)44 b(In)33 b(fact,)g(if)e(the)i(follo)m(wing)d(set)j (function)1052 3758 y Fs(\026)22 b Fo(\000)h Fs(\026)1292 3677 y Fh(\000)1337 3758 y Ft(supp)18 b(\()p Fs(\026)1652 3773 y Fp(1)1691 3758 y Ft(\))1729 3677 y Fh(\001)1774 3758 y Fs(\026)1833 3773 y Fp(1)1895 3758 y Fo(\000)k(\001)17 b(\001)g(\001)k(\000)h Fs(\026)2291 3677 y Fh(\000)2337 3758 y Ft(supp)17 b(\()p Fs(\026)2651 3773 y Fr(l)2677 3758 y Ft(\))2715 3677 y Fh(\001)2761 3758 y Fs(\026)2820 3773 y Fr(l)118 3978 y Ft(w)m(ere)36 b(nonzero,)f(then)g(its)f (normalization)c Fs(\026)1777 3942 y Fq(0)1835 3978 y Ft(w)m(ould)k(b)s(e)g(an)g(absolutely)g(con)m(tin)m(uous)h(stationary) 118 4099 y(measure,)48 b(and)d(the)h(ab)s(o)m(v)m(e)f(decomp)s(osition) e(could)h(b)s(e)h(applied)f(to)h Fs(\026)2828 4062 y Fq(0)2850 4099 y Ft(,)j(th)m(us)e(giving)d(another)118 4219 y(minimal)30 b(domain)i(inside)h(supp)18 b(\()p Fs(\026)p Ft(\).)47 b(Clearly)33 b(this)g(cannot)i(happ)s(en.)48 b(W)-8 b(e)34 b(then)g(ha)m(v)m(e)i(a)d(con)m(v)m(ex)118 4339 y(linear)e(decomp)s(osition)1489 4559 y Fs(\026)d Ft(=)f Fs(\013)1741 4574 y Fp(1)1781 4559 y Fs(\026)1840 4574 y Fp(1)1901 4559 y Ft(+)22 b Fo(\001)17 b(\001)g(\001)j Ft(+)i Fs(\013)2297 4574 y Fr(l)2324 4559 y Fs(\026)2383 4574 y Fr(l)3606 4559 y Ft(\(20\))118 4779 y(where)29 b Fs(\013)457 4794 y Fr(i)513 4779 y Ft(=)f Fs(\026)p Ft(\(supp)17 b(\()p Fs(\026)1028 4794 y Fr(i)1056 4779 y Ft(\)\))27 b Fo(\025)h Ft(0)g(and)g Fs(\013)1588 4794 y Fp(1)1640 4779 y Ft(+)12 b Fo(\001)17 b(\001)g(\001)11 b Ft(+)h Fs(\013)2007 4794 y Fr(l)2060 4779 y Ft(=)28 b(1.)42 b(W)-8 b(e)28 b(will)d(see)k(that)f(this)f(decomp)s(osition)118 4900 y(is)32 b(uniquely)h(de\014ned.)1900 5251 y(19)p eop %%Page: 20 20 20 19 bop 264 548 a Ft(W)-8 b(e)34 b(remark)f(that)g(so)g(far)g(w)m(e)h (did)f(not)g(use)h(more)f(than)g(the)h(con)m(tin)m(uit)m(y)f(of)g(the)h (map)e Fs(f)11 b Ft(.)45 b(F)-8 b(or)118 668 y(the)41 b(next)h(result)f(w)m(e)h(assume)f(that)g Fs(f)53 b Ft(:)42 b Fs(M)52 b Fo(!)42 b Fs(M)51 b Ft(is)41 b(a)g Fs(C)2396 632 y Fp(2)2476 668 y Ft(non-uniformly)d(expanding)j(map)118 789 y(whose)f(orbits)f(ha)m(v)m(e)h(slo)m(w)f(appro)m(ximation)e(to)i (the)g(critical)e Fo(C)45 b Ft(\(p)s(ossibly)38 b(the)i(empt)m(yset\))g (with)118 909 y Fs(m)p Ft(\()p Fo(C)6 b Ft(\))33 b(=)f(0.)51 b(This)36 b(result)f(con)m(tains)g(the)h(assertions)g(of)e(the)i (\014rst)g(t)m(w)m(o)g(items)e(of)h(Theorem)g(A)h(\(if)118 1029 y(w)m(e)e(think)e(of)g Fo(C)i Ft(=)28 b Fo(;)p Ft(\))k(and)h (Theorem)g(C.)118 1229 y Fc(Prop)s(osition)j(4.1.)49 b Fi(If)24 b Fs(\017)k(>)g Ft(0)c Fi(is)h(smal)5 b(l)24 b(enough,)i(then)f(ther)-5 b(e)25 b(exist)g(physic)-5 b(al)24 b(me)-5 b(asur)g(es)24 b Fs(\026)3459 1193 y Fr(\017)3459 1253 y Fp(1)3498 1229 y Fs(;)17 b(:)g(:)g(:)f(;)h(\026) 3776 1193 y Fr(\017)3776 1255 y(l)118 1349 y Fi(\(with)35 b Fs(l)i Fi(not)e(dep)-5 b(ending)33 b(on)h Fs(\017)p Fi(\))i(such)e(that)234 1548 y(1.)48 b(for)35 b Fs(x)28 b Fo(2)g Fs(M)46 b Fi(ther)-5 b(e)34 b(is)h(a)g Fs(\022)1315 1512 y Fe(N)1312 1573 y Fr(\017)1402 1548 y Fi(mo)-5 b(d)34 b Ft(0)h Fi(p)-5 b(artition)34 b Fs(T)2151 1563 y Fp(1)2191 1548 y Ft(\()p Fs(x)p Ft(\))p Fs(;)17 b(:)g(:)g(:)f(;)h(T) 2598 1563 y Fr(l)2624 1548 y Ft(\()p Fs(x)p Ft(\))35 b Fi(of)g Fs(T)2976 1512 y Fe(N)3062 1548 y Fi(such)g(that)921 1854 y Fs(\026)980 1813 y Fr(\017)980 1878 y(i)1040 1854 y Ft(=)28 b Fs(w)1217 1818 y Fq(\003)1256 1854 y Fi(-)40 b Ft(lim)1307 1914 y Fr(n)p Fq(!1)1539 1786 y Ft(1)p 1534 1831 59 4 v 1534 1922 a Fs(n)1624 1729 y Fr(n)p Fq(\000)p Fp(1)1619 1759 y Fh(X)1630 1969 y Fr(j)t Fp(=1)1779 1854 y Fs(\016)1822 1885 y Fr(f)1863 1852 y Fn(j)1856 1905 y(t)p 1856 1915 28 3 v 1896 1885 a Fp(\()p Fr(x)p Fp(\))2130 1854 y Fi(if)34 b(and)h(only)f(if)135 b Fs(t)p 2820 1870 36 4 v 28 w Fo(2)28 b Fs(T)3034 1869 y Fr(i)3062 1854 y Ft(\()p Fs(x)p Ft(\);)234 2205 y Fi(2.)48 b(for)35 b(e)-5 b(ach)34 b Fs(i)28 b Ft(=)f(1)p Fs(;)17 b(:)g(:)g(:)f(;)h(l)37 b Fi(we)d(have)1081 2505 y Fs(\026)1140 2464 y Fr(\017)1140 2530 y(i)1200 2505 y Ft(=)27 b Fs(w)1376 2469 y Fq(\003)1415 2505 y Fi(-)41 b Ft(lim)1466 2565 y Fr(n)p Fq(!1)1698 2438 y Ft(1)p 1694 2482 59 4 v 1694 2573 a Fs(n)1784 2380 y Fr(n)p Fq(\000)p Fp(1)1778 2410 y Fh(X)1789 2620 y Fr(j)t Fp(=0)1939 2369 y Fh(Z)2039 2505 y Ft(\()p Fs(f)2136 2458 y Fr(j)2125 2527 y(t)p 2125 2539 30 3 v 2172 2505 a Ft(\))2210 2520 y Fq(\003)2249 2424 y Fh(\000)2295 2505 y Fs(m)28 b Fo(j)f Fs(B)5 b Ft(\()p Fs(\026)2639 2464 y Fr(\017)2639 2530 y(i)2672 2505 y Ft(\))2710 2424 y Fh(\001)2772 2505 y Fs(d\022)2871 2464 y Fe(N)2868 2530 y Fr(\017)2923 2505 y Ft(\()p Fs(t)p 2961 2521 36 4 v Ft(\))p Fs(;)362 2821 y Fi(wher)-5 b(e)34 b Fs(m)28 b Fo(j)g Fs(B)5 b Ft(\()p Fs(\026)982 2785 y Fr(\017)982 2846 y(i)1014 2821 y Ft(\))35 b Fi(is)f(the)h(normalize)-5 b(d)34 b(r)-5 b(estriction)34 b(of)h(L)-5 b(eb)g(esgue)34 b(me)-5 b(asur)g(e)35 b(to)g Fs(B)5 b Ft(\()p Fs(\026)3506 2785 y Fr(\017)3506 2846 y(i)3538 2821 y Ft(\))p Fi(.)118 3020 y(Pr)-5 b(o)g(of.)49 b Ft(T)-8 b(ak)m(e)32 b Fs(x)d Fo(2)f Fs(M)42 b Ft(and)32 b(let)f Fs(\026)h Ft(b)s(e)g(a)f(w)m(eak) 1814 2984 y Fq(\003)1887 3020 y Ft(accum)m(ulation)f(p)s(oin)m(t)h(of)g (the)h(sequence)j(\()p Fs(\026)3517 3035 y Fr(n)3563 3020 y Ft(\()p Fs(x)p Ft(\)\))3732 3035 y Fr(n)118 3141 y Ft(de\014ned)46 b(in)d(\(17\).)78 b(W)-8 b(e)44 b(will)e(pro)m(v)m(e) k(that)e(this)g(is)f(the)i(only)f(accum)m(ulation)e(p)s(oin)m(t)h(of)h (\(17\))g(b)m(y)118 3261 y(sho)m(wing)33 b(that)g(the)g(v)-5 b(alues)33 b(of)f(the)i Fs(\013)1503 3276 y Fp(1)1542 3261 y Fs(;)17 b(:)g(:)g(:)f(;)h(\013)1823 3276 y Fr(l)1882 3261 y Ft(in)32 b(decomp)s(osition)f(\(20\))h(dep)s(end)i(only)e(on)h Fs(x)g Ft(and)118 3381 y(not)44 b(on)g(the)h(subsequence)j(that)c(con)m (v)m(erges)i(to)e Fs(\026)p Ft(.)78 b(The)46 b(de\014nition)d(of)h(the) g(a)m(v)m(erage)i(in)d(\(17\))118 3502 y(implies)h(that)j(there)g(is)f (a)g(subset)j(of)d(parameter)g(v)m(ectors)i Fs(t)p 2423 3518 V 52 w Fo(2)k Ft(supp)17 b(\()p Fs(\022)2931 3466 y Fe(N)2928 3526 y Fr(\017)2984 3502 y Ft(\))46 b(with)g(p)s(ositiv)m (e)g Fs(\022)3727 3466 y Fe(N)3724 3526 y Fr(\017)118 3622 y Ft(measure)29 b(for)f(whic)m(h)h(there)g(is)f Fs(j)34 b Fo(\025)28 b Ft(1)g(suc)m(h)i(that)e Fs(f)1991 3575 y Fr(j)1980 3645 y(t)p 1980 3657 30 3 v 2027 3622 a Ft(\()p Fs(x)p Ft(\))g Fo(2)h Ft(supp)17 b(\()p Fs(\026)2595 3637 y Fr(i)2623 3622 y Ft(\).)42 b(W)-8 b(e)29 b(de\014ne)h(for)e Fs(i)f Ft(=)h(1)p Fs(;)17 b(:)g(:)g(:)f(;)h(l)622 3854 y(T)679 3869 y Fr(i)707 3854 y Ft(\()p Fs(x)p Ft(\))28 b(=)970 3773 y Fh(\010)1028 3854 y Fs(t)p 1028 3870 36 4 v 28 w Fo(2)g Ft(supp)18 b(\()p Fs(\022)1489 3813 y Fe(N)1486 3878 y Fr(\017)1541 3854 y Ft(\))28 b(:)f Fs(f)1720 3807 y Fr(j)1709 3876 y(t)p 1709 3888 30 3 v 1757 3854 a Ft(\()p Fs(x)p Ft(\))h Fo(2)g Ft(supp)17 b(\()p Fs(\026)2324 3869 y Fr(i)2352 3854 y Ft(\))98 b(for)32 b(some)97 b Fs(j)34 b Fo(\025)28 b Ft(1)3174 3773 y Fh(\011)3248 3854 y Fs(:)118 4069 y Ft(W)-8 b(e)33 b(clearly)f(ha)m(v)m(e)413 4284 y Fs(T)470 4299 y Fr(i)498 4284 y Ft(\()p Fs(x)p Ft(\))c(=)761 4210 y Fh(S)844 4313 y Fr(j)t Fq(\025)p Fp(1)970 4284 y Fs(T)1041 4237 y Fr(j)1027 4310 y(i)1078 4284 y Ft(\()p Fs(x)p Ft(\))98 b(where)g Fs(T)1724 4237 y Fr(j)1710 4310 y(i)1761 4284 y Ft(\()p Fs(x)p Ft(\))28 b(=)f Fo(f)p Fs(t)p 2073 4300 36 4 v 28 w Fo(2)h Ft(supp)18 b(\()p Fs(\022)2534 4243 y Fe(N)2531 4309 y Fr(\017)2586 4284 y Ft(\))27 b(:)h Fs(f)2765 4237 y Fr(j)2754 4307 y(t)p 2754 4319 30 3 v 2801 4284 a Ft(\()p Fs(x)p Ft(\))g Fo(2)h Ft(supp)17 b(\()p Fs(\026)3369 4299 y Fr(i)3397 4284 y Ft(\))p Fo(g)118 4521 y Ft(and)42 b Fs(T)388 4473 y Fr(j)374 4546 y(i)424 4521 y Ft(\()p Fs(x)p Ft(\))h Fo(\032)g Fs(T)789 4473 y Fr(j)t Fp(+1)775 4546 y Fr(i)916 4521 y Ft(\()p Fs(x)p Ft(\))e(for)g(all)f Fs(i;)17 b(j)49 b Fo(\025)43 b Ft(1,)g(since)f(the)g(supp)s(orts)g(of)f(stationary)g (measures)h(are)118 4641 y(themselv)m(es)34 b(in)m(v)-5 b(arian)m(t.)42 b(In)33 b(addition,)e(since)i Fs(\026)f Ft(is)g(a)h(regular)e(\(Borel\))h(probabilit)m(y)e(measure,)j(w)m(e)118 4761 y(ma)m(y)27 b(\014nd)h(for)e(eac)m(h)i Fs(\021)j(>)d Ft(0)f(an)g(op)s(en)g(set)h Fs(U)37 b Ft(and)27 b(a)g(closed)g(set)h Fs(K)34 b Ft(suc)m(h)29 b(that)e Fs(K)34 b Fo(\032)29 b Ft(supp)17 b(\()p Fs(\026)3504 4776 y Fr(i)3532 4761 y Ft(\))28 b Fo(\032)g Fs(U)118 4882 y Ft(with)37 b Fs(\026)p Ft(\()p Fs(U)f Fo(n)25 b Fs(K)7 b Ft(\))35 b Fs(<)h(\021)41 b Ft(and)c Fs(\026)p Ft(\()p Fs(@)5 b(U)10 b Ft(\))37 b(=)e Fs(\026)p Ft(\()p Fs(@)5 b(K)i Ft(\))36 b(=)g(0.)57 b(In)38 b(fact,)g(there)g(is)f(an)g(at)g(most)f(coun)m(table)118 5002 y(n)m(um)m(b)s(er)44 b(of)e Fs(\016)t Ft(-neigh)m(b)s(orho)s(o)s (ds)g(of)h(supp)17 b(\()p Fs(\026)1782 5017 y Fr(i)1810 5002 y Ft(\))43 b(whose)h(b)s(oundaries)f(ha)m(v)m(e)i(p)s(ositiv)m(e)d Fs(\026)h Ft(measure,)1900 5251 y(20)p eop %%Page: 21 21 21 20 bop 118 548 a Ft(and)36 b(lik)m(ewise)f(for)g(the)h(compacts)g (coinciding)e(with)h(the)h(complemen)m(t)f(of)g(the)h Fs(\016)t Ft(-neigh)m(b)s(orho)s(o)s(d)118 668 y(of)c Fs(M)h Fo(n)22 b Ft(supp)c(\()p Fs(\026)743 683 y Fr(i)771 668 y Ft(\).)43 b(Then,)34 b(taking)e Fs(\013)1524 683 y Fr(i)1579 668 y Ft(=)c Fs(\026)p Ft(\(supp)17 b(\()p Fs(\026)2094 683 y Fr(i)2122 668 y Ft(\)\))32 b(w)m(e)i(ha)m(v)m(e)742 972 y Fs(\013)804 987 y Fr(i)855 972 y Ft(+)22 b Fs(\021)31 b Fo(\025)d Fs(\026)p Ft(\()p Fs(U)10 b Ft(\))84 b(=)133 b(lim)1592 1035 y Fr(k)r Fq(!)p Fp(+)p Fq(1)1879 905 y Ft(1)p 1853 949 101 4 v 1853 1041 a Fs(n)1911 1056 y Fr(k)1981 845 y(n)2024 857 y Fn(k)2061 845 y Fq(\000)p Fp(1)1994 878 y Fh(X)2005 1088 y Fr(j)t Fp(=0)2168 972 y Fs(\022)2216 931 y Fe(N)2213 997 y Fr(\017)2268 972 y Fo(f)p Fs(t)p 2318 988 36 4 v 28 w Fo(2)28 b Fs(T)2546 931 y Fe(N)2626 972 y Ft(:)g Fs(f)2740 925 y Fr(j)2729 995 y(t)p 2729 1007 30 3 v 2776 972 a Ft(\()p Fs(x)p Ft(\))g Fo(2)g Fs(U)10 b Fo(g)1431 1324 y(\025)84 b Ft(lim)17 b(sup)1624 1406 y Fr(k)r Fq(!)p Fp(+)p Fq(1)1944 1256 y Ft(1)p 1917 1301 101 4 v 1917 1392 a Fs(n)1975 1407 y Fr(k)2045 1196 y(n)2088 1208 y Fn(k)2126 1196 y Fq(\000)p Fp(1)2059 1229 y Fh(X)2069 1439 y Fr(j)t Fp(=0)2233 1324 y Fs(\022)2281 1283 y Fe(N)2278 1348 y Fr(\017)2333 1243 y Fh(\000)2379 1324 y Fs(T)2450 1277 y Fr(j)2436 1349 y(i)2486 1324 y Ft(\()p Fs(x)p Ft(\))2617 1243 y Fh(\001)118 1630 y Ft(for)32 b(some)h(sequence)i(of)d(in)m(tegers)h Fs(n)1446 1645 y Fp(1)1513 1630 y Fs(<)28 b(n)1675 1645 y Fp(2)1742 1630 y Fs(<)g(n)1904 1645 y Fp(3)1971 1630 y Fs(<)g Fo(\001)17 b(\001)g(\001)d Ft(,)33 b(and)g(lik)m(ewise)f(for) 728 1929 y Fs(\013)790 1944 y Fr(i)840 1929 y Fo(\000)23 b Fs(\021)31 b Fo(\024)d Fs(\026)p Ft(\()p Fs(K)7 b Ft(\))84 b(=)133 b(lim)1593 1991 y Fr(k)r Fq(!)p Fp(+)p Fq(1)1880 1861 y Ft(1)p 1854 1906 V 1854 1997 a Fs(n)1912 2012 y Fr(k)1981 1801 y(n)2024 1813 y Fn(k)2062 1801 y Fq(\000)p Fp(1)1995 1834 y Fh(X)2006 2044 y Fr(j)t Fp(=0)2169 1929 y Fs(\022)2217 1887 y Fe(N)2214 1953 y Fr(\017)2269 1929 y Fo(f)p Fs(t)p 2319 1944 36 4 v 28 w Fo(2)28 b Fs(T)2547 1887 y Fe(N)2627 1929 y Ft(:)f Fs(f)2740 1881 y Fr(j)2729 1951 y(t)p 2729 1963 30 3 v 2777 1929 a Ft(\()p Fs(x)p Ft(\))h Fo(2)g Fs(K)7 b Fo(g)1432 2280 y(\024)84 b Ft(lim)17 b(inf)1611 2343 y Fr(k)r Fq(!)p Fp(+)p Fq(1)1916 2213 y Ft(1)p 1889 2257 101 4 v 1889 2348 a Fs(n)1947 2363 y Fr(k)2017 2153 y(n)2060 2165 y Fn(k)2098 2153 y Fq(\000)p Fp(1)2031 2185 y Fh(X)2041 2395 y Fr(j)t Fp(=0)2205 2280 y Fs(\022)2253 2239 y Fe(N)2250 2305 y Fr(\017)2305 2199 y Fh(\000)2351 2280 y Fs(T)2422 2233 y Fr(j)2408 2306 y(i)2458 2280 y Ft(\()p Fs(x)p Ft(\))2589 2199 y Fh(\001)2635 2280 y Fs(;)118 2586 y Ft(where)34 b Fs(\021)d(>)d Ft(0)k(is)g (arbitrary)-8 b(.)43 b(This)32 b(sho)m(ws)1026 2885 y Fs(\013)1088 2900 y Fr(i)1144 2885 y Ft(=)27 b Fs(\026)p Ft(\(supp)17 b(\()p Fs(\026)1658 2900 y Fr(i)1686 2885 y Ft(\)\))28 b(=)49 b(lim)1893 2947 y Fr(k)r Fq(!1)2126 2817 y Ft(1)p 2100 2862 V 2100 2953 a Fs(n)2158 2968 y Fr(k)2227 2757 y(n)2270 2769 y Fn(k)2308 2757 y Fq(\000)p Fp(1)2241 2790 y Fh(X)2251 3000 y Fr(j)t Fp(=0)2415 2885 y Fs(\022)2463 2844 y Fe(N)2460 2909 y Fr(\017)2515 2804 y Fh(\000)2561 2885 y Fs(T)2632 2838 y Fr(j)2618 2910 y(i)2668 2885 y Ft(\()p Fs(x)p Ft(\))2799 2804 y Fh(\001)2845 2885 y Fs(:)118 3191 y Ft(W)-8 b(e)33 b(also)f(ha)m(v)m(e)750 3467 y Fs(\022)798 3426 y Fe(N)795 3492 y Fr(\017)850 3386 y Fh(\000)896 3467 y Fs(T)953 3482 y Fr(i)981 3467 y Ft(\()p Fs(x)p Ft(\))1112 3386 y Fh(\001)1186 3467 y Ft(=)46 b(lim)1289 3528 y Fr(j)t Fq(!1)1479 3467 y Fs(\022)1527 3426 y Fe(N)1524 3492 y Fr(\017)1580 3386 y Fh(\000)1625 3467 y Fs(T)1696 3420 y Fr(j)1682 3493 y(i)1732 3467 y Ft(\()p Fs(x)p Ft(\))1863 3386 y Fh(\001)1937 3467 y Ft(=)52 b(lim)2041 3527 y Fr(n)p Fq(!1)2256 3400 y Ft(1)p 2251 3444 59 4 v 2251 3536 a Fs(n)2341 3343 y Fr(n)p Fq(\000)p Fp(1)2336 3373 y Fh(X)2347 3582 y Fr(j)t Fp(=0)2496 3467 y Fs(\022)2544 3426 y Fe(N)2541 3492 y Fr(\017)2597 3386 y Fh(\000)2642 3467 y Fs(T)2713 3420 y Fr(j)2699 3493 y(i)2750 3467 y Ft(\()p Fs(x)p Ft(\))2881 3386 y Fh(\001)2954 3467 y Ft(=)28 b Fs(\013)3120 3482 y Fr(i)118 3773 y Ft(whic)m(h)23 b(sho)m(ws)h(that)e(the)h Fs(\013)1078 3788 y Fr(i)1128 3773 y Ft(dep)s(end)h(only)e(on)g(the)h (random)e(orbits)h(of)g Fs(x)h Ft(and)f(not)g(on)g(the)h(particular)118 3894 y(sequence)42 b(\()p Fs(n)625 3909 y Fr(k)667 3894 y Ft(\))705 3909 y Fr(k)748 3894 y Ft(.)62 b(Th)m(us)41 b(w)m(e)e(see)h(that)f(the)g(sequence)j(of)c(measures)i(in)e(\(17\))g (con)m(v)m(erges)j(in)d(the)118 4014 y(w)m(eak)328 3978 y Fq(\003)405 4014 y Ft(top)s(ology)-8 b(.)52 b(Moreo)m(v)m(er)37 b(the)g(sets)g Fs(T)1700 4029 y Fp(1)1739 4014 y Ft(\()p Fs(x)p Ft(\))p Fs(;)17 b(:)g(:)g(:)g(;)g(T)2147 4029 y Fr(l)2172 4014 y Ft(\()p Fs(x)p Ft(\))37 b(are)f(pairwise)f(disjoin)m (t)g(b)m(y)h(de\014nition)118 4135 y(and)i(their)g(total)e Fs(\022)838 4098 y Fe(N)835 4159 y Fr(\017)928 4135 y Ft(measure)j(equals)f Fs(\013)1677 4150 y Fp(1)1742 4135 y Ft(+)26 b Fo(\001)17 b(\001)g(\001)24 b Ft(+)i Fs(\013)2150 4150 y Fr(l)2213 4135 y Ft(=)36 b(1,)j(th)m(us)g(forming)d(a)i Fs(\022)3161 4098 y Fe(N)3158 4159 y Fr(\017)3251 4135 y Ft(mo)s(dulo)e(zero)118 4255 y(partition)28 b(of)h Fs(T)701 4219 y Fe(N)753 4255 y Ft(.)42 b(W)-8 b(e)31 b(observ)m(e)g(that)f(if)e Fs(t)p 1627 4271 36 4 v 28 w Fo(2)g Fs(T)1841 4270 y Fr(i)1870 4255 y Ft(\()p Fs(x)p Ft(\),)i(then)h Fs(f)2337 4219 y Fr(n)2326 4280 y(t)p 2326 4292 30 3 v 2383 4255 a Ft(\()p Fs(x)p Ft(\))d Fo(2)h Ft(supp)17 b(\()p Fs(\026)2951 4270 y Fr(i)2979 4255 y Ft(\))28 b Fo(\032)g Fs(B)5 b Ft(\()p Fs(\026)3326 4270 y Fr(i)3354 4255 y Ft(\))29 b(for)h(some)118 4385 y Fs(n)36 b Fo(\025)g Ft(1)g(and)i Fs(i)d Ft(=)g(1)p Fs(;)17 b(:)g(:)g(:)f(;)h(l)r Ft(.)57 b(This)37 b(means)g(this)g Fs(\022)1940 4349 y Fe(N)1937 4409 y Fr(\017)2029 4385 y Ft(mo)s(dulo)e(zero)j(partition)d(of)h Fs(T)3191 4349 y Fe(N)3280 4385 y Ft(satis\014es)i(the)118 4505 y(\014rst)33 b(item)e(of)h(the)h(prop)s(osition.)264 4626 y(No)m(w)h(\014xing)e Fs(i)c Ft(=)f(1)p Fs(;)17 b(:)g(:)g(:)f(;)h(l)r Ft(,)33 b(for)f(all)e Fs(x)e Fo(2)g Fs(B)5 b Ft(\()p Fs(\026)1915 4641 y Fr(i)1943 4626 y Ft(\))33 b(\(the)g(ergo)s(dic)e(basin)i(of)f Fs(\026)2984 4641 y Fr(i)3011 4626 y Ft(\))h(it)e(holds)i(that)1329 4927 y(lim)1278 4986 y Fr(n)p Fq(!)p Fp(+)p Fq(1)1548 4859 y Ft(1)p 1543 4904 59 4 v 1543 4995 a Fs(n)1633 4802 y Fr(n)p Fq(\000)p Fp(1)1628 4832 y Fh(X)1643 5042 y Fr(i)p Fp(=0)1788 4927 y Fs(')p Ft(\()p Fs(f)1949 4879 y Fr(j)1938 4949 y(t)p 1938 4961 30 3 v 1985 4927 a Ft(\()p Fs(x)p Ft(\)\))28 b(=)2286 4791 y Fh(Z)2402 4927 y Fs(')17 b(d\026)2593 4942 y Fr(i)1900 5251 y Ft(21)p eop %%Page: 22 22 22 21 bop 118 548 a Ft(for)39 b Fs(\022)322 512 y Fe(N)319 573 y Fr(\017)414 548 y Ft(almost)f(ev)m(ery)j Fs(t)p 1000 564 36 4 v 40 w Fo(2)f Fs(T)1252 512 y Fe(N)1304 548 y Ft(.)65 b(Recall)38 b(that)h Fs(m)p Ft(\()p Fs(B)5 b Ft(\()p Fs(\026)2214 563 y Fr(i)2242 548 y Ft(\)\))40 b Fs(>)f Ft(0)g(b)m(y)i(the)f(de\014nition)e(of)i(ph)m(ysical)118 668 y(measure.)h(Using)23 b(dominated)f(con)m(v)m(ergence)k(and)d(in)m (tegrating)f(b)s(oth)h(sides)g(of)g(the)h(ab)s(o)m(v)m(e)g(equalit)m(y) 118 789 y(t)m(wice,)36 b(\014rst)f(with)f(resp)s(ect)i(to)f(the)g(Leb)s (esgue)h(measure)f Fs(m)p Ft(,)g(and)g(then)g(with)g(resp)s(ect)h(to)e Fs(\022)3554 753 y Fe(N)3551 813 y Fr(\017)3606 789 y Ft(,)h(w)m(e)118 909 y(arriv)m(e)e(at)f(the)h(statemen)m(t)g(of)f(item) f(2.)264 1029 y(Recall)38 b(that)i(up)g(un)m(til)e(no)m(w)j(the)f (noise)f(lev)m(el)g Fs(\017)i(>)e Ft(0)h(w)m(as)g(k)m(ept)h(\014xed.)66 b(F)-8 b(or)39 b(small)e(enough)118 1150 y Fs(\017)28 b(>)g Ft(0)j(the)g(measures)h Fs(\026)1011 1165 y Fr(i)1067 1150 y Ft(=)27 b Fs(\026)1229 1114 y Fr(\017)1229 1174 y(i)1293 1150 y Ft(dep)s(end)32 b(on)f(the)h(noise)f(lev)m(el,)g(but)g (w)m(e)h(will)d(see)k(that)d(the)i(n)m(um)m(b)s(er)118 1270 y(of)g(ph)m(ysical)h(measures)g(is)f(constan)m(t.)264 1391 y(Fixing)40 b Fs(i)k Fo(2)g(f)p Ft(1)p Fs(;)17 b(:)g(:)g(:)e(;)i (l)r Fo(g)42 b Ft(w)m(e)g(let)f Fs(x)h Ft(in)f(the)i(in)m(terior)d(of)h (supp)18 b(\()p Fs(\026)2693 1354 y Fr(\017)2693 1415 y(i)2725 1391 y Ft(\))42 b(b)s(e)g(suc)m(h)h(that)e(the)i(orbit)118 1511 y(\()p Fs(f)215 1475 y Fr(j)251 1511 y Ft(\()p Fs(x)p Ft(\)\))420 1526 y Fr(j)498 1511 y Ft(has)d(in\014nitely)g(man)m(y)g(h) m(yp)s(erb)s(olic)g(times.)67 b(Recall)39 b(that)h Fs(f)52 b Fo(\021)42 b Fs(f)2966 1526 y Fr(t)2991 1507 y Ff(\003)3072 1511 y Ft(is)e(non-uniformly)118 1631 y(expanding)c(\(p)s(ossibly)f (with)g(criticalities\).)50 b(Then)36 b(there)h(is)e(a)g(big)g(enough)h (h)m(yp)s(erb)s(olic)f(time)f Fs(n)118 1752 y Ft(so)40 b(that)g Fs(V)521 1767 y Fr(n)568 1752 y Ft(\()p Fs(t)p 606 1768 V -36 x Fq(\003)681 1752 y Fs(;)17 b(x)p Ft(\))40 b Fo(\032)h Ft(supp)18 b(\()p Fs(\026)1291 1716 y Fr(\017)1291 1776 y(i)1323 1752 y Ft(\),)42 b(b)m(y)f(Prop)s(osition)e(2.6,)i(where) h(w)m(e)f(tak)m(e)g Fs(t)p 2959 1768 V -36 x Fq(\003)3074 1752 y Ft(=)f(\()p Fs(t)3263 1716 y Fq(\003)3303 1752 y Fs(;)17 b(t)3382 1716 y Fq(\003)3421 1752 y Fs(;)g(t)3500 1716 y Fq(\003)3540 1752 y Fs(;)g(:)g(:)g(:)e Ft(\).)118 1872 y(Since)37 b Fs(t)412 1836 y Fq(\003)486 1872 y Fo(2)f Ft(supp)17 b(\()p Fs(\022)888 1887 y Fr(\017)921 1872 y Ft(\))37 b(and)g(supp)17 b(\()p Fs(\026)1504 1836 y Fr(\017)1504 1897 y(i)1537 1872 y Ft(\))36 b(is)h(in)m(v)-5 b(arian)m(t)35 b(under)j Fs(f)2456 1887 y Fr(t)2522 1872 y Ft(for)e(all)f Fs(t)g Fo(2)g Ft(supp)18 b(\()p Fs(\022)3287 1887 y Fr(\017)3320 1872 y Ft(\),)38 b(w)m(e)f(m)m(ust)118 1993 y(ha)m(v)m(e)1061 2113 y Fs(f)1120 2072 y Fr(n)1109 2138 y(t)p 1109 2150 26 3 v -21 x Ff(\003)1174 2032 y Fh(\000)1220 2113 y Fs(V)1277 2128 y Fr(n)1324 2113 y Ft(\()p Fs(t)p 1362 2129 36 4 v -41 x Fq(\003)1436 2113 y Fs(;)17 b(x)p Ft(\))1573 2032 y Fh(\001)1647 2113 y Ft(=)27 b Fs(B)1829 2032 y Fh(\000)1875 2113 y Fs(f)1934 2072 y Fr(n)1923 2138 y(t)1948 2119 y Ff(\003)1989 2113 y Ft(\()p Fs(x)p Ft(\))p Fs(;)17 b(\016)2207 2128 y Fp(1)2247 2032 y Fh(\001)2320 2113 y Fo(\032)28 b Ft(supp)18 b(\()p Fs(\026)2740 2072 y Fr(\017)2740 2138 y(i)2772 2113 y Ft(\))p Fs(;)118 2299 y Ft(where)37 b Fs(\016)446 2314 y Fp(1)519 2299 y Fs(>)c Ft(0)i(is)g(the)h(constan)m(t)h(giv)m(en)f(b)m (y)g(Prop)s(osition)e(2.6)i(and)f Fs(B)2737 2219 y Fh(\000)2783 2299 y Fs(f)2842 2263 y Fr(n)2831 2324 y(t)2856 2305 y Ff(\003)2897 2299 y Ft(\()p Fs(x)p Ft(\))p Fs(;)17 b(\016)3115 2314 y Fp(1)3155 2219 y Fh(\001)3236 2299 y Ft(is)35 b(the)h(ball)e(of)118 2420 y(radius)e Fs(\016)454 2435 y Fp(1)526 2420 y Ft(around)h Fs(f)916 2384 y Fr(n)905 2444 y(t)930 2426 y Ff(\003)971 2420 y Ft(\()p Fs(x)p Ft(\).)264 2540 y(On)38 b(the)g(one)f(hand,)i(w)m(e)f(deduce)h(that)e (the)h(n)m(um)m(b)s(er)g Fs(l)g Ft(=)e Fs(l)r Ft(\()p Fs(\017)p Ft(\))h(is)g(b)s(ounded)h(from)e(ab)s(o)m(v)m(e)j(b)m(y)118 2661 y(some)28 b(uniform)f(constan)m(t)i Fs(N)38 b Ft(since)29 b Fs(M)39 b Ft(is)28 b(compact.)41 b(On)29 b(the)f(other)h(hand,)g (since)g(eac)m(h)g(in)m(v)-5 b(arian)m(t)118 2781 y(set)39 b(m)m(ust)f(con)m(tain)f(some)h(ph)m(ysical)f(measure)i(\(b)m(y)f (Lemma)f(3.4\),)i(w)m(e)f(see)h(that)f(for)f(0)g Fs(<)f(\017)3567 2745 y Fq(0)3628 2781 y Fs(<)g(\017)118 2901 y Ft(there)31 b(m)m(ust)f(b)s(e)h(some)f(ph)m(ysical)g(measure)h Fs(\026)1786 2865 y Fr(\017)1815 2842 y Ff(0)1870 2901 y Ft(with)f(supp)18 b(\()p Fs(\026)2405 2865 y Fr(\017)2434 2842 y Ff(0)2460 2901 y Ft(\))27 b Fo(\032)h Ft(supp)18 b(\()p Fs(\026)2945 2865 y Fr(\017)2977 2901 y Ft(\).)43 b(In)31 b(fact)f(supp)17 b(\()p Fs(\026)3709 2865 y Fr(\017)3742 2901 y Ft(\))118 3022 y(is)34 b(in)m(v)-5 b(arian)m(t)34 b(under)h Fs(f)956 3037 y Fr(t)1021 3022 y Ft(for)f(ev)m(ery)j Fs(t)32 b Fo(2)f Ft(supp)18 b(\()p Fs(\022)1897 3037 y Fr(\017)1926 3018 y Ff(0)1953 3022 y Ft(\))31 b Fo(\032)h Ft(supp)18 b(\()p Fs(\022)2432 3037 y Fr(\017)2465 3022 y Ft(\).)50 b(This)35 b(means)g(the)g(n)m(um)m(b)s(er)g Fs(l)r Ft(\()p Fs(\017)p Ft(\))118 3142 y(of)29 b(ph)m(ysical)h(measures)g(is)f(a)g (nonincreasing)g(function)g(of)g Fs(\017)f(>)g Ft(0.)42 b(Th)m(us)31 b(w)m(e)f(conclude)g(that)g(there)118 3262 y(m)m(ust)37 b(b)s(e)f Fs(\017)539 3277 y Fp(0)614 3262 y Fs(>)e Ft(0)i(suc)m(h)i(that)e Fs(l)h Ft(=)d Fs(l)r Ft(\()p Fs(\017)p Ft(\))j(is)f(constan)m(t)i(for)d(0)g Fs(<)f(\017)h(<)f(\017)2676 3277 y Fp(0)2716 3262 y Ft(,)j(ending)f (the)h(pro)s(of)f(of)g(the)118 3383 y(prop)s(osition.)p 3709 3383 4 66 v 3713 3320 59 4 v 3713 3383 V 3771 3383 4 66 v 118 3586 a Fc(Remark)h(4.2.)49 b Fi(Observe)31 b(that)h(if)f(the)g(map)g Fs(f)39 b Ft(:)27 b Fs(M)39 b Fo(!)27 b Fs(M)42 b Fi(is)32 b(tr)-5 b(ansitive,)32 b(then)f(every)g(stationary)118 3707 y(me)-5 b(asur)g(e)36 b(must)h(b)-5 b(e)37 b(supp)-5 b(orte)g(d)37 b(on)f(the)h(whole)f(of)g Fs(M)10 b Fi(,)38 b(sinc)-5 b(e)36 b(the)h(supp)-5 b(ort)37 b(is)g(invariant)f(and)g(has)118 3827 y(nonempty)45 b(interior.)74 b(A)-5 b(c)g(c)g(or)g(ding)45 b(to)g(the)g(discussion)f(ab)-5 b(ove,)47 b(ther)-5 b(e)45 b(must)g(b)-5 b(e)44 b(only)h(one)g(such)118 3947 y(stationary)35 b(me)-5 b(asur)g(e,)34 b(which)g(must)h(b)-5 b(e)35 b(physic)-5 b(al.)264 4151 y Ft(W)d(e)34 b(note)f(that)g(the)h (n)m(um)m(b)s(er)f Fs(l)j Ft(of)c(ph)m(ysical)h(measures)h(for)f(small) e Fs(\017)e(>)f Ft(0)33 b(and)g(the)h(n)m(um)m(b)s(er)g Fs(p)118 4271 y Ft(of)c(SRB)g(measures)h(for)f Fs(f)41 b Ft(are)30 b(obtained)g(b)m(y)h(di\013eren)m(t)f(existen)m(tial)g (argumen)m(ts.)43 b(It)30 b(is)g(natural)f(to)118 4392 y(ask)k(if)f(there)h(is)f(an)m(y)h(relation)e(b)s(et)m(w)m(een)k Fs(l)g Ft(and)d Fs(p)p Ft(.)118 4595 y Fc(Prop)s(osition)k(4.3.)49 b Fi(If)c Fs(p)h Fo(\025)i Ft(1)d Fi(is)g(the)g(numb)-5 b(er)45 b(of)g(SRB)g(me)-5 b(asur)g(es)45 b(of)g Fs(f)56 b Fi(and)45 b Fs(l)k Fo(\025)e Ft(1)e Fi(is)h(the)118 4715 y(numb)-5 b(er)43 b(of)h(physic)-5 b(al)42 b(me)-5 b(asur)g(es)43 b(of)h(the)f(r)-5 b(andom)43 b(p)-5 b(erturb)g(ation)44 b(of)f Fs(f)11 b Fi(,)45 b(then)f(for)f Fs(\017)h(>)g Ft(0)f Fi(smal)5 b(l)118 4836 y(enough)34 b(we)h(have)f Fs(l)c Fo(\024)e Fs(p)p Fi(.)1900 5251 y Ft(22)p eop %%Page: 23 23 23 22 bop 118 548 a Fi(Pr)-5 b(o)g(of.)49 b Ft(W)-8 b(e)35 b(start)g(b)m(y)i(observing)e(that)g(if)f Fs(p)e Ft(=)g(1)j(then)h(ev)m (ery)h(w)m(eak)2682 512 y Fq(\003)2758 548 y Ft(accum)m(ulation)d(p)s (oin)m(t)g(of)h(a)118 668 y(family)e(\()p Fs(\026)516 632 y Fr(\017)516 693 y(i)549 668 y Ft(\))587 683 y Fr(\017>)p Fp(0)745 668 y Ft(of)i(ph)m(ysical)h(measures)g(when)h Fs(\017)c Fo(!)g Ft(0)i(m)m(ust)h(equal)f(the)h(unique)g(SRB)g(measure) 118 789 y Fs(\026)177 804 y Fp(1)257 789 y Ft(for)k Fs(f)11 b Ft(.)68 b(Hence)42 b(the)g(w)m(eak)1253 753 y Fq(\003)1334 789 y Ft(limit)37 b(of)k(\()p Fs(\026)1792 753 y Fr(\017)1792 813 y(i)1824 789 y Ft(\))1862 804 y Fr(\017>)p Fp(0)2026 789 y Ft(when)h Fs(\017)g Fo(!)f Ft(0)g(exists)g(and)g(equals)g Fs(\026)3439 804 y Fp(1)3519 789 y Ft(for)g(all)118 909 y Fs(i)h Ft(=)g(1)p 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/Times-Roman-iso ff 150.00 scf sf 584 3358 m gs 1 -1 sc (-6) dup sw pop neg 0 rm col-1 sh gr /Times-Roman-iso ff 150.00 scf sf 584 3002 m gs 1 -1 sc (-5) dup sw pop neg 0 rm col-1 sh gr /Times-Roman-iso ff 150.00 scf sf 584 2646 m gs 1 -1 sc (-4) dup sw pop neg 0 rm col-1 sh gr /Times-Roman-iso ff 150.00 scf sf 584 2290 m gs 1 -1 sc (-3) dup sw pop neg 0 rm col-1 sh gr /Times-Roman-iso ff 150.00 scf sf 584 1934 m gs 1 -1 sc (-2) dup sw pop neg 0 rm col-1 sh gr /Times-Roman-iso ff 150.00 scf sf 584 1578 m gs 1 -1 sc (-1) dup sw pop neg 0 rm col-1 sh gr /Times-Roman-iso ff 150.00 scf sf 584 1222 m gs 1 -1 sc (0) dup sw pop neg 0 rm col-1 sh gr /Times-Roman-iso ff 150.00 scf sf 584 866 m gs 1 -1 sc (1) dup sw pop neg 0 rm col-1 sh gr /Times-Roman-iso ff 150.00 scf sf 584 510 m gs 1 -1 sc (2) dup sw pop neg 0 rm col-1 sh gr /Times-Roman-iso ff 150.00 scf sf 658 3838 m gs 1 -1 sc (-7) dup sw pop 2 div neg 0 rm col-1 sh gr /Times-Roman-iso ff 150.00 scf sf 1014 3838 m gs 1 -1 sc (-6) dup sw pop 2 div neg 0 rm col-1 sh gr /Times-Roman-iso ff 150.00 scf sf 1370 3838 m gs 1 -1 sc (-5) dup sw pop 2 div neg 0 rm col-1 sh gr /Times-Roman-iso ff 150.00 scf sf 1726 3838 m gs 1 -1 sc (-4) dup sw pop 2 div neg 0 rm col-1 sh gr /Times-Roman-iso ff 150.00 scf sf 2082 3838 m gs 1 -1 sc (-3) dup sw pop 2 div neg 0 rm col-1 sh gr /Times-Roman-iso ff 150.00 scf sf 2438 3838 m gs 1 -1 sc (-2) dup sw pop 2 div neg 0 rm col-1 sh gr /Times-Roman-iso ff 150.00 scf sf 2794 3838 m gs 1 -1 sc (-1) dup sw pop 2 div neg 0 rm col-1 sh gr /Times-Roman-iso ff 150.00 scf sf 3150 3838 m gs 1 -1 sc (0) dup sw pop 2 div neg 0 rm col-1 sh gr /Times-Roman-iso ff 150.00 scf sf 3506 3838 m gs 1 -1 sc (1) dup sw pop 2 div neg 0 rm col-1 sh gr /Times-Roman-iso ff 150.00 scf sf 3862 3838 m gs 1 -1 sc (2) dup sw pop 2 div neg 0 rm col-1 sh gr $F2psEnd rs showpage %%EndDocument @endspecial 1217 2505 a Ft(Figure)32 b(2:)43 b(map)32 b(for)g(whic)m(h)h Fs(l)d Ft(=)d Fs(p)h Ft(=)g(2)118 2898 y(ev)m(en)m(tually)43 b(arriv)m(es)g(at)f(one)h(of)e(these)j(in)m (terv)-5 b(als.)72 b(Then)44 b(b)m(y)f([BC1)q(])f(and)h([BY])f(the)h (map)f Fs(f)53 b Ft(is)118 3019 y(non-uniformly)30 b(expanding)j(and)g (has)g(t)m(w)m(o)g(SRB)g(measures)g(with)f(supp)s(orts)i(con)m(tained)f (in)f(eac)m(h)118 3139 y(trapping)j(region.)53 b(Finally)33 b Fs(f)47 b Ft(admits)35 b(t)m(w)m(o)h(distinct)g(ph)m(ysical)g (measures)g(whose)i(supp)s(orts)e(are)118 3260 y(con)m(tained)d(in)f Fs(I)717 3275 y Fp(1)789 3260 y Ft(and)g Fs(I)1021 3275 y Fp(2)1093 3260 y Ft(resp)s(ectiv)m(ely)-8 b(,)34 b(for)e Fs(\017)1832 3275 y Fp(0)1899 3260 y Fs(>)c Ft(0)k(small)e(enough,)k (see)f([BaV].)118 3592 y Fu(5)161 b(Sto)t(c)l(hastic)53 b(stabilit)l(y)118 3811 y Ft(In)32 b(this)g(section)g(w)m(e)g(will)e (pro)m(v)m(e)j(the)f(\014rst)g(item)f(of)g(Theorem)h(B)g(and)g(Theorem) g(D.)42 b(The)33 b(second)118 3932 y(item)f(of)i(Theorem)g(B)f(ma)m(y)h (b)s(e)g(obtained)f(in)g(the)h(same)f(w)m(a)m(y)i(as)f(Theorem)g(D,)f (if)g(w)m(e)i(think)e(of)g 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Ft(for)g(whic)m(h)947 4844 y Fs(\026)1006 4803 y Fr(\017)1006 4868 y(i)1066 4844 y Ft(=)c Fs(w)1243 4808 y Fq(\003)1281 4844 y Ft(-)41 b(lim)1331 4904 y Fr(n)p Fq(!1)1562 4776 y Ft(1)p 1558 4821 59 4 v 1558 4912 a Fs(n)1648 4719 y Fr(n)p Fq(\000)p Fp(1)1643 4749 y Fh(X)1653 4959 y Fr(j)t Fp(=1)1803 4844 y Fs(\016)1846 4875 y Fr(f)1887 4842 y Fn(j)1880 4895 y(t)p 1880 4905 28 3 v 1920 4875 a Fp(\()p Fr(x)p Fp(\))2116 4844 y Ft(for)32 b(eac)m(h)99 b Fs(t)p 2550 4860 36 4 v 28 w Fo(2)28 b Fs(T)2764 4859 y Fr(i)2792 4844 y Ft(\()p Fs(x)p Ft(\))p Fs(:)1900 5251 y Ft(25)p eop %%Page: 26 26 26 25 bop 118 548 a Ft(F)-8 b(urthermore,)47 b(since)e(w)m(e)h(are)f (taking)f Fs(f)55 b Ft(a)44 b(lo)s(cal)f(di\013eomorphism,)i(then)g (log)17 b Fo(k)p Ft(\()p Fs(D)s(f)11 b Ft(\))3433 512 y Fq(\000)p Fp(1)3526 548 y Fo(k)45 b Ft(is)f(a)118 668 y(con)m(tin)m(uous)33 b(map.)43 b(Th)m(us,)34 b(w)m(e)g(ha)m(v)m(e)g (for)e(eac)m(h)h Fs(x)28 b Fo(2)h Fs(M)43 b Ft(and)33 b Fs(\022)2425 632 y Fe(N)2422 693 y Fr(\017)2509 668 y Ft(almost)e(ev)m(ery)k Fs(t)p 3082 684 36 4 v 28 w Fo(2)28 b Fs(T)3310 632 y Fe(N)875 978 y Ft(lim)850 1038 y Fr(n)p Fq(!1)1066 911 y Ft(1)p 1061 955 59 4 v 1061 1047 a Fs(n)1151 854 y Fr(n)p Fq(\000)p Fp(1)1146 884 y Fh(X)1156 1093 y Fr(j)t Fp(=0)1306 978 y Ft(log)17 b Fo(k)p Fs(D)s(f)1642 897 y Fh(\000)1687 978 y Fs(f)1746 931 y Fr(j)1735 1001 y(t)p 1735 1013 30 3 v 1782 978 a Ft(\()p Fs(x)p Ft(\))1913 897 y Fh(\001)1959 920 y Fq(\000)p Fp(1)2053 978 y Fo(k)28 b Ft(=)2234 843 y Fh(Z)2350 978 y Ft(log)17 b Fo(k)p Ft(\()p Fs(D)s(f)11 b Ft(\))2762 937 y Fq(\000)p Fp(1)2855 978 y Fo(k)p Fs(d\026)3015 937 y Fr(\017)3015 1003 y(i)118 1293 y Ft(for)43 b(some)f(ph)m(ysical)h (measure)h Fs(\026)1367 1257 y Fr(\017)1367 1318 y(i)1442 1293 y Ft(with)e(1)k Fo(\024)g Fs(i)f Fo(\024)h Fs(l)r Ft(.)75 b(Hence,)47 b(for)42 b(pro)m(ving)h(the)h(non-uniform)118 1414 y(expansion)39 b(of)e Fs(f)49 b Ft(on)38 b(random)f(orbits)g(it)g (su\016ces)j(to)e(sho)m(w)h(that)f(there)g(is)g 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Fp(+)2303 3028 y Fi(\))c(lie)g(in)g Fs(K)7 b Fi(.)85 b(Then)48 b(for)g(every)g(op)-5 b(en)118 3148 y(neighb)g(orho)g(o)g(d)36 b Fs(U)48 b Fi(of)38 b Fs(K)44 b Fi(ther)-5 b(e)38 b(is)f Fs(t)1447 3163 y Fp(0)1520 3148 y Fs(>)32 b Ft(0)38 b Fi(such)f(that)h Fs(x)2196 3163 y Fr(t)2259 3148 y Fo(2)33 b Fs(U)48 b Fi(for)38 b(every)f Ft(0)c Fs(<)f(t)h(<)g(t)3293 3163 y Fp(0)3333 3148 y Fi(.)105 b Ft(Indeed,)118 3269 y(supp)s(osing)29 b(not,)h(there)f(is)g(a)f(sequence)k(\()p Fs(t)1657 3284 y Fr(n)1704 3269 y Ft(\))1742 3284 y Fr(n)1818 3269 y Ft(with)c Fs(t)2071 3284 y Fr(n)2146 3269 y Fo(!)f Ft(0)2322 3233 y Fp(+)2410 3269 y Ft(when)j Fs(n)e Fo(!)f(1)h Ft(suc)m(h)j(that)d Fs(x)3481 3284 y Fr(t)3506 3292 y Fn(n)3582 3269 y Fo(62)g Fs(U)10 b Ft(.)118 3389 y(Since)41 b Fs(X)48 b Ft(is)39 b(compact)h(this)g(means)h(that)f(\()p Fs(x)1830 3404 y Fr(t)1860 3389 y Ft(\))1898 3404 y Fr(t>)p Fp(0)2058 3389 y Ft(has)h(some)f(accum)m(ulation)e(p)s(oin)m(t)i(in)f Fs(X)d Fo(n)27 b Fs(U)10 b Ft(,)118 3510 y(th)m(us)34 b(outside)e Fs(K)7 b Ft(,)33 b(con)m(trary)g(to)f(the)h(assumption.)264 3630 y(No)m(w,)40 b(the)f(space)g Fs(X)44 b Ft(=)36 b Fg(P)p Ft(\()p Fs(M)10 b Ft(\))41 b(of)c(all)f(probabilit)m(y)g (measures)i(in)g Fs(M)48 b Ft(is)38 b(a)f(compact)h(metric)118 3750 y(space)46 b(with)e(the)h(w)m(eak)1015 3714 y Fq(\003)1100 3750 y Ft(top)s(ology)-8 b(,)46 b(and)f(the)g(con)m(v)m(ex)i(h)m(ull)c Fs(K)52 b Ft(of)44 b(the)h(\(\014nitely)f(man)m(y\))g(SRB)118 3871 y(measures)f(of)e Fs(f)52 b Ft(is)41 b(closed.)71 b(Hence,)46 b(considering)41 b(the)h(curv)m(e)h(\()p Fs(\026)2617 3886 y Fr(\017)2649 3871 y Ft(\))2687 3886 y Fr(\017)2761 3871 y Ft(in)e Fg(P)p Ft(\()p Fs(M)10 b Ft(\),)47 b(w)m(e)42 b(are)g(in)f(the)118 3991 y(con)m(text)g(of)f (the)g(ab)s(o)m(v)m(e)h(result,)h(since)e(w)m(e)h(are)f(supp)s(osing)g Fs(f)51 b Ft(to)39 b(b)s(e)h(sto)s(c)m(hastically)f(stable.)65 b(A)118 4112 y(metric)32 b(on)g Fs(X)40 b Ft(top)s(ologically)29 b(equiv)-5 b(alen)m(t)32 b(to)g(the)h(w)m(eak)2213 4075 y Fq(\003)2287 4112 y Ft(top)s(ology)e(ma)m(y)h(b)s(e)h(giv)m(en)g(b)m (y)1126 4402 y(d)1180 4417 y Fe(P)1225 4402 y Ft(\()p Fs(\026;)17 b(\027)6 b Ft(\))27 b(=)1625 4277 y Fq(1)1589 4307 y Fh(X)1596 4519 y Fr(k)r Fp(=1)1783 4335 y Ft(1)p 1759 4379 96 4 v 1759 4470 a(2)1808 4441 y Fr(n)1882 4257 y Fh(\014)1882 4317 y(\014)1882 4377 y(\014)1882 4437 y(\014)1915 4266 y(Z)2031 4402 y Fs(')2095 4417 y Fr(n)2159 4402 y Fs(d\026)21 b Fo(\000)2390 4266 y Fh(Z)2506 4402 y Fs(')2570 4417 y Fr(n)2633 4402 y Fs(d\027)2738 4257 y Fh(\014)2738 4317 y(\014)2738 4377 y(\014)2738 4437 y(\014)118 4716 y Ft(where)29 b Fs(\026;)17 b(\027)34 b Fo(2)28 b Fg(P)p Ft(\()p Fs(M)10 b Ft(\))30 b(and)f(\()p Fs(')1231 4731 y Fr(n)1277 4716 y Ft(\))1315 4731 y Fr(n)p Fq(\025)p Fp(1)1480 4716 y Ft(is)f(a)g(dense)h(sequence)i(of)c (functions)h(in)g Fs(C)3021 4680 y Fp(0)3060 4716 y Ft(\()p Fs(M)5 b(;)17 b Fg(R)5 b Ft(\),)35 b(see)29 b([Ma].)264 4837 y(Let)d Fs(')i Ft(:)g Fs(M)38 b Fo(!)28 b Fg(R)36 b Ft(con)m(tin)m(uous)27 b(b)s(e)f(giv)m(en)g(and)g(let)f(us)h(\014x)h (some)e Fs(\016)32 b(>)c Ft(0.)41 b(There)27 b(m)m(ust)f(b)s(e)g Fs(n)h Fo(2)i Fg(N)118 4957 y Ft(suc)m(h)36 b(that)e Fo(k)p Fs(')23 b Fo(\000)h Fs(')855 4972 y Fr(n)902 4957 y Fo(k)952 4972 y Fp(0)1021 4957 y Fs(<)31 b(\016)t(=)p Ft(3)j(and,)g(b)m(y)i(the)e(auxiliary)e(result)j(in)e(the)i(b)s (eginning)e(of)h(the)g(pro)s(of,)1900 5251 y(26)p eop %%Page: 27 27 27 26 bop 118 548 a Ft(there)37 b(exists,)g(for)e(some)h Fs(\017)1110 563 y Fp(0)1183 548 y Fs(>)d Ft(0)i(and)h(ev)m(ery)i(0)33 b Fs(<)g(\017)g(<)g(\017)2241 563 y Fp(0)2281 548 y Ft(,)k(a)e (probabilit)m(y)f(measure)i Fs(\026)3373 563 y Fr(\017)3438 548 y Fo(2)e Fg(P)p Ft(\()p Fs(M)10 b Ft(\))118 668 y(for)32 b(whic)m(h)h(d)600 683 y Fe(P)645 668 y Ft(\()p Fs(\026)742 632 y Fr(\017)774 668 y Fs(;)17 b(\026)877 683 y Fr(\017)910 668 y Ft(\))27 b Fs(<)h(\016)t Ft(\(3)22 b Fo(\001)f Ft(2)1333 632 y Fr(n)1380 668 y Ft(\))1418 632 y Fq(\000)p Fp(1)1512 668 y Ft(.)44 b(This)33 b(in)e(particular)g(means)i(that)1239 865 y(1)p 1215 910 96 4 v 1215 1001 a(2)1264 972 y Fr(n)1338 788 y Fh(\014)1338 848 y(\014)1338 908 y(\014)1338 967 y(\014)1371 797 y(Z)1487 933 y Fs(')1551 948 y Fr(n)1615 933 y Fs(d\026)1725 891 y Fr(\017)1779 933 y Fo(\000)1878 797 y Fh(Z)1995 933 y Fs(')2059 948 y Fr(n)2122 933 y Fs(d\026)2232 948 y Fr(\017)2264 788 y Fh(\014)2264 848 y(\014)2264 908 y(\014)2264 967 y(\014)2325 933 y Fs(<)2523 865 y(\016)p 2439 910 217 4 v 2439 1001 a Ft(3)22 b Fo(\001)f Ft(2)2608 972 y Fr(n)2665 933 y Fs(;)118 1191 y Ft(b)m(y)34 b(the)f(de\014nition)e(of)h(the)h(distance)g(d)1568 1206 y Fe(P)1613 1191 y Ft(,)g(whic)m(h)g(implies)1355 1305 y Fh(\014)1355 1365 y(\014)1355 1425 y(\014)1355 1485 y(\014)1389 1314 y(Z)1505 1450 y Fs(')1569 1465 y Fr(n)1632 1450 y Fs(d\026)1742 1409 y Fr(\017)1797 1450 y Fo(\000)1896 1314 y Fh(Z)2012 1450 y Fs(')2076 1465 y Fr(n)2140 1450 y Fs(d\026)2250 1465 y Fr(\017)2282 1305 y Fh(\014)2282 1365 y(\014)2282 1425 y(\014)2282 1485 y(\014)2343 1450 y Fs(<)2457 1382 y(\016)p 2456 1427 49 4 v 2456 1518 a Ft(3)2515 1450 y Fs(:)118 1707 y Ft(Hence)h(w)m(e)g(get)247 1822 y Fh(\014)247 1881 y(\014)247 1941 y(\014)247 2001 y(\014)280 1830 y(Z)396 1966 y Fs(')17 b(d\026)587 1925 y Fr(\017)641 1966 y Fo(\000)741 1830 y Fh(Z)857 1966 y Fs(')g(d\026)1048 1981 y Fr(\017)1080 1822 y Fh(\014)1080 1881 y(\014)1080 1941 y(\014)1080 2001 y(\014)1140 1966 y Fo(\024)448 2238 y(\024)609 2094 y Fh(\014)609 2154 y(\014)609 2213 y(\014)609 2273 y(\014)642 2103 y(Z)758 2238 y Fs(')g(d\026)949 2197 y Fr(\017)1003 2238 y Fo(\000)1102 2103 y Fh(Z)1219 2238 y Fs(')1283 2253 y Fr(n)1346 2238 y Fs(d\026)1456 2197 y Fr(\017)1488 2094 y Fh(\014)1488 2154 y(\014)1488 2213 y(\014)1488 2273 y(\014)1543 2238 y Ft(+)1641 2094 y Fh(\014)1641 2154 y(\014)1641 2213 y(\014)1641 2273 y(\014)1675 2103 y(Z)1791 2238 y Fs(')1855 2253 y Fr(n)1918 2238 y Fs(d\026)2028 2197 y Fr(\017)2083 2238 y Fo(\000)2182 2103 y Fh(Z)2298 2238 y Fs(')2362 2253 y Fr(n)2426 2238 y Fs(d\026)2536 2253 y Fr(\017)2568 2094 y Fh(\014)2568 2154 y(\014)2568 2213 y(\014)2568 2273 y(\014)2623 2238 y Ft(+)2721 2094 y Fh(\014)2721 2154 y(\014)2721 2213 y(\014)2721 2273 y(\014)2755 2103 y(Z)2871 2238 y Fs(')2935 2253 y Fr(n)2998 2238 y Fs(d\026)3108 2253 y Fr(\017)3162 2238 y Fo(\000)3262 2103 y Fh(Z)3378 2238 y Fs(')g(d\026)3569 2253 y Fr(\017)3601 2094 y Fh(\014)3601 2154 y(\014)3601 2213 y(\014)3601 2273 y(\014)449 2503 y Fs(<)619 2435 y(\016)p 619 2480 V 619 2571 a Ft(3)699 2503 y(+)808 2435 y Fs(\016)p 807 2480 V 807 2571 a Ft(3)888 2503 y(+)997 2435 y Fs(\016)p 996 2480 V 996 2571 a Ft(3)1083 2503 y(=)27 b Fs(\016)n(;)118 2735 y Ft(whic)m(h)33 b(completes)g(the)g (pro)s(of)e(of)h(the)h(lemma.)p 3709 2735 4 66 v 3713 2673 59 4 v 3713 2735 V 3771 2735 4 66 v 264 2926 a(No)m(w)h(w)m(e)f (tak)m(e)h Fs(')28 b Ft(=)f(log)17 b Fo(k)p Ft(\()p Fs(D)s(f)11 b Ft(\))1449 2890 y Fq(\000)p Fp(1)1542 2926 y Fo(k)33 b Ft(and)g Fs(\016)e Ft(=)d Fs(c=)p Ft(2)k(in)g(the)h(previous)h (lemma,)c(where)k Fs(c)28 b(>)g Ft(0)k(is)118 3047 y(the)25 b(constan)m(t)h(giv)m(en)f(b)m(y)h(the)f(non-uniform)e(expansion)i(of)g Fs(f)35 b Ft(\(recall)24 b(\(3\)\).)40 b(F)-8 b(or)24 b(eac)m(h)i Fs(\017)i Fo(\024)g Fs(\017)3490 3062 y Fp(0)3555 3047 y Ft(let)c Fs(\026)3747 3062 y Fr(\017)118 3167 y Ft(b)s(e)31 b(the)g(measure)g(giv)m(en)g(b)m(y)g(Lemma)f(5.1.)42 b(Since)31 b(prop)s(ert)m(y)g(\(P\))g(holds,)g(there)g(are)g(real)f(n)m (um)m(b)s(ers)118 3288 y Fs(w)188 3303 y Fp(1)227 3288 y Ft(\()p Fs(\017)p Ft(\))p Fs(;)17 b(:)g(:)g(:)f(;)h(w)631 3303 y Fr(p)670 3288 y Ft(\()p Fs(\017)p Ft(\))28 b Fo(\025)h Ft(0)c(with)h Fs(w)1279 3303 y Fp(1)1318 3288 y Ft(\()p Fs(\017)p Ft(\))9 b(+)g Fo(\001)17 b(\001)g(\001)6 b Ft(+)j Fs(w)1806 3303 y Fr(p)1845 3288 y Ft(\()p Fs(\017)p Ft(\))28 b(=)g(1)e(for)f(whic)m(h)i Fs(\026)2641 3303 y Fr(\017)2701 3288 y Ft(=)g Fs(w)2874 3303 y Fp(1)2913 3288 y Ft(\()p Fs(\017)p Ft(\))p Fs(\026)3087 3303 y Fp(1)3135 3288 y Ft(+)9 b Fo(\001)17 b(\001)g(\001)7 b Ft(+)i Fs(w)3500 3303 y Fr(p)3539 3288 y Ft(\()p Fs(\017)p Ft(\))p Fs(\026)3713 3303 y Fr(p)3752 3288 y Ft(.)118 3408 y(Since)40 b(eac)m(h)g Fs(\026)665 3423 y Fr(i)732 3408 y Ft(is)f(an)g(SRB)g(measure)h(for)f(1)g Fo(\024)g Fs(i)h Fo(\024)g Fs(p)p Ft(,)h(w)m(e)f(ha)m(v)m(e)h(for)d(Leb)s(esgue)j (almost)d(ev)m(ery)118 3528 y Fs(x)28 b Fo(2)g Fs(B)5 b Ft(\()p Fs(\026)471 3543 y Fr(i)499 3528 y Ft(\))603 3690 y Fh(Z)719 3825 y Ft(log)17 b Fo(k)p Ft(\()p Fs(D)s(f)11 b Ft(\))1131 3784 y Fq(\000)p Fp(1)1224 3825 y Fo(k)p Fs(d\026)1384 3840 y Fr(i)1439 3825 y Ft(=)80 b(lim)1543 3885 y Fr(n)p Fq(!)p Fp(+)p Fq(1)1813 3758 y Ft(1)p 1808 3802 59 4 v 1808 3894 a Fs(n)1899 3701 y Fr(n)p Fq(\000)p Fp(1)1893 3731 y Fh(X)1904 3941 y Fr(j)t Fp(=0)2054 3825 y Ft(log)16 b Fo(k)p Fs(D)s(f)11 b Ft(\()p Fs(f)2486 3784 y Fr(j)2522 3825 y Ft(\()p Fs(x)p Ft(\)\))2691 3784 y Fq(\000)p Fp(1)2785 3825 y Fo(k)28 b(\024)g(\000)p Fs(c)g(<)g Ft(0)p Fs(:)118 4128 y Ft(This)33 b(implies)1403 4155 y Fh(Z)1519 4291 y Ft(log)16 b Fo(k)p Ft(\()p Fs(D)s(f)11 b Ft(\))1930 4250 y Fq(\000)p Fp(1)2024 4291 y Fo(k)p Fs(d\026)2184 4306 y Fr(\017)2243 4291 y Fo(\024)29 b(\000)p Fs(c;)118 4502 y Ft(and)k(so,)g(b)m(y)g(Lemma)e(5.1)h(and)h(the)g(c)m (hoice)g(of)f Fs(\016)t Ft(,)1354 4617 y Fh(Z)1470 4752 y Ft(log)17 b Fo(k)p Ft(\()p Fs(D)s(f)11 b Ft(\))1882 4711 y Fq(\000)p Fp(1)1975 4752 y Fo(k)p Fs(d\026)2135 4711 y Fr(\017)2195 4752 y Fo(\024)28 b(\000)p Fs(c=)p Ft(2)p Fs(:)118 5002 y Ft(This)33 b(completes)f(the)h(pro)s(of)f(of)g (the)h(\014rst)g(item)e(of)h(Theorem)h(B.)1900 5251 y(27)p eop %%Page: 28 28 28 27 bop 264 548 a Ft(No)m(w)34 b(w)m(e)f(go)f(in)m(to)g(the)h(pro)s (of)e(of)h(Theorem)h(D.)43 b(In)33 b(order)f(to)h(pro)m(v)m(e)g(that)g Fs(f)43 b Ft(is)32 b(sto)s(c)m(hastically)118 668 y(stable,)h(and)g (taking)g(in)m(to)f(accoun)m(t)i(prop)s(ert)m(y)g(\(P\),)f(it)f (su\016ces)j(to)e(pro)m(v)m(e)h(that)f(the)h(w)m(eak)3484 632 y Fq(\003)3557 668 y Ft(accu-)118 789 y(m)m(ulation)g(p)s(oin)m(ts) i(of)g(an)m(y)h(family)d(\()p Fs(\026)1530 753 y Fr(\017)1562 789 y Ft(\))1600 804 y Fr(\017>)p Fp(0)1723 789 y Ft(,)j(where)h(eac)m (h)f Fs(\026)2355 753 y Fr(\017)2423 789 y Ft(is)f(a)g(ph)m(ysical)h (measure)f(of)g(lev)m(el)g Fs(\017)p Ft(,)118 909 y(are)d(absolutely)g (con)m(tin)m(uous)h(with)e(resp)s(ect)j(to)e(the)g(Leb)s(esgue)i (measure.)45 b(Let)34 b Fs(\026)3157 873 y Fr(\017)3222 909 y Ft(b)s(e)g(a)f(ph)m(ysical)118 1029 y(measure)g(of)f(lev)m(el)g Fs(\017)h Ft(for)f(some)h(small)d Fs(\017)e(>)g Ft(0)k(and)g(de\014ne)i (for)e(eac)m(h)i Fs(n)28 b Fo(\025)g Ft(1)1137 1318 y Fs(\026)1196 1277 y Fr(\017)1196 1343 y(n)1270 1318 y Ft(=)1388 1251 y(1)p 1383 1296 59 4 v 1383 1387 a Fs(n)1474 1194 y Fr(n)p Fq(\000)p Fp(1)1468 1224 y Fh(X)1479 1434 y Fr(j)t Fp(=0)1629 1183 y Fh(Z)1728 1318 y Ft(\()p Fs(f)1825 1277 y Fr(n)1814 1343 y(t)p 1814 1355 30 3 v 1872 1318 a Ft(\))1910 1333 y Fq(\003)1949 1238 y Fh(\000)1995 1318 y Fs(m)g Fo(j)f Fs(B)5 b Ft(\()p Fs(\026)2339 1277 y Fr(\017)2372 1318 y Ft(\))2410 1238 y Fh(\001)2472 1318 y Fs(d\022)2571 1277 y Fe(N)2568 1343 y Fr(\017)2623 1318 y Ft(\()p Fs(t)p 2661 1334 36 4 v Ft(\))p Fs(:)118 1630 y Ft(W)-8 b(e)36 b(kno)m(w)h(from)e(Prop)s(osition)f(4.1)i(that)g (eac)m(h)g Fs(\026)1966 1593 y Fr(\017)2035 1630 y Ft(is)f(the)h(w)m (eak)2517 1593 y Fq(\003)2594 1630 y Ft(limit)c(of)k(the)g(sequence)j (\()p Fs(\026)3621 1593 y Fr(\017)3621 1654 y(n)3668 1630 y Ft(\))3706 1645 y Fr(n)3752 1630 y Ft(.)118 1750 y(W)-8 b(e)37 b(will)e(pro)m(v)m(e)j(Theorem)f(D)f(b)m(y)i(pro)m (viding)d(some)i(useful)g(estimates)f(on)h(the)g(densities)g(of)f(the) 118 1870 y(measures)d Fs(\026)595 1834 y Fr(\017)595 1895 y(n)642 1870 y Ft(.)44 b(De\014ne)33 b(for)f(eac)m(h)h Fs(t)p 1383 1886 V 28 w Fo(2)28 b Fs(T)1611 1834 y Fe(N)1696 1870 y Ft(and)k Fs(n)c Fo(\025)g Ft(1)625 2081 y Fs(H)706 2096 y Fr(n)753 2081 y Ft(\()p Fs(t)p 791 2096 V Ft(\))g(=)f Fo(f)p Fs(x)h Fo(2)g Fs(B)5 b Ft(\()p Fs(\026)1398 2039 y Fr(\017)1431 2081 y Ft(\))11 b(:)66 b Fs(n)32 b Ft(is)h(a)f(\()p Fs(\013)q(;)17 b(\016)t Ft(\)-h)m(yp)s(erb)s(olic)30 b(time)i(for)g(\()p Fs(t)p 2991 2096 V(;)17 b(x)p Ft(\))32 b Fo(g)p Fs(;)118 2291 y Ft(and)481 2411 y Fs(H)570 2370 y Fq(\003)562 2436 y Fr(n)609 2411 y Ft(\()p Fs(t)p 647 2427 V Ft(\))c(=)g Fo(f)p Fs(x)g Fo(2)g Fs(B)5 b Ft(\()p Fs(\026)1255 2370 y Fr(\017)1287 2411 y Ft(\))11 b(:)66 b Fs(n)33 b Ft(is)f(the)h(\014rst)g(\()p Fs(\013)q(;)17 b(\016)t Ft(\)-h)m(yp)s(erb)s(olic)30 b(time)i(for)g(\()p Fs(t)p 3135 2427 V(;)17 b(x)p Ft(\))32 b Fo(g)p Fs(:)118 2581 y(H)207 2545 y Fq(\003)199 2606 y Fr(n)246 2581 y Ft(\()p Fs(t)p 284 2597 V Ft(\))k(is)g(precisely)h(the)g(set)g(of)f (those)h(p)s(oin)m(ts)e Fs(x)g Fo(2)f Fs(B)5 b Ft(\()p Fs(\026)2262 2545 y Fr(\017)2295 2581 y Ft(\))36 b(for)g(whic)m(h)h Fs(h)2861 2596 y Fr(\017)2894 2581 y Ft(\()p Fs(t)p 2932 2597 V(;)17 b(x)p Ft(\))34 b(=)g Fs(n)i Ft(\(recall)f(the)118 2702 y(de\014nition)30 b(of)h(the)h(map)e Fs(h)1098 2717 y Fr(\017)1131 2702 y Ft(\).)43 b(F)-8 b(or)30 b Fs(n;)17 b(k)31 b Fo(\025)d Ft(1)j(w)m(e)h(also)e(de\014ne)j Fs(R)2472 2717 y Fr(n;k)2577 2702 y Ft(\()p Fs(t)p 2615 2718 V Ft(\))e(as)h(the)f(set)h(of)f(those)h(p)s(oin)m(ts)118 2822 y Fs(x)i Fo(2)g Fs(M)46 b Ft(for)36 b(whic)m(h)g Fs(n)g Ft(is)g(a)g(\()p Fs(\013)q(;)17 b(\016)t Ft(\)-h)m(yp)s(erb)s (olic)34 b(time)g(and)i Fs(n)25 b Ft(+)f Fs(k)39 b Ft(is)d(the)g (\014rst)g(\()p Fs(\013)q(;)17 b(\016)t Ft(\)-h)m(yp)s(erb)s(olic)118 2942 y(time)31 b(after)i Fs(n)p Ft(,)f(i.e.)1040 3063 y Fs(R)1114 3078 y Fr(n;k)1220 3063 y Ft(\()p Fs(t)p 1258 3079 V Ft(\))27 b(=)1462 2982 y Fh(\010)1520 3063 y Fs(x)h Fo(2)g Fs(H)1778 3078 y Fr(n)1825 3063 y Ft(\()p Fs(t)p 1863 3079 V Ft(\))11 b(:)56 b Fs(f)2089 3022 y Fr(n)2078 3087 y(t)p 2078 3099 30 3 v 2135 3063 a Ft(\()p Fs(x)p Ft(\))28 b Fo(2)g Fs(H)2477 3022 y Fq(\003)2469 3087 y Fr(k)2516 3063 y Ft(\()p Fs(\033)2613 3022 y Fr(n)2660 3063 y Fs(t)p 2660 3079 36 4 v Ft(\))2756 2982 y Fh(\011)2830 3063 y Fs(;)118 3242 y Ft(where)37 b Fs(\033)15 b Ft(:)34 b Fs(T)605 3206 y Fe(N)690 3242 y Fo(!)e Fs(T)893 3206 y Fe(N)981 3242 y Ft(is)j(the)h(shift)f(map)f Fs(\033)t Ft(\()p Fs(t)1827 3257 y Fp(1)1867 3242 y Fs(;)17 b(t)1946 3257 y Fp(2)1985 3242 y Fs(;)g(:)g(:)g(:)f Ft(\))33 b(=)f(\()p Fs(t)2412 3257 y Fp(2)2452 3242 y Fs(;)17 b(t)2531 3257 y Fp(3)2570 3242 y Fs(;)g(:)g(:)g(:)f Ft(\).)53 b(Considering)35 b(the)h(mea-)118 3363 y(sures)1295 3507 y Fs(\027)1349 3466 y Fr(\017)1343 3531 y(n)1418 3507 y Ft(=)1521 3371 y Fh(Z)1621 3507 y Ft(\()p Fs(f)1718 3466 y Fr(n)1707 3531 y(t)p 1707 3543 30 3 v 1765 3507 a Ft(\))1803 3522 y Fq(\003)1842 3426 y Fh(\000)1888 3507 y Fs(m)28 b Fo(j)f Fs(H)2137 3522 y Fr(n)2184 3507 y Ft(\()p Fs(t)p 2222 3523 36 4 v Ft(\))2295 3426 y Fh(\001)2341 3507 y Fs(d\022)2440 3466 y Fe(N)2437 3531 y Fr(\017)2492 3507 y Ft(\()p Fs(t)p 2530 3523 V Ft(\))118 3720 y(and)1052 3909 y Fs(\021)1104 3868 y Fr(\017)1100 3933 y(n)1174 3909 y Ft(=)1315 3784 y Fq(1)1278 3814 y Fh(X)1285 4026 y Fr(k)r Fp(=2)1446 3784 y Fr(k)r Fq(\000)p Fp(1)1438 3814 y Fh(X)1449 4024 y Fr(j)t Fp(=1)1599 3773 y Fh(Z)1699 3909 y Ft(\()p Fs(f)1796 3861 y Fr(n)p Fp(+)p Fr(j)1785 3931 y(t)p 1785 3943 30 3 v 1929 3909 a Ft(\))1967 3924 y Fq(\003)2007 3828 y Fh(\000)2052 3909 y Fs(m)h Fo(j)g Fs(R)2295 3924 y Fr(n;k)2400 3909 y Ft(\()p Fs(t)p 2438 3925 36 4 v Ft(\))2511 3828 y Fh(\001)2557 3909 y Fs(d\022)2656 3868 y Fe(N)2653 3933 y Fr(\017)2708 3909 y Ft(\()p Fs(t)p 2746 3925 V Ft(\))p Fs(;)118 4171 y Ft(w)m(e)34 b(ma)m(y)e(write)1513 4376 y Fs(\026)1572 4335 y Fr(\017)1572 4401 y(n)1646 4376 y Fo(\024)1766 4309 y Ft(1)p 1761 4353 59 4 v 1761 4444 a Fs(n)1851 4251 y Fr(n)p Fq(\000)p Fp(1)1846 4281 y Fh(X)1857 4491 y Fr(j)t Fp(=0)1990 4376 y Ft(\()p Fs(\027)2082 4335 y Fr(\017)2076 4401 y(j)2137 4376 y Ft(+)22 b Fs(\021)2287 4335 y Fr(\017)2283 4401 y(j)2320 4376 y Ft(\))p Fs(:)118 4679 y Fc(Prop)s(osition)36 b(5.2.)49 b Fi(Ther)-5 b(e)34 b(is)h(a)f(c)-5 b(onstant)35 b Fs(C)1875 4694 y Fp(2)1942 4679 y Fs(>)27 b Ft(0)35 b Fi(such)g(that)g(for)f(every)h Fs(n)28 b Fo(\025)g Ft(0)35 b Fi(and)f Fs(t)p 3427 4695 36 4 v 28 w Fo(2)28 b Fs(T)3655 4643 y Fe(N)1429 4866 y Fs(d)p 1386 4911 137 4 v 1386 5002 a(dm)1532 4934 y Ft(\()p Fs(f)1629 4893 y Fr(n)1618 4958 y(t)p 1618 4970 30 3 v 1676 4934 a Ft(\))1714 4949 y Fq(\003)1753 4853 y Fh(\000)1799 4934 y Fs(m)g Fo(j)f Fs(H)2048 4949 y Fr(n)2095 4934 y Ft(\()p Fs(t)p 2133 4950 36 4 v Ft(\))2206 4853 y Fh(\001)2280 4934 y Fo(\024)h Fs(C)2455 4949 y Fp(2)2494 4934 y Fs(:)1900 5251 y Ft(28)p eop %%Page: 29 29 29 28 bop 118 548 a Fi(Pr)-5 b(o)g(of.)49 b Ft(T)-8 b(ak)m(e)29 b Fs(\016)698 563 y Fp(1)766 548 y Fs(>)e Ft(0)i(giv)m(en)g(b)m(y)g (Prop)s(osition)e(2.6.)42 b(It)29 b(is)f(su\016cien)m(t)i(to)e(pro)m(v) m(e)i(that)f(there)g(is)g(some)118 668 y(uniform)h(constan)m(t)i Fs(C)j(>)28 b Ft(0)j(suc)m(h)i(that)e(if)g Fs(A)g Ft(is)h(a)f(Borel)g (set)h(in)f Fs(M)42 b Ft(with)31 b(diameter)g(smaller)e(than)118 789 y Fs(\016)161 804 y Fp(1)201 789 y Fs(=)p Ft(2)j(then)1295 909 y Fs(m)1380 828 y Fh(\000)1426 909 y Fs(f)1485 868 y Fq(\000)p Fr(n)1474 934 y(t)p 1474 946 30 3 v 1587 909 a Ft(\()p Fs(A)p Ft(\))22 b Fo(\\)h Fs(H)1928 924 y Fr(n)1974 909 y Ft(\()p Fs(t)p 2012 925 36 4 v 1 w Ft(\))2086 828 y Fh(\001)2159 909 y Fo(\024)28 b Fs(C)7 b(m)p Ft(\()p Fs(A)p Ft(\))p Fs(:)118 1072 y Ft(Let)42 b Fs(A)f Ft(b)s(e)h(a)f(Borel)f(set)i(in)f Fs(M)52 b Ft(with)41 b(diameter)f(smaller)g(than)h Fs(\016)2616 1087 y Fp(1)2656 1072 y Fs(=)p Ft(2)g(and)g Fs(B)47 b Ft(an)41 b(op)s(en)g(ball)f(of)118 1192 y(radius)32 b Fs(\016)454 1207 y Fp(1)494 1192 y Fs(=)p Ft(2)g(con)m(taining)f Fs(A)p Ft(.)44 b(W)-8 b(e)33 b(ma)m(y)f(write)1581 1403 y Fs(f)1640 1362 y Fq(\000)p Fr(n)1629 1427 y(t)p 1629 1439 30 3 v 1741 1403 a Ft(\()p Fs(B)5 b Ft(\))28 b(=)2037 1308 y Fh([)2028 1520 y Fr(k)r Fq(\025)p Fp(1)2173 1403 y Fs(B)2247 1418 y Fr(k)2290 1403 y Fs(;)118 1697 y Ft(where)50 b(\()p Fs(B)528 1712 y Fr(k)571 1697 y Ft(\))609 1712 y Fr(k)r Fq(\025)p Fp(1)790 1697 y Ft(is)e(a)h(\(p)s(ossibly)f (\014nite\))g(family)e(of)i(t)m(w)m(o-b)m(y-t)m(w)m(o)j(disjoin)m(t)c (op)s(en)i(sets)h(in)e Fs(M)10 b Ft(.)118 1818 y(Discarding)40 b(those)j Fs(B)953 1833 y Fr(k)1038 1818 y Ft(that)e(do)h(not)g(in)m (tersect)g Fs(H)2069 1833 y Fr(n)2116 1818 y Ft(\()p Fs(t)p 2154 1834 36 4 v Ft(\),)j(w)m(e)e(c)m(ho)s(ose)f(for)g(eac)m(h)g Fs(k)47 b Fo(\025)d Ft(1)e(a)f(p)s(oin)m(t)118 1938 y Fs(x)173 1953 y Fr(k)256 1938 y Fo(2)g Fs(H)444 1953 y Fr(n)490 1938 y Ft(\()p Fs(t)p 528 1954 V 1 w Ft(\))27 b Fo(\\)g Fs(B)796 1953 y Fr(k)839 1938 y Ft(.)65 b(F)-8 b(or)39 b Fs(k)k Fo(\025)e Ft(1)e(let)g Fs(V)1618 1953 y Fr(n)1665 1938 y Ft(\()p Fs(t)p 1703 1954 V(;)17 b(x)1837 1953 y Fr(k)1880 1938 y Ft(\))40 b(b)s(e)g(the)g(neigh)m(b)s(orho)s(o)s (d)f(of)g Fs(x)3066 1953 y Fr(k)3149 1938 y Ft(in)g Fs(M)51 b Ft(giv)m(en)40 b(b)m(y)118 2058 y(Prop)s(osition)e(2.6.)65 b(Since)40 b Fs(B)45 b Ft(is)39 b(con)m(tained)h(in)f Fs(B)2002 1978 y Fh(\000)2048 2058 y Fs(f)2107 2022 y Fr(n)2096 2083 y(t)p 2096 2095 30 3 v 2154 2058 a Ft(\()p Fs(x)2247 2073 y Fr(k)2290 2058 y Ft(\))p Fs(;)17 b(\016)2415 2073 y Fp(1)2454 1978 y Fh(\001)2500 2058 y Ft(,)41 b(the)f(ball)e(of)i (radius)f Fs(\016)3402 2073 y Fp(1)3481 2058 y Ft(around)118 2191 y Fs(f)177 2155 y Fr(n)166 2216 y(t)p 166 2228 V 224 2191 a Ft(\()p Fs(x)317 2206 y Fr(k)360 2191 y Ft(\),)h(and)f Fs(f)720 2155 y Fr(n)709 2216 y(t)p 709 2228 V 805 2191 a Ft(is)f(a)h(di\013eomorphism)d(from)h Fs(V)1982 2206 y Fr(n)2029 2191 y Ft(\()p Fs(t)p 2067 2207 36 4 v(;)17 b(x)2201 2206 y Fr(k)2244 2191 y Ft(\))38 b(on)m(to)h Fs(B)2625 2110 y Fh(\000)2671 2191 y Fs(f)2730 2155 y Fr(n)2719 2216 y(t)p 2719 2228 30 3 v 2776 2191 a Ft(\()p Fs(x)2869 2206 y Fr(k)2913 2191 y Ft(\))p Fs(;)17 b(\016)3038 2206 y Fp(1)3077 2110 y Fh(\001)3122 2191 y Ft(,)41 b(w)m(e)e(m)m(ust)g (ha)m(v)m(e)118 2311 y Fs(B)192 2326 y Fr(k)263 2311 y Fo(\032)28 b Fs(V)425 2326 y Fr(n)472 2311 y Ft(\()p Fs(t)p 510 2327 36 4 v(;)17 b(x)644 2326 y Fr(k)687 2311 y Ft(\))26 b(\(recall)f(that)h(b)m(y)h(our)f(c)m(hoice)h(of)f Fs(B)2007 2326 y Fr(k)2076 2311 y Ft(w)m(e)h(ha)m(v)m(e)h Fs(f)2491 2275 y Fr(n)2480 2336 y(t)p 2480 2348 30 3 v 2538 2311 a Ft(\()p Fs(B)2650 2326 y Fr(k)2692 2311 y Ft(\))g Fo(\032)g Fs(B)5 b Ft(\).)41 b(As)27 b(a)f(consequence)118 2432 y(of)41 b(this)f(and)i(Corollary)d(2.7,)k(w)m(e)f(ha)m(v)m(e)g (for)f(ev)m(ery)i Fs(k)h Ft(that)d(the)g(map)g Fs(f)2854 2396 y Fr(n)2843 2456 y(t)p 2843 2468 V 2942 2432 a Fo(j)h Fs(B)3086 2447 y Fr(k)3140 2432 y Ft(:)36 b Fs(B)3277 2447 y Fr(k)3362 2432 y Fo(!)42 b Fs(B)k Ft(is)41 b(a)118 2552 y(di\013eomorphism)30 b(with)i(b)s(ounded)i(distortion:)1457 2738 y(1)p 1427 2783 110 4 v 1427 2874 a Fs(C)1497 2889 y Fp(1)1573 2806 y Fo(\024)1689 2727 y(j)17 b Ft(det)f Fs(D)s(f)2028 2691 y Fr(n)2017 2752 y(t)p 2017 2764 30 3 v 2074 2727 a Ft(\()p Fs(y)t Ft(\))p Fo(j)p 1689 2783 541 4 v 1690 2874 a(j)h Ft(det)f Fs(D)s(f)2029 2840 y Fr(n)2018 2896 y(t)p 2018 2908 30 3 v 2075 2874 a Ft(\()p Fs(z)t Ft(\))p Fo(j)2267 2806 y(\024)28 b Fs(C)2442 2821 y Fp(1)118 3062 y Ft(for)k(all)f Fs(y)t(;)17 b(z)31 b Fo(2)d Fs(B)743 3077 y Fr(k)786 3062 y Ft(.)43 b(This)33 b(\014nally)e(giv)m(es)877 3267 y Fs(m)962 3186 y Fh(\000)1008 3267 y Fs(f)1067 3226 y Fq(\000)p Fr(n)1056 3291 y(t)p 1056 3303 V 1169 3267 a Ft(\()p Fs(A)p Ft(\))22 b Fo(\\)g Fs(H)1509 3282 y Fr(n)1556 3267 y Ft(\()p Fs(t)p 1594 3283 36 4 v Ft(\))1667 3186 y Fh(\001)1796 3267 y Fo(\024)1956 3172 y Fh(X)2009 3384 y Fr(k)2117 3267 y Fs(m)2202 3186 y Fh(\000)2248 3267 y Fs(f)2307 3226 y Fq(\000)p Fr(n)2296 3291 y(t)p 2296 3303 30 3 v 2409 3267 a Ft(\()p Fs(A)g Fo(\\)g Fs(B)5 b Ft(\))23 b Fo(\\)f Fs(B)2932 3282 y Fr(k)2975 3186 y Fh(\001)1796 3569 y Fo(\024)1956 3475 y Fh(X)2009 3687 y Fr(k)2117 3569 y Fs(C)2187 3584 y Fp(1)2236 3502 y Fs(m)p Ft(\()p Fs(A)h Fo(\\)f Fs(B)5 b Ft(\))p 2236 3547 425 4 v 2328 3638 a Fs(m)p Ft(\()p Fs(B)g Ft(\))2670 3569 y Fs(m)p Ft(\()p Fs(B)2867 3584 y Fr(k)2910 3569 y Ft(\))1796 3805 y Fo(\024)83 b Fs(C)2026 3820 y Fp(2)2066 3805 y Fs(m)p Ft(\()p Fs(A)p Ft(\))p Fs(;)118 3998 y Ft(where)28 b Fs(C)464 4013 y Fp(2)531 3998 y Fs(>)g Ft(0)e(is)h(a)g(constan)m(t)h(only)e(dep)s(ending)h(on)g Fs(C)2138 4013 y Fp(1)2177 3998 y Ft(,)i(on)d(the)i(v)m(olume)e(of)h (the)g(ball)e Fs(B)32 b Ft(of)27 b(radius)118 4118 y Fs(\016)161 4133 y Fp(1)201 4118 y Fs(=)p Ft(2,)32 b(and)h(on)f(the)h (v)m(olume)f(of)g Fs(M)10 b Ft(.)p 3709 4118 4 66 v 3713 4056 59 4 v 3713 4118 V 3771 4118 4 66 v 264 4306 a(It)33 b(follo)m(ws)e(from)h(Prop)s(osition)f(5.2)h(that)1755 4475 y Fs(d\027)1860 4439 y Fr(\017)1854 4500 y(n)p 1755 4520 146 4 v 1760 4611 a Fs(dm)1938 4543 y Fo(\024)d Fs(C)2114 4558 y Fp(2)3606 4543 y Ft(\(21\))118 4761 y(for)e(ev)m(ery)j Fs(n)d Fo(\025)i Ft(0)e(and)h(small)d Fs(\017)j(>)g Ft(0.)41 b(Our)28 b(goal)e(no)m(w)i(is)f(to)g(con)m(trol) g(the)h(densit)m(y)h(of)e(the)h(measures)118 4882 y Fs(\021)170 4846 y Fr(\017)166 4906 y(n)240 4882 y Ft(in)f(suc)m(h)h(a)f(w)m(a)m(y) h(that)f(w)m(e)h(ma)m(y)f(assure)i(the)e(absolute)g(con)m(tin)m(uit)m (y)g(of)g(the)g(w)m(eak)3145 4846 y Fq(\003)3213 4882 y Ft(accum)m(ulation)118 5002 y(p)s(oin)m(ts)32 b(of)g(the)h(measures)h Fs(\026)1168 4966 y Fr(\017)1233 5002 y Ft(when)f Fs(\017)g Ft(go)s(es)g(to)f(zero.)1900 5251 y(29)p eop %%Page: 30 30 30 29 bop 118 548 a Fc(Prop)s(osition)36 b(5.3.)49 b Fi(Given)29 b Fs(\020)34 b(>)28 b Ft(0)p Fi(,)i(ther)-5 b(e)29 b(is)g Fs(C)1922 563 y Fp(3)1961 548 y Ft(\()p Fs(\020)8 b Ft(\))27 b Fs(>)g Ft(0)i Fi(such)g(that)h(for)f(every)g Fs(n)f Fo(\025)g Ft(0)h Fi(and)f Fs(\017)g(>)g Ft(0)118 668 y Fi(we)35 b(may)f(b)-5 b(ound)35 b Fs(\021)808 632 y Fr(\017)804 693 y(n)886 668 y Fi(by)g(the)g(sum)f(of)h(two)g(non-ne) -5 b(gative)33 b(me)-5 b(asur)g(es,)34 b Fs(\021)2747 632 y Fr(\017)2743 693 y(n)2818 668 y Fo(\024)28 b Fs(!)3003 632 y Fr(\017)3058 668 y Ft(+)22 b Fs(\032)3222 632 y Fr(\017)3255 668 y Fi(,)34 b(with)1262 867 y Fs(d!)1393 831 y Fr(\017)p 1262 912 164 4 v 1276 1003 a Fs(dm)1463 935 y Fo(\024)29 b Fs(C)1639 950 y Fp(3)1678 935 y Ft(\()p Fs(\020)8 b Ft(\))98 b Fi(and)h Fs(\032)2223 894 y Fr(\017)2256 935 y Ft(\()p Fs(M)10 b Ft(\))28 b Fs(<)g(\020)8 b(:)118 1178 y Fi(Pr)-5 b(o)g(of.)49 b Ft(Let)32 b Fs(A)h Ft(b)s(e)g(some)f (Borel)g(set)h(in)f Fs(M)10 b Ft(.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)h(for) e(eac)m(h)i Fs(n)28 b Fo(\025)g Ft(0)582 1483 y Fs(\021)634 1442 y Fr(\017)630 1508 y(n)678 1483 y Ft(\()p Fs(A)p Ft(\))83 b(=)1107 1359 y Fq(1)1070 1388 y Fh(X)1078 1601 y Fr(k)r Fp(=2)1238 1359 y Fr(k)r Fq(\000)p Fp(1)1231 1388 y Fh(X)1241 1598 y Fr(j)t Fp(=1)1391 1347 y Fh(Z)1507 1483 y Fs(m)1592 1402 y Fh(\000)1638 1483 y Fs(f)1697 1436 y Fq(\000)p Fr(n)p Fq(\000)p Fr(j)1686 1506 y(t)p 1686 1518 30 3 v 1886 1483 a Ft(\()p Fs(A)p Ft(\))22 b Fo(\\)h Fs(R)2220 1498 y Fr(n;k)2325 1483 y Ft(\()p Fs(t)p 2363 1499 36 4 v Ft(\))2436 1402 y Fh(\001)2482 1483 y Fs(d\022)2581 1442 y Fe(N)2578 1508 y Fr(\017)2633 1483 y Ft(\()p Fs(t)p 2671 1499 V Ft(\))910 1835 y Fo(\024)1107 1711 y Fq(1)1070 1740 y Fh(X)1078 1953 y Fr(k)r Fp(=2)1238 1711 y Fr(k)r Fq(\000)p Fp(1)1231 1740 y Fh(X)1241 1950 y Fr(j)t Fp(=1)1391 1699 y Fh(Z)1507 1835 y Fs(m)1592 1754 y Fh(\000)1638 1835 y Fs(f)1697 1794 y Fq(\000)p Fr(n)1686 1860 y(t)p 1686 1872 30 3 v 1799 1754 a Fh(\000)1845 1835 y Fs(f)1904 1788 y Fq(\000)p Fr(j)1893 1858 y(\033)1935 1839 y Fn(n)1978 1858 y Fr(t)p 1978 1870 26 3 v 2007 1835 a Ft(\()p Fs(A)p Ft(\))g Fo(\\)f Fs(H)2356 1794 y Fq(\003)2348 1860 y Fr(k)2395 1835 y Ft(\()p Fs(\033)2492 1794 y Fr(n)2539 1835 y Fs(t)p 2539 1851 36 4 v Ft(\))2612 1754 y Fh(\001)2680 1835 y Fo(\\)h Fs(H)2850 1850 y Fr(n)2896 1835 y Ft(\()p Fs(t)p 2934 1851 V 1 w Ft(\))3008 1754 y Fh(\001)3053 1835 y Fs(d\022)3152 1794 y Fe(N)3149 1860 y Fr(\017)3204 1835 y Ft(\()p Fs(t)p 3242 1851 V Ft(\))910 2187 y(=)1107 2063 y Fq(1)1070 2092 y Fh(X)1078 2305 y Fr(k)r Fp(=2)1238 2063 y Fr(k)r Fq(\000)p Fp(1)1231 2092 y Fh(X)1241 2302 y Fr(j)t Fp(=1)1391 2052 y Fh(Z)1507 2187 y Fs(\027)1561 2146 y Fr(\017)1555 2212 y(n)1602 2106 y Fh(\000)1648 2187 y Fs(f)1707 2140 y Fq(\000)p Fr(j)1696 2210 y(\033)1738 2191 y Fn(n)1781 2210 y Fr(t)p 1781 2222 26 3 v 1811 2187 a Ft(\()p Fs(A)p Ft(\))f Fo(\\)h Fs(H)2160 2146 y Fq(\003)2152 2212 y Fr(k)2199 2187 y Ft(\()p Fs(\033)2296 2146 y Fr(n)2343 2187 y Fs(t)p 2343 2203 36 4 v Ft(\))2416 2106 y Fh(\001)2461 2187 y Fs(d\022)2560 2146 y Fe(N)2557 2212 y Fr(\017)2612 2187 y Ft(\()p Fs(t)p 2650 2203 V 1 w Ft(\))910 2539 y Fo(\024)1107 2415 y Fq(1)1070 2444 y Fh(X)1078 2657 y Fr(k)r Fp(=2)1238 2415 y Fr(k)r Fq(\000)p Fp(1)1231 2444 y Fh(X)1241 2654 y Fr(j)t Fp(=1)1391 2539 y Fs(C)1461 2554 y Fp(2)1517 2404 y Fh(Z)1633 2539 y Fs(m)1718 2458 y Fh(\000)1764 2539 y Fs(f)1823 2492 y Fq(\000)p Fr(j)1812 2562 y(t)p 1812 2574 30 3 v 1914 2539 a Ft(\()p Fs(A)p Ft(\))f Fo(\\)h Fs(H)2263 2498 y Fq(\003)2255 2564 y Fr(k)2302 2539 y Ft(\()p Fs(t)p 2340 2555 36 4 v Ft(\))2413 2458 y Fh(\001)2459 2539 y Fs(d\022)2558 2498 y Fe(N)2555 2564 y Fr(\017)2610 2539 y Ft(\()p Fs(t)p 2648 2555 V Ft(\))p Fs(:)118 2865 y Ft(\(in)38 b(this)g(last)f(inequalit)m(y)g(w)m(e)j(used)f(that)f Fs(\022)1763 2828 y Fe(N)1760 2889 y Fr(\017)1854 2865 y Ft(is)g Fs(\033)t Ft(-in)m(v)-5 b(arian)m(t)36 b(and)j(estimate)e (\(21\))h(ab)s(o)m(v)m(e\).)61 b(Let)118 2985 y(no)m(w)42 b Fs(\020)50 b(>)43 b Ft(0)e(b)s(e)h(some)f(\014xed)i(small)c(n)m(um)m (b)s(er.)71 b(Since)41 b(w)m(e)i(are)f(assuming)e(\()p Fs(h)3070 3000 y Fr(\017)3103 2985 y Ft(\))3141 3000 y Fr(\017)3215 2985 y Ft(with)h(uniform)118 3105 y Fs(L)184 3069 y Fp(1)224 3105 y Ft(-tail)30 b(this)i(means)h(that)f(there)h(is)f (some)h(in)m(teger)f Fs(N)39 b Ft(=)27 b Fs(N)10 b Ft(\()p Fs(\020)e Ft(\))32 b(for)g(whic)m(h)1329 3274 y Fq(1)1292 3304 y Fh(X)1289 3515 y Fr(j)t Fp(=)p Fr(N)1456 3399 y Fs(k)1527 3263 y Fh(Z)1643 3399 y Fs(m)1728 3318 y Fh(\000)1774 3399 y Fs(H)1863 3358 y Fq(\003)1855 3423 y Fr(k)1902 3399 y Ft(\()p Fs(t)p 1940 3415 V Ft(\))2013 3318 y Fh(\001)2059 3399 y Fs(d\022)2158 3358 y Fe(N)2155 3423 y Fr(\017)2210 3399 y Ft(\()p Fs(t)p 2248 3415 V Ft(\))c Fs(<)2492 3331 y(\020)p 2462 3376 110 4 v 2462 3467 a(C)2532 3482 y Fp(2)2581 3399 y Fs(:)118 3713 y Ft(W)-8 b(e)33 b(tak)m(e)1063 3902 y Fs(!)1143 3861 y Fr(\017)1204 3902 y Ft(=)27 b Fs(C)1377 3917 y Fp(2)1433 3777 y Fr(N)7 b Fq(\000)p Fp(1)1438 3807 y Fh(X)1445 4019 y Fr(k)r Fp(=2)1610 3777 y Fr(k)r Fq(\000)p Fp(1)1603 3807 y Fh(X)1614 4017 y Fr(j)t Fp(=1)1763 3766 y Fh(Z)1863 3902 y Ft(\()p Fs(f)1960 3855 y Fr(j)1949 3924 y(t)p 1949 3936 30 3 v 1996 3902 a Ft(\))2034 3917 y Fq(\003)2074 3821 y Fh(\000)2119 3902 y Fs(m)28 b Fo(j)g Fs(H)2377 3861 y Fq(\003)2369 3926 y Fr(k)2416 3902 y Ft(\()p Fs(t)p 2454 3918 36 4 v Ft(\))2527 3821 y Fh(\001)2572 3902 y Fs(d\022)2671 3861 y Fe(N)2668 3926 y Fr(\017)2723 3902 y Ft(\()p Fs(t)p 2761 3918 V Ft(\))118 4170 y(and)1055 4359 y Fs(\032)1121 4318 y Fr(\017)1181 4359 y Ft(=)g Fs(C)1355 4374 y Fp(2)1454 4235 y Fq(1)1417 4265 y Fh(X)1411 4477 y Fr(k)r Fp(=)p Fr(N)1592 4235 y(k)r Fq(\000)p Fp(1)1584 4265 y Fh(X)1595 4475 y Fr(j)t Fp(=1)1744 4224 y Fh(Z)1844 4359 y Ft(\()p Fs(f)1941 4312 y Fr(j)1930 4382 y(t)p 1930 4394 30 3 v 1977 4359 a Ft(\))2015 4374 y Fq(\003)2055 4279 y Fh(\000)2100 4359 y Fs(m)g Fo(j)g Fs(H)2358 4318 y Fq(\003)2350 4384 y Fr(k)2397 4359 y Ft(\()p Fs(t)p 2435 4375 36 4 v Ft(\))2508 4279 y Fh(\001)2553 4359 y Fs(d\022)2652 4318 y Fe(N)2649 4384 y Fr(\017)2704 4359 y Ft(\()p Fs(t)p 2742 4375 V Ft(\))p Fs(:)118 4628 y Ft(F)-8 b(or)32 b(this)g(last)g(measure)h(w)m(e)g(ha)m(v)m(e)364 4914 y Fs(\032)430 4873 y Fr(\017)463 4914 y Ft(\()p Fs(M)10 b Ft(\))28 b(=)g Fs(C)845 4929 y Fp(2)944 4789 y Fq(1)907 4819 y Fh(X)901 5031 y Fr(k)r Fp(=)p Fr(N)1081 4789 y(k)r Fq(\000)p Fp(1)1074 4819 y Fh(X)1085 5029 y Fr(j)t Fp(=1)1234 4778 y Fh(Z)1351 4914 y Fs(m)1436 4833 y Fh(\000)1482 4914 y Fs(H)1571 4873 y Fq(\003)1563 4938 y Fr(k)1610 4914 y Ft(\()p Fs(t)p 1648 4930 V Ft(\))1721 4833 y Fh(\001)1766 4914 y Fs(d\022)1865 4873 y Fe(N)1862 4938 y Fr(\017)1917 4914 y Ft(\()p Fs(t)p 1955 4930 V Ft(\))g Fo(\024)g Fs(C)2231 4929 y Fp(2)2330 4789 y Fq(1)2293 4819 y Fh(X)2287 5031 y Fr(k)r Fp(=)p Fr(N)2460 4914 y Fs(k)2531 4778 y Fh(Z)2647 4914 y Fs(m)2732 4833 y Fh(\000)2778 4914 y Fs(H)2867 4873 y Fq(\003)2859 4938 y Fr(k)2906 4914 y Ft(\()p Fs(t)p 2944 4930 V Ft(\))3017 4833 y Fh(\001)3063 4914 y Fs(d\022)3162 4873 y Fe(N)3159 4938 y Fr(\017)3214 4914 y Ft(\()p Fs(t)p 3252 4930 V Ft(\))f Fs(<)h(\020)8 b(:)1900 5251 y Ft(30)p eop %%Page: 31 31 31 30 bop 118 548 a Ft(On)34 b(the)g(other)g(hand,)h(it)e(follo)m(ws)f (from)h(the)h(de\014nition)f(of)g(\()p Fs(\013)q(;)17 b(\016)t Ft(\)-h)m(yp)s(erb)s(olic)32 b(times)h(that)h(there)118 668 y(is)g(some)g(constan)m(t)h Fs(a)c Ft(=)g Fs(a)p Ft(\()p Fs(N)10 b Ft(\))31 b Fs(>)f Ft(0)k(suc)m(h)i(that)e(dist)2076 588 y Fh(\000)2122 668 y Fs(H)2203 683 y Fr(k)2245 668 y Ft(\()p Fs(t)p 2283 684 36 4 v Ft(\))p Fs(;)17 b Fo(C)2458 588 y Fh(\001)2534 668 y Fo(\025)32 b Fs(a)i Ft(for)g(1)c Fo(\024)h Fs(k)j Fo(\024)d Fs(N)10 b Ft(.)49 b(De\014ning)118 789 y(\001)28 b Fo(\032)g Fs(M)42 b Ft(as)32 b(the)g(set)g(of)f(those)h (p)s(oin)m(ts)f(in)f Fs(M)42 b Ft(whose)33 b(distance)f(to)f Fo(C)37 b Ft(is)31 b(greater)g(than)h Fs(a)p Ft(,)g(w)m(e)g(ha)m(v)m(e) 1124 1097 y Fs(!)1204 1055 y Fr(\017)1265 1097 y Fo(\024)c Fs(C)1440 1112 y Fp(2)1496 972 y Fr(N)7 b Fq(\000)p Fp(1)1500 1002 y Fh(X)1508 1214 y Fr(k)r Fp(=2)1673 972 y Fr(k)r Fq(\000)p Fp(1)1666 1002 y Fh(X)1676 1212 y Fr(j)t Fp(=1)1826 961 y Fh(Z)1926 1097 y Ft(\()p Fs(f)2023 1049 y Fr(j)2012 1119 y(t)p 2012 1131 30 3 v 2059 1097 a Ft(\))2097 1112 y Fq(\003)2136 1097 y Ft(\()p Fs(m)28 b Fo(j)g Ft(\001\))22 b Fs(d\022)2583 1055 y Fe(N)2580 1121 y Fr(\017)2635 1097 y Ft(\()p Fs(t)p 2673 1113 36 4 v Ft(\))p Fs(;)118 1412 y Ft(and)38 b(this)g(last)g(measure)g(has)h(densit)m(y)g(b)s (ounded)g(b)m(y)g(some)f(uniform)e(constan)m(t,)41 b(as)d(long)f(as)h (w)m(e)118 1532 y(tak)m(e)33 b(the)g(maps)g Fs(f)801 1547 y Fr(t)863 1532 y Ft(in)f(a)g(su\016cien)m(tly)h(small)e(neigh)m (b)s(orho)s(o)s(d)g(of)h Fs(f)43 b Ft(in)32 b(the)h Fs(C)2975 1496 y Fp(1)3047 1532 y Ft(top)s(ology)-8 b(.)p 3709 1532 4 66 v 3713 1470 59 4 v 3713 1532 V 3771 1532 4 66 v 264 1727 a(It)38 b(follo)m(ws)d(from)h(this)h(last)f(prop)s (osition)f(and)i(\(21\))g(that)f(the)i(w)m(eak)2838 1691 y Fq(\003)2916 1727 y Ft(accum)m(ulation)d(p)s(oin)m(ts)118 1848 y(of)40 b Fs(\026)296 1812 y Fr(\017)369 1848 y Ft(when)i Fs(\017)f Fo(!)g Ft(0)f(cannot)h(ha)m(v)m(e)h(singular)d (part,)j(th)m(us)g(b)s(eing)e(absolutely)g(con)m(tin)m(uous)h(with)118 1968 y(resp)s(ect)29 b(to)e(the)h(Leb)s(esgue)h(measure.)42 b(Moreo)m(v)m(er,)30 b(the)e(w)m(eak)2381 1932 y Fq(\003)2450 1968 y Ft(accum)m(ulation)d(p)s(oin)m(ts)j(of)f(a)g(family)118 2089 y(of)i(stationary)f(measures)i(are)f(alw)m(a)m(ys)h Fs(f)11 b Ft(-in)m(v)-5 b(arian)m(t)26 b(measures,)31 b(cf.)43 b(Remark)28 b(3.1.)42 b(This)29 b(together)118 2209 y(with)j(\(P\))h(giv)m(es)g(the)g(sto)s(c)m(hastic)g(stabilit)m(y) e(of)h Fs(f)11 b Ft(.)118 2542 y Fu(6)161 b(Applications)118 2761 y Ft(In)33 b(this)f(section)h(w)m(e)h(will)c(apply)j(Theorems)g(B) g(and)f(D)h(to)f(certain)g(examples)h(of)f(non-uniformly)118 2881 y(expanding)24 b(maps.)40 b(Before)24 b(w)m(e)g(explicit)e(the)i (examples)g(w)m(e)g(ha)m(v)m(e)h(in)e(mind)f(let)h(us)h(giv)m(e)g(a)f (practical)118 3002 y(criterion)28 b(for)g(pro)m(ving)g(that)h(the)g (family)d(of)j(h)m(yp)s(erb)s(olic)f(time)f(maps)i(\()p Fs(h)2809 3017 y Fr(\017)2842 3002 y Ft(\))2880 3017 y Fr(\017)2941 3002 y Ft(has)g(uniform)e Fs(L)3539 2965 y Fp(1)3579 3002 y Ft(-tail.)264 3122 y(If)d(w)m(e)g(lo)s(ok)f(at)g (the)h(pro)s(of)e(of)i(Prop)s(osition)d(2.3)i(w)m(e)i(realize)d(that)i (what)f(w)m(e)i(did)e(w)m(as)h(\014xing)g(some)118 3242 y(p)s(ositiv)m(e)30 b(n)m(um)m(b)s(er)h Fs(c)872 3257 y Fp(0)941 3242 y Ft(smaller)e(than)h Fs(c)p Ft(,)h(and)g(then,)g(for)f Fs(\022)2231 3206 y Fe(N)2228 3267 y Fr(\017)2301 3242 y Fo(\002)18 b Fs(m)31 b Ft(almost)e(ev)m(ery)j(\()p Fs(t)p 3118 3258 36 4 v(;)17 b(x)p Ft(\))28 b Fo(2)g Fs(T)3483 3206 y Fe(N)3552 3242 y Fo(\002)19 b Fs(M)10 b Ft(,)118 3363 y(w)m(e)34 b(to)s(ok)e(a)g(p)s(ositiv)m(e)g(in)m(teger) h Fs(N)1327 3378 y Fr(\017)1387 3363 y Ft(=)28 b Fs(N)1569 3378 y Fr(\017)1601 3363 y Ft(\()p Fs(t)p 1639 3379 V 1 w(;)17 b(x)p Ft(\))32 b(for)g(whic)m(h)398 3551 y Fr(N)454 3559 y Fn(\017)484 3551 y Fq(\000)p Fp(1)414 3581 y Fh(X)425 3791 y Fr(j)t Fp(=0)591 3675 y Ft(log)16 b Fo(k)p Fs(D)s(f)11 b Ft(\()p Fs(f)1023 3628 y Fr(j)1012 3698 y(t)p 1012 3710 30 3 v 1059 3675 a Ft(\()p Fs(x)p Ft(\)\))1228 3634 y Fq(\000)p Fp(1)1322 3675 y Fo(k)28 b(\024)g(\000)p Fs(c)1624 3690 y Fp(0)1664 3675 y Fs(N)1742 3690 y Fr(\017)1872 3675 y Ft(and)2144 3551 y Fr(N)2200 3559 y Fn(\017)2230 3551 y Fq(\000)p Fp(1)2160 3581 y Fh(X)2170 3791 y Fr(j)t Fp(=0)2337 3675 y Fo(\000)17 b Ft(log)f(dist)2731 3690 y Fr(\016)2769 3675 y Ft(\()p Fs(f)2866 3628 y Fr(j)2855 3698 y(t)p 2855 3710 V 2902 3675 a Ft(\()p Fs(x)p Ft(\))p Fs(;)h Fo(C)6 b Ft(\))28 b Fo(\024)g Fs(\015)5 b(N)3440 3690 y Fr(\017)3473 3675 y Fs(;)118 3991 y Ft(for)32 b(suitable)g(c)m(hoices)h(of)f Fs(\016)g(>)27 b Ft(0)33 b(and)f Fs(\015)h(>)28 b Ft(0.)43 b(This)33 b(p)s(ermits)e(us)i(to)g (in)m(tro)s(duce)f(a)g(map)1543 4211 y Fs(N)1621 4226 y Fr(\017)1665 4211 y Ft(:)h Fs(T)1796 4170 y Fe(N)1870 4211 y Fo(\002)23 b Fs(M)38 b Fo(!)27 b Fg(Z)2298 4170 y Fp(+)118 4431 y Ft(whose)34 b(existence)g(pro)m(vides)f(a)g(\014rst)g (h)m(yp)s(erb)s(olic)f(time)f(map)1195 4651 y Fs(h)1251 4666 y Fr(\017)1295 4651 y Ft(:)i Fs(T)1426 4610 y Fe(N)1500 4651 y Fo(\002)23 b Fs(M)38 b Fo(!)28 b Fg(Z)1929 4610 y Fp(+)2083 4651 y Ft(with)97 b Fs(h)2426 4666 y Fr(\017)2486 4651 y Fo(\024)29 b Fs(N)2670 4666 y Fr(\017)118 4871 y Ft(\(recall)34 b(the)i(pro)s(of)f(of)g(Prop)s(osition)f(2.3\).)52 b(Th)m(us,)38 b(the)e(in)m(tegrabilit)m(y)e(of)h(the)h(map)f Fs(h)3296 4886 y Fr(\017)3364 4871 y Ft(is)g(implied)118 4991 y(b)m(y)f(the)f(in)m(tegrabilit)m(y)d(of)i(the)h(map)f Fs(N)1546 5006 y Fr(\017)1578 4991 y Ft(,)h(whic)m(h)g(is)f(in)g (practice)g(easier)h(to)f(handle.)1900 5251 y(31)p eop %%Page: 32 32 32 31 bop 264 548 a Ft(In)33 b(the)f(examples)g(w)m(e)i(are)e(going)e (to)i(study)h(b)s(elo)m(w)f(w)m(e)i(will)29 b(sho)m(w)34 b(that)d(there)i(is)f(a)g(sequence)118 668 y(of)g(p)s(ositiv)m(e)g (real)g(n)m(um)m(b)s(ers)h(\()p Fs(a)1262 632 y Fr(\017)1262 694 y(k)1305 668 y Ft(\))1343 683 y Fr(k)1418 668 y Ft(for)f(whic)m(h) 435 962 y(\()p Fs(\022)521 921 y Fe(N)518 986 y Fr(\017)595 962 y Fo(\002)23 b Fs(m)p Ft(\))835 881 y Fh(\000)o(\010)938 962 y Ft(\()p Fs(t)p 976 978 36 4 v(;)17 b(x)p Ft(\))28 b Fo(2)g Fs(T)1341 921 y Fe(N)1415 962 y Fo(\002)23 b Fs(M)f Ft(:)33 b Fs(N)1769 977 y Fr(\017)1802 962 y Ft(\()p Fs(t)p 1840 978 V(;)17 b(x)p Ft(\))28 b Fs(>)f(k)2197 881 y Fh(\011\001)2329 962 y Fo(\024)h Fs(a)2485 921 y Fr(\017)2485 986 y(k)2625 962 y Ft(and)2933 837 y Fq(1)2897 867 y Fh(X)2904 1079 y Fr(k)r Fp(=1)3057 962 y Fs(k)s(a)3162 921 y Fr(\017)3162 986 y(k)3233 962 y Fs(<)f Fo(1)p Fs(;)118 1275 y Ft(This)40 b(giv)m(es)g(the)g(in)m(tegrabilit)m(y)d(of)i Fs(h)1500 1290 y Fr(\017)1572 1275 y Ft(with)g(resp)s(ect)i(to)e(the)h (measure)f Fs(\022)2877 1239 y Fe(N)2874 1300 y Fr(\017)2957 1275 y Fo(\002)27 b Fs(m)p Ft(.)64 b(The)41 b(fact)e(the)118 1396 y(family)26 b(\()p Fs(h)506 1411 y Fr(\017)539 1396 y Ft(\))577 1411 y Fr(\017)637 1396 y Ft(has)j(uniform)d Fs(L)1234 1360 y Fp(1)1274 1396 y Ft(-tail)g(can)i(b)s(e)h(pro)m(v)m (ed)g(b)m(y)g(sho)m(wing)g(that)f(the)g(sequence)j(\()p Fs(a)3449 1360 y Fr(\017)3449 1422 y(k)3492 1396 y Ft(\))3530 1411 y Fr(k)3601 1396 y Ft(ma)m(y)118 1516 y(b)s(e)i(c)m(hosen)h(not)e (dep)s(ending)h(on)g Fs(\017)28 b(>)f Ft(0.)264 1637 y(No)m(w)44 b(w)m(e)g(are)f(ready)g(for)f(the)i(applications)c(of)j (Theorems)g(B)g(and)g(D.)74 b(W)-8 b(e)43 b(will)e(describ)s(e)118 1757 y(\014rstly)31 b(a)g(class)h(of)e(lo)s(cal)f(di\013eomorphisms)h (in)m(tro)s(duced)h(in)g([ABV,)h(App)s(endix)f(A])h(that)f(satis\014es) 118 1877 y(the)h(h)m(yp)s(otheses)i(of)d(Theorem)h(B,)g(and)g(then)g(a) g(class)g(of)f(maps)g(\(with)g(critical)f(sets\))i(in)m(tro)s(duced)118 1998 y(in)g([Vi1)o(])h(satisfying)e(the)i(h)m(yp)s(otheses)i(of)d (Theorem)h(D.)118 2286 y Fj(6.1)135 b(Lo)t(cal)46 b(di\013eomorphisms) 118 2471 y Ft(No)m(w)27 b(w)m(e)h(follo)m(w)d([ABV,)j(App)s(endix)f(A]) g(and)g(describ)s(e)g(robust)g(classes)h(of)e(maps)g(\(op)s(en)h(in)f (the)h Fs(C)3740 2435 y Fp(2)118 2591 y Ft(top)s(ology\))35 b(that)g(are)i(non-uniformly)c(expanding)k(lo)s(cal)c (di\013eomorphisms)h(and)j(sto)s(c)m(hastically)118 2712 y(stable.)43 b(Let)33 b Fs(M)43 b Ft(b)s(e)33 b(a)g(compact)f (Riemannian)e(manifold)g(and)i(consider)1426 2920 y(\010)c(:)83 b Fs(T)97 b Fo(\000)-16 b(!)83 b Fs(C)2109 2883 y Fp(2)2148 2920 y Ft(\()p Fs(M)5 b(;)17 b(M)10 b Ft(\))1652 3040 y Fs(t)101 b Fo(7\000)-16 b(!)83 b Fs(f)2080 3055 y Fr(t)118 3259 y Ft(a)23 b(con)m(tin)m(uous)h(family)d(of)i Fs(C)1137 3223 y Fp(2)1200 3259 y Ft(maps,)i(where)g Fs(T)37 b Ft(is)23 b(a)g(metric)f(space.)42 b(W)-8 b(e)24 b(b)s(egin)e(with)i(an) f(essen)m(tially)118 3380 y(com)m(binatorial)29 b(lemma.)118 3580 y Fc(Lemma)37 b(6.1.)49 b Fi(L)-5 b(et)38 b Fs(p;)17 b(q)35 b Fo(\025)d Ft(1)37 b Fi(b)-5 b(e)37 b(inte)-5 b(gers)36 b(and)h Fs(\033)e(>)d(q)41 b Fi(a)c(r)-5 b(e)g(al)37 b(numb)-5 b(er.)50 b(Assume)37 b Fs(M)48 b Fi(admits)37 b(a)118 3700 y(me)-5 b(asur)g(able)34 b(c)-5 b(over)34 b Fo(f)p Fs(B)994 3715 y Fp(1)1034 3700 y Fs(;)17 b(:)g(:)g(:)32 b(;)17 b(B)1343 3715 y Fr(p)1383 3700 y Fs(;)g(B)1501 3715 y Fr(p)p Fp(+1)1630 3700 y Fs(;)g(:)g(:)g(:)33 b(;)17 b(B)1940 3715 y Fr(p)p Fp(+)p Fr(q)2068 3700 y Fo(g)35 b Fi(such)f(that)h(for)g(al)5 b(l)35 b Fs(t)28 b Fo(2)g Fs(T)48 b Fi(it)35 b(holds)234 3901 y(1.)48 b Fo(j)17 b Ft(det)f Fs(D)s(f)690 3916 y Fr(t)720 3901 y Ft(\()p Fs(x)p Ft(\))p Fo(j)27 b(\025)i Fs(\033)38 b Fi(for)d(al)5 b(l)35 b Fs(x)28 b Fo(2)g Fs(B)1652 3916 y Fr(p)p Fp(+1)1804 3901 y Fo([)22 b Fs(:)17 b(:)g(:)22 b Fo([)h Fs(B)2192 3916 y Fr(p)p Fp(+)p Fr(q)2320 3901 y Fi(;)234 4103 y(2.)48 b Ft(\()p Fs(f)448 4118 y Fr(t)505 4103 y Fo(j)28 b Fs(B)635 4118 y Fr(i)663 4103 y Ft(\))35 b Fi(is)f(inje)-5 b(ctive)34 b(for)h(al)5 b(l)35 b Fs(i)28 b Ft(=)f(1)p Fs(;)17 b(:)g(:)g(:)32 b(;)17 b(p)p Fi(.)118 4304 y(Then)34 b(ther)-5 b(e)35 b(is)g Fs(\020)f(>)28 b Ft(0)35 b Fi(such)f(that)i(for)e(every)h(Bor)-5 b(el)34 b(pr)-5 b(ob)g(ability)35 b Fs(\022)j Fi(on)c Fs(T)49 b Fi(we)34 b(have)1032 4520 y Ft(#)p Fo(f)p Ft(0)28 b Fo(\024)g Fs(j)34 b(<)27 b(n)h Ft(:)g Fs(f)1722 4473 y Fr(j)1711 4542 y(t)p 1711 4554 30 3 v 1758 4520 a Ft(\()p Fs(x)p Ft(\))g Fo(2)g Fs(B)2085 4535 y Fp(1)2147 4520 y Fo([)22 b Fs(:)17 b(:)g(:)22 b Fo([)h Fs(B)2535 4535 y Fr(p)2574 4520 y Fo(g)28 b(\025)g Fs(\020)8 b(n)740 b Ft(\(22\))118 4736 y Fi(for)34 b Fs(\022)321 4700 y Fe(N)393 4736 y Fo(\002)20 b Fs(m)35 b 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Ft(1)p Fi(.)1900 5251 y Ft(32)p eop %%Page: 33 33 33 32 bop 118 548 a Fi(Pr)-5 b(o)g(of.)49 b Ft(Let)29 b(us)h(\014x)g Fs(n)e Fo(\025)g Ft(1)h(and)g Fs(t)p 1307 564 36 4 v 28 w Fo(2)f Fs(T)1535 512 y Fe(N)1587 548 y Ft(.)42 b(F)-8 b(or)29 b(a)f(sequence)k Fs(i)p 2306 564 34 4 v 28 w Ft(=)c(\()p Fs(i)2542 563 y Fp(0)2582 548 y Fs(;)17 b(:)g(:)g(:)32 b(;)17 b(i)2850 563 y Fr(n)p Fq(\000)p Fp(1)2987 548 y Ft(\))28 b Fo(2)g(f)p Ft(1)p Fs(;)17 b(:)g(:)g(:)32 b(;)17 b(p)e Ft(+)g Fs(q)t Fo(g)3733 512 y Fr(n)118 668 y Ft(w)m(e)34 b(write)974 789 y([)p Fs(i)p 1001 805 V Ft(])28 b(=)f Fs(B)1266 804 y Fr(i)1290 813 y Fd(0)1351 789 y Fo(\\)c Ft(\()p Fs(f)1537 748 y Fp(1)1526 813 y Fr(t)p 1526 825 30 3 v 1576 789 a Ft(\))1614 748 y Fq(\000)p Fp(1)1708 789 y Ft(\()p Fs(B)1820 804 y Fr(i)1844 813 y Fd(1)1883 789 y Ft(\))f Fo(\\)h(\001)17 b(\001)g(\001)j(\\)j Ft(\()p Fs(f)2356 748 y Fr(n)p Fq(\000)p Fp(1)2345 813 y Fr(t)p 2345 825 V 2493 789 a Ft(\))2531 748 y Fq(\000)p Fp(1)2625 789 y Ft(\()p Fs(B)2737 804 y Fr(i)2761 813 y Fn(n)p Ff(\000)p Fd(1)2886 789 y Ft(\))118 961 y(and)33 b(de\014ne)h Fs(g)t Ft(\()p Fs(i)p 679 976 34 4 v -1 w Ft(\))28 b(=)f(#)p Fo(f)p Ft(0)h Fo(\024)g Fs(j)34 b(<)27 b(n)h Ft(:)g Fs(i)1544 976 y Fr(j)1608 961 y Fo(\024)g Fs(p)p Fo(g)p Ft(.)264 1081 y(W)-8 b(e)29 b(start)g(b)m(y)h(observing)f(that)f(for)g Fs(\020)35 b(>)28 b Ft(0)g(the)h(n)m(um)m(b)s(er)g(of)f(sequences)k Fs(i)p 2895 1097 V 29 w Ft(suc)m(h)e(that)f Fs(g)t Ft(\()p Fs(i)p 3470 1097 V -1 w Ft(\))f Fs(<)f(\020)8 b(n)118 1201 y Ft(is)32 b(b)s(ounded)i(b)m(y)1237 1279 y Fh(X)1223 1491 y Fr(k)r(<\020)5 b(n)1412 1233 y Fh(\022)1485 1306 y Fs(n)1487 1442 y(k)1543 1233 y Fh(\023)1617 1373 y Fs(p)1666 1332 y Fr(k)1708 1373 y Fs(q)1755 1332 y Fr(n)p Fq(\000)p Fr(k)1923 1373 y Fo(\024)2042 1279 y Fh(X)2028 1491 y Fr(k)r Fq(\024)p Fr(\020)g(n)2217 1233 y Fh(\022)2290 1306 y Fs(n)2292 1442 y(k)2348 1233 y Fh(\023)2422 1373 y Fs(p)2471 1332 y Fr(\020)g(n)2553 1373 y Fs(q)2600 1332 y Fr(n)2647 1373 y Fs(:)118 1648 y Ft(Using)30 b(Stirling's)f (form)m(ula)f(\(cf.)j([BV,)g(Section)g(6.3]\))f(the)h(expression)g(on)g (the)g(righ)m(t)f(hand)g(side)h(is)118 1769 y(b)s(ounded)i(b)m(y)h(\()p Fs(e)735 1733 y Fr(\015)779 1769 y Fs(p)828 1733 y Fr(\020)868 1769 y Fs(q)t Ft(\))953 1733 y Fr(n)1000 1769 y Ft(,)f(where)g Fs(\015)g(>)28 b Ft(0)k(dep)s(ends)i(only)e(on)h Fs(\020)39 b Ft(and)33 b Fs(\015)5 b Ft(\()p Fs(\020)j Ft(\))27 b Fo(!)g Ft(0)32 b(when)i Fs(\020)h Fo(!)27 b Ft(0.)264 1889 y(Assumptions)35 b(1)f(and)h(2)f(ensure)i Fs(m)p Ft(\([)p Fs(i)p 1659 1905 V 1 w Ft(]\))30 b Fo(\024)i Fs(\033)1956 1853 y Fq(\000)p Fp(\(1)p Fq(\000)p Fr(\020)5 b Fp(\))p Fr(n)2273 1889 y Ft(\(recall)33 b(that)h Fs(m)p Ft(\()p Fs(M)10 b Ft(\))32 b(=)e(1\).)49 b(Hence)36 b(the)118 2009 y(measure)d(of)f(the)h(union)f Fs(I)1091 2024 y Fr(n)1138 2009 y Ft(\()p Fs(t)p 1176 2025 36 4 v Ft(\))h(of)f(all)e (the)j(sets)h([)p Fs(i)p 1914 2025 34 4 v Ft(])f(with)f Fs(g)t Ft(\()p Fs(i)p 2318 2025 V Ft(\))27 b Fs(<)h(\020)8 b(n)32 b Ft(is)g(b)s(ounded)h(b)m(y)1591 2224 y Fs(\033)1650 2182 y Fq(\000)p Fp(\(1)p Fq(\000)p Fr(\020)5 b Fp(\))p Fr(n)1932 2224 y Ft(\()p Fs(e)2015 2182 y Fr(\015)2060 2224 y Fs(p)2109 2182 y Fr(\020)2148 2224 y Fs(q)t Ft(\))2233 2182 y Fr(n)2280 2224 y Fs(:)118 2438 y Ft(Since)35 b Fs(\033)h(>)c(q)39 b Ft(w)m(e)d(ma)m(y)f(c)m(ho)s(ose)h Fs(\020)42 b Ft(so)35 b(small)e(that)i Fs(e)2052 2402 y Fr(\015)2097 2438 y Fs(p)2146 2402 y Fr(\020)2185 2438 y Fs(q)h(<)c(\033)2431 2402 y Fp(\(1)p Fq(\000)p Fr(\020)5 b Fp(\))2616 2438 y Ft(.)51 b(Then)36 b Fs(m)p Ft(\()p Fs(I)3117 2453 y Fr(n)3164 2438 y Ft(\()p Fs(t)p 3202 2454 36 4 v Ft(\)\))c Fo(\024)g Fs(\034)3507 2402 y Fr(n)3590 2438 y Ft(with)118 2558 y Fs(\034)41 b Ft(=)29 b Fs(e)351 2522 y Fr(\015)t Fp(+)p Fr(\020)5 b Fq(\000)p Fp(1)599 2558 y Fo(\001)23 b Fs(p)699 2522 y Fr(\020)761 2558 y Fo(\001)g Fs(q)33 b(<)c Ft(1)k(for)g(big)g(enough)h Fs(n)29 b Fo(\025)h Fs(N)10 b Ft(.)47 b(Note)33 b(that)h Fs(\034)45 b Ft(and)33 b Fs(N)44 b Ft(do)34 b(not)f(dep)s(end)i(on)e Fs(t)p 3717 2574 V Ft(.)118 2678 y(Setting)1438 2799 y Fs(I)1481 2814 y Fr(n)1556 2799 y Ft(=)1660 2724 y Fh(S)1743 2828 y Fr(t)p 1743 2840 26 3 v Fq(2)p Fr(T)1866 2809 y Fb(N)1910 2718 y Fh(\000)1956 2799 y Fo(f)p Fs(t)p 2006 2815 36 4 v Fo(g)22 b(\002)h Fs(I)2256 2814 y Fr(n)2303 2799 y Ft(\()p Fs(t)p 2341 2815 V Ft(\))2414 2718 y Fh(\001)118 2985 y Ft(w)m(e)39 b(also)f(ha)m(v)m(e)h(\()p Fs(\022)785 2948 y Fe(N)864 2985 y Fo(\002)26 b Fs(m)p Ft(\)\()p Fs(I)1171 3000 y Fr(n)1218 2985 y Ft(\))38 b Fo(\024)g Fs(\034)1462 2948 y Fr(n)1547 2985 y Ft(for)g(all)e(big)i Fs(n)f Fo(\025)h Fs(N)49 b Ft(and)39 b(for)e(ev)m(ery)k(Borel)c (probabilit)m(y)f Fs(\022)118 3105 y Ft(on)c Fs(T)14 b Ft(,)32 b(b)m(y)h(F)-8 b(ubini's)31 b(Theorem.)43 b(Since)1591 3030 y Fh(P)1696 3134 y Fr(n)1743 3105 y Ft(\()p Fs(\022)1829 3069 y Fe(N)1902 3105 y Fo(\002)22 b Fs(m)p Ft(\)\()p Fs(I)2205 3120 y Fr(n)2252 3105 y Ft(\))28 b Fs(<)f Fo(1)32 b Ft(then)g(Borel-Can)m(telli's)e(Lemma)118 3225 y(implies)1349 3346 y(\()p Fs(\022)1435 3305 y Fe(N)1509 3346 y Fo(\002)23 b Fs(m)p Ft(\))1749 3265 y Fh(\000)1794 3271 y(T)1877 3375 y Fr(n)p Fq(\025)p Fp(1)2014 3271 y Fh(S)2097 3375 y Fr(k)r Fq(\025)p Fr(n)2238 3346 y Fs(I)2281 3361 y Fr(k)2323 3265 y Fh(\001)2397 3346 y Ft(=)k(0)118 3528 y(and)33 b(this)f(means)h(that)f Fs(\022)1056 3491 y Fe(N)1130 3528 y Fo(\002)23 b Fs(m)33 b Ft(almost)e(ev)m(ery)j(\()p Fs(t)p 1958 3544 V(;)17 b(x)p Ft(\))28 b Fo(2)g Fs(T)2323 3491 y Fe(N)2397 3528 y Fo(\002)23 b Fs(M)43 b Ft(satis\014es)33 b(\(22\).)p 3709 3528 4 66 v 3713 3465 59 4 v 3713 3528 V 3771 3528 4 66 v 118 3726 a Fc(Lemma)k(6.2.)49 b Fi(L)-5 b(et)33 b Fo(f)p Fs(B)1028 3741 y Fp(1)1068 3726 y Fs(;)17 b(:)g(:)g(:)32 b(;)17 b(B)1377 3741 y Fr(p)1417 3726 y Fs(;)g(B)1535 3741 y Fr(p)p Fp(+1)1664 3726 y Fs(;)g(:)g(:)g(:)33 b(;)17 b(B)1974 3741 y Fr(p)p Fp(+)p Fr(q)2102 3726 y Fo(g)32 b Fi(b)-5 b(e)32 b(a)h(me)-5 b(asur)g(able)31 b(c)-5 b(over)32 b(of)h Fs(M)43 b Fi(satisfying)118 3846 y(c)-5 b(onditions)31 b(1)h(and)g(2)g(of)g(L)-5 b(emma)32 b(6.1.)43 b(F)-7 b(or)31 b Ft(0)d Fs(<)f(\025)h(<)f Ft(1)32 b Fi(ther)-5 b(e)33 b(ar)-5 b(e)32 b Fs(\021)f(>)d Ft(0)k Fi(and)f Fs(c)3114 3861 y Fp(0)3182 3846 y Fs(>)c Ft(0)32 b Fi(such)g(that,)118 3967 y(if)j Fs(f)261 3982 y Fr(t)325 3967 y Fi(also)g(satis\014es)f(for)g(al)5 b(l)35 b Fs(t)28 b Fo(2)g Fs(T)234 4165 y Fi(3.)48 b Fo(k)p Fs(D)s(f)544 4180 y Fr(t)573 4165 y Ft(\()p Fs(x)p Ft(\))704 4129 y Fq(\000)p Fp(1)799 4165 y Fo(k)27 b(\024)h Fs(\025)g(<)g Ft(1)34 b Fi(for)h Fs(x)28 b Fo(2)g Fs(B)1660 4180 y Fp(1)1700 4165 y Fs(;)17 b(:)g(:)g(:)32 b(;)17 b(B)2009 4180 y Fr(p)2049 4165 y Fi(;)234 4367 y(4.)48 b Fo(k)p Fs(D)s(f)544 4382 y Fr(t)573 4367 y Ft(\()p Fs(x)p Ft(\))704 4331 y Fq(\000)p Fp(1)799 4367 y Fo(k)27 b(\024)h Ft(1)22 b(+)g Fs(\021)39 b Fi(for)c Fs(x)28 b Fo(2)g Fs(B)1644 4382 y Fr(p)p Fp(+1)1774 4367 y Fs(;)17 b(:)g(:)g(:)32 b(;)17 b(B)2083 4382 y Fr(p)p Fp(+)p Fr(q)2211 4367 y Fi(;)118 4565 y(then)35 b(we)f(have)g(for)h Fs(f)j Fo(\021)29 b Fs(f)1099 4580 y Fr(t)1124 4561 y Ff(\003)1164 4565 y Fi(,)35 b(wher)-5 b(e)34 b Fs(t)1539 4529 y Fq(\003)1614 4565 y Fi(is)g(some)h(given)f(p)-5 b(oint)34 b(in)h Fs(T)14 b Fi(,)1127 4864 y Ft(lim)j(sup)1157 4943 y Fr(n)p Fq(!)p Fp(+)p Fq(1)1457 4796 y Ft(1)p 1452 4841 59 4 v 1452 4932 a Fs(n)1543 4739 y Fr(n)p Fq(\000)p Fp(1)1537 4769 y Fh(X)1548 4979 y Fr(j)t Fp(=0)1698 4864 y Ft(log)f Fo(k)p Fs(D)s(f)11 b Ft(\()p Fs(f)2130 4817 y Fr(j)2119 4886 y(t)p 2119 4898 30 3 v 2166 4864 a Ft(\()p Fs(x)p Ft(\)\))2335 4823 y Fq(\000)p Fp(1)2429 4864 y Fo(k)28 b(\024)g(\000)p Fs(c)2731 4879 y Fp(0)3606 4864 y Ft(\(23\))1900 5251 y(33)p eop %%Page: 34 34 34 33 bop 118 548 a Fi(for)33 b Fs(\022)320 512 y Fe(N)392 548 y Fo(\002)20 b Fs(m)33 b Fi(almost)g(al)5 b(l)33 b Ft(\()p Fs(t)p 1093 564 36 4 v 1 w(;)17 b(x)p Ft(\))27 b Fo(2)h Fs(T)1458 512 y Fe(N)1529 548 y Fo(\002)20 b Fs(M)10 b Fi(,)35 b(wher)-5 b(e)32 b Fs(\022)37 b Fi(is)c(any)h(Bor)-5 b(el)33 b(pr)-5 b(ob)g(ability)33 b(me)-5 b(asur)g(e)33 b(on)g Fs(T)14 b Fi(.)118 668 y(Mor)-5 b(e)g(over)35 b(the)g(\014rst)g(hyp)-5 b(erb)g(olic)34 b(time)g(map)h Fs(h)27 b Ft(:)h Fs(T)2009 632 y Fe(N)2083 668 y Fo(\002)23 b Fs(M)38 b Fo(!)27 b Fg(Z)2511 632 y Fp(+)2603 668 y Fi(satis\014es)531 955 y Ft(\()p Fs(\022)617 914 y Fe(N)692 955 y Fo(\002)22 b Fs(m)p Ft(\))p Fo(f)p Ft(\()p Fs(t)p 1002 971 V 1 w(;)17 b(x)p Ft(\))27 b Fo(2)i Fs(T)1368 914 y Fe(N)1442 955 y Fo(\002)22 b Fs(M)39 b Ft(:)27 b Fs(h)p Ft(\()p Fs(t)p 1822 971 V 1 w(;)17 b(x)p Ft(\))27 b Fs(>)h(k)s Fo(g)g(\024)g Fs(a)2414 970 y Fr(k)2556 955 y Fi(and)2864 831 y Fq(1)2827 861 y Fh(X)2834 1073 y Fr(k)r Fp(=1)2987 955 y Fs(k)s(a)3092 970 y Fr(k)3163 955 y Fs(<)g Fo(1)118 1262 y Fi(with)35 b Ft(\()p Fs(a)419 1277 y Fr(k)462 1262 y Ft(\))500 1277 y Fr(k)577 1262 y Fi(indep)-5 b(endent)34 b(of)g(the)h(choic)-5 b(e)34 b(of)h Fs(\022)s Fi(.)118 1462 y(Pr)-5 b(o)g(of.)49 b Ft(Let)38 b Fs(\020)44 b(>)38 b Ft(0)g(b)s(e)g(the)h(constan)m(t)g(pro) m(vided)g(b)m(y)g(Lemma)e(6.1.)60 b(W)-8 b(e)39 b(\014x)f Fs(\021)k(>)37 b Ft(0)h(su\016cien)m(tly)118 1583 y(small)f(so)i(that)f Fs(\025)780 1546 y Fr(\020)820 1583 y Ft(\(1)26 b(+)g Fs(\021)t Ft(\))38 b Fo(\024)h Fs(e)1324 1546 y Fq(\000)p Fr(c)1410 1555 y Fd(0)1487 1583 y Ft(holds)f(for)h(some)f Fs(c)2196 1598 y Fp(0)2274 1583 y Fs(>)g Ft(0)h(and)g(tak)m(e)g(\()p Fs(t)p 2927 1599 V(;)17 b(x)p Ft(\))39 b(satisfying)f(\(22\).)118 1703 y(Conditions)32 b(3)g(and)h(4)f(no)m(w)h(imply)970 1879 y Fr(n)p Fq(\000)p Fp(1)973 1909 y Fh(Y)976 2119 y Fr(j)t Fp(=0)1120 2004 y Fo(k)p Fs(D)s(f)11 b Ft(\()p Fs(f)1410 1957 y Fr(j)1399 2027 y(t)p 1399 2039 30 3 v 1445 2004 a Ft(\()p Fs(x)p Ft(\)\))1614 1963 y Fq(\000)p Fp(1)1709 2004 y Fo(k)27 b(\024)i Fs(\025)1949 1963 y Fr(\020)5 b(n)2031 2004 y Ft(\(1)22 b(+)g Fs(\021)t Ft(\))2328 1963 y Fp(\(1)p Fq(\000)p Fr(\020)5 b Fp(\))p Fr(n)2583 2004 y Fo(\024)28 b Fs(e)2733 1963 y Fq(\000)p Fr(c)2819 1972 y Fd(0)2853 1963 y Fr(n)2900 2004 y Fs(:)679 b Ft(\(24\))118 2329 y(for)32 b(large)g(enough)h Fs(n)p Ft(.)43 b(This)33 b(means)g(\(24\))f(holds)g(for)g Fs(\022)2150 2292 y Fe(N)2224 2329 y Fo(\002)23 b Fs(m)33 b Ft(almost)e(ev)m(ery)j(\()p Fs(t)p 3052 2344 36 4 v(;)17 b(x)p Ft(\))28 b Fo(2)g Fs(T)3417 2292 y Fe(N)3491 2329 y Fo(\002)23 b Fs(M)10 b Ft(.)264 2449 y(W)-8 b(e)28 b(observ)m(e)g(that)f(if)e Fs(h)p Ft(\()p Fs(t)p 1151 2465 V 1 w(;)17 b(x)p Ft(\))27 b(=)h Fs(k)s Ft(,)g(then)f(1)h Fo(\024)g Fs(n)g(<)f(k)j Ft(cannot)d(b)s(e)g(h)m(yp)s(erb)s(olic)f(times)g(for)g(\()p Fs(t)p 3580 2465 V(;)17 b(x)p Ft(\).)118 2569 y(Hence)34 b(\()p Fs(t)p 446 2585 V(;)17 b(x)p Ft(\))28 b Fo(2)g Fs(I)783 2584 y Fr(n)863 2569 y Ft(for)k(all)e Fs(n)e Ft(=)f(1)p Fs(;)17 b(:)g(:)g(:)33 b(;)17 b(k)25 b Fo(\000)d Ft(1.)43 b(In)33 b(particular)602 2786 y(\()p Fs(\022)688 2745 y Fe(N)763 2786 y Fo(\002)22 b Fs(m)p Ft(\))p Fo(f)p Ft(\()p Fs(t)p 1073 2802 V 1 w(;)17 b(x)p Ft(\))27 b Fo(2)h Fs(T)1438 2745 y Fe(N)1512 2786 y Fo(\002)23 b Fs(M)38 b Ft(:)28 b Fs(h)p Ft(\()p Fs(t)p 1893 2802 V 1 w(;)17 b(x)p Ft(\))27 b(=)h Fs(k)s Fo(g)f(\024)i Ft(\()p Fs(\022)2520 2745 y Fe(N)2594 2786 y Fo(\002)23 b Fs(m)p Ft(\)\()p Fs(I)2898 2801 y Fr(k)r Fq(\000)p Fp(1)3031 2786 y Ft(\))k Fo(\021)h Fs(a)3252 2801 y Fr(k)118 3002 y Ft(and)308 2928 y Fh(P)413 3031 y Fr(k)472 3002 y Fs(k)s(a)577 3017 y Fr(k)648 3002 y Fo(\024)753 2928 y Fh(P)858 3031 y Fr(k)917 3002 y Fs(k)s(\034)1024 2966 y Fr(k)r Fq(\000)p Fp(1)1186 3002 y Fs(<)f Fo(1)p Ft(.)p 3709 3002 4 66 v 3713 2940 59 4 v 3713 3002 V 3771 3002 4 66 v 264 3196 a(No)m(w)43 b(w)m(e)g(will)c(sho)m(w)k(that)f(families)d(of)i Fs(C)1876 3160 y Fp(2)1957 3196 y Ft(maps)h(satisfying)f(conditions)g (1)h(through)f(4)h(of)118 3317 y(Lemmas)g(6.1)g(and)g(6.2)g(con)m(tain) g(op)s(en)h(sets)h(of)e(families)d(in)j(the)h Fs(C)2714 3281 y Fp(2)2796 3317 y Ft(top)s(ology)-8 b(.)71 b(Let)43 b Fs(M)53 b Ft(b)s(e)43 b(a)118 3437 y Fs(n)p Ft(-dimensional)38 b(torus)j Fg(T)1075 3401 y Fr(n)1166 3437 y Ft(and)g Fs(f)1412 3452 y Fp(0)1493 3437 y Ft(:)h Fs(M)52 b Fo(!)42 b Fs(M)51 b Ft(a)41 b(uniformly)d(expanding)j(map:)59 b(there)42 b(exists)118 3557 y(0)36 b Fs(<)g(\025)g(<)g Ft(1)h(suc)m(h)i(that)e Fo(k)p Fs(D)s(f)1229 3572 y Fp(0)1268 3557 y Ft(\()p Fs(x)p Ft(\))p Fs(v)t Fo(k)f(\025)g Fs(\025)1706 3521 y Fq(\000)p Fp(1)1801 3557 y Fo(k)p Fs(v)t Fo(k)g Ft(for)h(all)f Fs(x)g Fo(2)h Fs(M)48 b Ft(and)37 b Fs(v)j Fo(2)d Fs(T)3060 3572 y Fr(x)3104 3557 y Fs(M)10 b Ft(.)59 b(Let)37 b(also)g Fs(W)118 3678 y Ft(b)s(e)g(some)f(small)d(compact)j (domain)f(in)g Fs(M)47 b Ft(where)38 b Fs(f)2098 3693 y Fp(0)2171 3678 y Fo(j)33 b Fs(W)50 b Ft(is)36 b(injectiv)m(e.)54 b(Observ)m(e)38 b(that)e Fs(f)3553 3693 y Fp(0)3629 3678 y Ft(is)g(a)118 3798 y(v)m(olume)c(expanding)h(lo)s(cal)d (di\013eomorphism)g(due)k(to)e(the)h(uniform)e(expansion.)264 3919 y(Mo)s(difying)d Fs(f)775 3934 y Fp(0)844 3919 y Ft(b)m(y)i(an)f(isotop)m(y)g(inside)f Fs(W)43 b Ft(w)m(e)30 b(ma)m(y)f(obtain)f(a)h(map)g Fs(f)2837 3934 y Fp(1)2905 3919 y Ft(whic)m(h)h(coincides)f(with)118 4039 y Fs(f)166 4054 y Fp(0)236 4039 y Ft(outside)i Fs(W)14 b Ft(,)31 b(is)g(v)m(olume)f(expanding)h(in)f Fs(M)10 b Ft(,)32 b(i.e.,)f Fo(j)17 b Ft(det)g Fs(D)s(f)2420 4054 y Fp(1)2459 4039 y Ft(\()p Fs(x)p Ft(\))p Fo(j)27 b Fs(>)h Ft(1)j(for)f(all)f Fs(x)f Fo(2)g Fs(M)10 b Ft(,)32 b(and)f(has)118 4159 y(b)s(ounded)40 b(con)m(traction)f(on)g Fs(W)53 b Ft(near)40 b(1:)56 b Fo(k)p Fs(D)s(f)1867 4174 y Fp(1)1906 4159 y Ft(\()p Fs(x)p Ft(\))2037 4123 y Fq(\000)p Fp(1)2132 4159 y Fo(k)39 b(\024)g Ft(1)27 b(+)f Fs(\021)43 b Ft(for)c(ev)m(ery)i Fs(x)f Fo(2)f Fs(W)53 b Ft(and)40 b(some)118 4280 y Fs(\021)31 b(>)d Ft(0)k(small.)41 b(This)32 b(new)h(map)e Fs(f)1361 4295 y Fp(1)1433 4280 y Ft(ma)m(y)g(b)s(e)i(tak)m(en)g Fs(C)2118 4244 y Fp(1)2189 4280 y Ft(close)f(to)g Fs(f)2589 4295 y Fp(0)2660 4280 y Ft(and)g(w)m(e)h(ma)m(y)f(consider)g(a)g Fs(C)3740 4244 y Fp(2)118 4400 y Ft(map)g Fs(f)383 4415 y Fp(2)455 4400 y Ft(arbitrarily)e Fs(C)1001 4364 y Fp(1)1073 4400 y Ft(close)i(to)g Fs(f)1473 4415 y Fp(1)1513 4400 y Ft(.)264 4521 y(No)m(w)j(an)m(y)g(map)f Fs(f)45 b Ft(in)33 b(a)h(small)e(enough)j Fs(C)1856 4484 y Fp(2)1929 4521 y Ft(neigh)m(b)s(orho)s(o)s(d)e(of)h Fs(f)2704 4536 y Fp(2)2778 4521 y Ft(admits)f Fs(\033)i(>)30 b Ft(1)k(suc)m(h)i(that)118 4641 y Fo(j)17 b Ft(det)f Fs(D)s(f)11 b Ft(\()p Fs(x)p Ft(\))p Fo(j)31 b(\025)h Fs(\033)39 b Ft(for)34 b(all)f Fs(x)e Fo(2)h Fs(M)46 b Ft(and,)35 b(for)f Fs(x)h Ft(outside)g Fs(W)14 b Ft(,)35 b(w)m(e)h(ha)m(v)m(e)g Fo(k)p Fs(D)s(f)11 b Ft(\()p Fs(x)p Ft(\))3127 4605 y Fq(\000)p Fp(1)3221 4641 y Fo(k)31 b(\024)h Fs(\025)p Ft(.)49 b(If)35 b(the)118 4761 y Fs(C)195 4725 y Fp(2)270 4761 y Ft(neigh)m(b)s(orho)s(o)s(d)g (is)g(tak)m(en)i(su\016cien)m(tly)g(small)c(then)j(w)m(e)h(main)m(tain) d Fo(k)p Fs(D)s(f)11 b Ft(\()p Fs(x)p Ft(\))3115 4725 y Fq(\000)p Fp(1)3209 4761 y Fo(k)33 b(\024)g Ft(1)24 b(+)h Fs(\021)39 b Ft(for)118 4882 y Fs(x)g Fo(2)f Fs(W)52 b Ft(and)39 b(for)f(some)h(small)d Fs(\021)42 b(>)c Ft(0.)62 b(Let)39 b(us)g(tak)m(e)g Fs(B)2268 4897 y Fp(1)2308 4882 y Fs(;)17 b(:)g(:)g(:)32 b(;)17 b(B)2617 4897 y Fr(p)2657 4882 y Fs(;)g(B)2775 4897 y Fr(p)p Fp(+1)2943 4882 y Ft(=)38 b Fs(W)52 b Ft(a)38 b(partition)f(of)118 5002 y Fs(M)49 b Ft(in)m(to)37 b(measurable)h(sets)h(where)g(the)g (restriction)e Fs(f)47 b Fo(j)37 b Fs(B)2343 5017 y Fr(i)2409 5002 y Ft(is)h(injectiv)m(e)g(for)f Fs(i)g Ft(=)g(1)p Fs(;)17 b(:)g(:)g(:)32 b(;)17 b(p)26 b Ft(+)g(1.)1900 5251 y(34)p eop %%Page: 35 35 35 34 bop 118 548 a Ft(Then)40 b(an)m(y)g(con)m(tin)m(uous)g(family)c (of)j Fs(C)1564 512 y Fp(2)1642 548 y Ft(maps)g(\010)g(:)g Fs(T)53 b Fo(!)38 b Fs(C)2404 512 y Fp(2)2443 548 y Ft(\()p Fs(M)5 b(;)17 b(M)10 b Ft(\))40 b(together)f(with)g(a)g(family)118 668 y(\()p Fs(\022)201 683 y Fr(\017)234 668 y Ft(\))272 683 y Fr(\017>)p Fp(0)423 668 y Ft(of)28 b(Borel)f(probabilit)m(y)g (measures)i(in)e(the)i(metric)e(space)i Fs(T)14 b Ft(,)29 b(satisfying)f(supp)17 b(\()p Fs(\022)3379 683 y Fr(\017)3412 668 y Ft(\))28 b Fo(!)f(f)p Fs(t)3690 632 y Fq(\003)3730 668 y Fo(g)118 789 y Ft(when)37 b Fs(\017)c Fo(!)f Ft(0)j(and)g Fs(f)904 804 y Fr(t)929 785 y Ff(\003)1002 789 y Fo(\021)e Fs(f)11 b Ft(,)36 b(for)f(some)g Fs(t)1668 753 y Fq(\003)1741 789 y Fo(2)e Fs(T)14 b Ft(,)35 b(is)g(suc)m(h)i(that)e Fs(f)46 b Ft(is)35 b(non-uniformly)e(expanding)118 909 y(for)41 b(random)g(orbits)g(and)g(\()p Fs(h)1221 924 y Fr(\017)1254 909 y Ft(\))1292 924 y Fr(\017>)p Fp(0)1456 909 y Ft(has)h(uniform)e Fs(L)2080 873 y Fp(1)2120 909 y Ft(-tail)f({)i(b)m(y)h(Lemma)e(6.2)h(with)h Fs(q)k Ft(=)d(1)e(and)118 1029 y Fs(T)h Ft(=)27 b(supp)18 b(\()p Fs(\022)621 1044 y Fr(\017)654 1029 y Ft(\))32 b(for)g(small)f(enough)i Fs(\017)28 b(>)f Ft(0.)43 b(Theorem)33 b(B)g(then)g(sho)m(ws)118 1233 y Fc(Corollary)j(6.3.)49 b Fi(Ther)-5 b(e)36 b(ar)-5 b(e)37 b(op)-5 b(en)37 b(sets)g Fo(U)k(\032)33 b Fs(C)2005 1197 y Fp(2)2044 1233 y Ft(\()p Fs(M)5 b(;)17 b(M)10 b Ft(\))38 b Fi(of)f(maps)f(in)g(the)i Fs(C)3141 1197 y Fp(2)3217 1233 y Fi(top)-5 b(olo)g(gy)37 b(such)118 1353 y(that)k(every)f(map)f Fs(f)48 b Fo(2)38 b(U)50 b Fi(is)40 b(a)f(sto)-5 b(chastic)g(al)5 b(ly)40 b(stable)g (non-uniformly)f(exp)-5 b(anding)38 b(lo)-5 b(c)g(al)40 b(di\013e)-5 b(o-)118 1474 y(morphism.)118 1762 y Fj(6.2)135 b(Viana)46 b(maps)118 1947 y Ft(In)36 b(what)g(follo)m(ws)e(w)m(e)j (describ)s(e)f(the)g(class)g(of)f(non-uniformly)e(expanding)j(maps)f (\(with)g(critical)118 2068 y(sets\))j(in)m(tro)s(duced)e(b)m(y)i(M.)f (Viana,)f(referring)g(the)h(reader)g(to)f([Vi1)o(],)h([Al])f(and)h([A) -11 b(V])37 b(for)e(details.)118 2188 y(Then)f(w)m(e)f(sho)m(w)h(that)f (those)g(maps)f(satisfy)h(the)g(h)m(yp)s(otheses)i(of)d(Theorem)h(C)f (and)h(Theorem)g(D.)264 2308 y(Let)28 b Fs(a)485 2323 y Fp(0)553 2308 y Fo(2)g Ft(\(1)p Fs(;)17 b Ft(2\))27 b(b)s(e)h(suc)m(h)h(that)e(the)i(critical)c(p)s(oin)m(t)i Fs(x)h Ft(=)f(0)h(is)f(pre-p)s(erio)s(dic)f(for)i(the)g(quadratic)118 2429 y(map)j Fs(Q)p Ft(\()p Fs(x)p Ft(\))d(=)f Fs(a)724 2444 y Fp(0)783 2429 y Fo(\000)20 b Fs(x)935 2393 y Fp(2)975 2429 y Ft(.)43 b(Let)31 b Fs(S)1284 2393 y Fp(1)1351 2429 y Ft(=)d Fg(R)5 b Fs(=)p Fg(Z)34 b Ft(and)e Fs(b)c Ft(:)f Fs(S)2051 2393 y Fp(1)2118 2429 y Fo(!)h Fg(R)42 b Ft(b)s(e)31 b(a)g(Morse)h(function,)f(for)g(instance,)118 2549 y Fs(b)p Ft(\()p Fs(s)p Ft(\))d(=)g(sin)o(\(2)p Fs(\031)t(s)p Ft(\).)43 b(F)-8 b(or)32 b(\014xed)i(small)c Fs(\013)e(>)g Ft(0,)k(consider)h(the)g(map)1272 2751 y(^)1250 2777 y Fs(f)39 b Ft(:)83 b Fs(S)1513 2741 y Fp(1)1574 2777 y Fo(\002)23 b Fg(R)94 b Fo(\000)-16 b(!)220 b Fs(S)2276 2741 y Fp(1)2338 2777 y Fo(\002)22 b Fg(R)1486 2898 y Ft(\()p Fs(s;)17 b(x)p Ft(\))122 b Fo(7\000)-16 b(!)2072 2818 y Fh(\000)2121 2898 y Ft(^)-52 b Fs(g)s Ft(\()p Fs(s)p Ft(\))p Fs(;)24 b Ft(^)-56 b Fs(q)t Ft(\()p Fs(s;)17 b(x)p Ft(\))2602 2818 y Fh(\001)118 3117 y Ft(where)34 b(^)-52 b Fs(g)32 b Ft(is)d(the)h(uniformly)d(expanding)j(map)e(of)h (the)h(circle)f(de\014ned)i(b)m(y)i(^)-52 b Fs(g)s Ft(\()p Fs(s)p Ft(\))28 b(=)f Fs(ds)i Ft(\(mo)s(d)f Fg(Z)p Ft(\))f(for)118 3237 y(some)35 b Fs(d)30 b Fo(\025)i Ft(16,)j(and)41 b(^)-56 b Fs(q)t Ft(\()p Fs(s;)17 b(x)p Ft(\))31 b(=)g Fs(a)p Ft(\()p Fs(s)p Ft(\))24 b Fo(\000)g Fs(x)1665 3201 y Fp(2)1739 3237 y Ft(with)34 b Fs(a)p Ft(\()p Fs(s)p Ft(\))d(=)g Fs(a)2325 3252 y Fp(0)2388 3237 y Ft(+)24 b Fs(\013)q(b)p Ft(\()p Fs(s)p Ft(\).)49 b(It)35 b(is)f(easy)i(to)e(c)m (hec)m(k)j(that)118 3358 y(for)27 b Fs(\013)i(>)e Ft(0)h(small)d (enough)k(there)f(is)g(an)g(in)m(terv)-5 b(al)26 b Fs(I)36 b Fo(\032)28 b Ft(\()p Fo(\000)p Ft(2)p Fs(;)17 b Ft(2\))27 b(for)h(whic)m(h)2877 3331 y(^)2856 3358 y Fs(f)11 b Ft(\()p Fs(S)3019 3322 y Fp(1)3070 3358 y Fo(\002)i Fs(I)8 b Ft(\))28 b(is)f(con)m(tained)118 3478 y(in)36 b(the)h(in)m(terior)e (of)h Fs(S)937 3442 y Fp(1)1001 3478 y Fo(\002)25 b Fs(I)8 b Ft(.)56 b(Th)m(us,)39 b(an)m(y)e(map)f Fs(f)47 b Ft(su\016cien)m(tly) 37 b(close)g(to)2894 3452 y(^)2873 3478 y Fs(f)47 b Ft(in)36 b(the)h Fs(C)3335 3442 y Fp(0)3411 3478 y Ft(top)s(ology)118 3599 y(has)31 b Fs(S)356 3562 y Fp(1)412 3599 y Fo(\002)18 b Fs(I)38 b Ft(as)30 b(a)g(forw)m(ard)h(in)m(v)-5 b(arian)m(t)28 b(region.)42 b(W)-8 b(e)31 b(consider)f(from)f(here)i(on)f(these)i (maps)e Fs(f)40 b Ft(close)118 3719 y(to)259 3693 y(^)237 3719 y Fs(f)k Ft(restricted)33 b(to)f Fs(S)948 3683 y Fp(1)1009 3719 y Fo(\002)23 b Fs(I)8 b Ft(.)264 3839 y(T)-8 b(aking)33 b(in)m(to)f(accoun)m(t)h(the)g(expression)h(of)1918 3813 y(^)1897 3839 y Fs(f)43 b Ft(it)32 b(is)g(not)h(di\016cult)e(to)i (c)m(hec)m(k)h(that)3330 3813 y(^)3309 3839 y Fs(f)43 b Ft(\(and)33 b(an)m(y)118 3960 y(map)e Fs(f)42 b Ft(close)32 b(to)797 3933 y(^)775 3960 y Fs(f)43 b Ft(in)31 b(the)h Fs(C)1223 3924 y Fp(2)1294 3960 y Ft(top)s(ology\))e(b)s(eha)m(v)m(es)k (lik)m(e)d(a)g(p)s(o)m(w)m(er)i(of)e(the)h(distance)g(close)g(to)f(the) 118 4080 y(critical)f(set.)118 4340 y Fc(6.2.1)113 b(Non-uniform)36 b(expansion)118 4525 y Ft(The)31 b(results)f(in)g([Vi1)o(])g(sho)m(w)h (that)f(if)e(the)j(map)e Fs(f)41 b Ft(is)29 b(su\016cien)m(tly)i(close) f(to)2921 4498 y(^)2899 4525 y Fs(f)41 b Ft(in)29 b(the)i Fs(C)3342 4488 y Fp(3)3411 4525 y Ft(top)s(ology)118 4645 y(then)39 b Fs(f)48 b Ft(has)39 b(t)m(w)m(o)f(p)s(ositiv)m(e)g(Ly) m(apuno)m(v)h(exp)s(onen)m(ts)h(almost)c(ev)m(erywhere:)58 b(there)38 b(is)g(a)g(constan)m(t)118 4765 y Fs(\025)28 b(>)f Ft(0)33 b(for)f(whic)m(h)1312 4908 y(lim)17 b(inf)1327 4968 y Fr(n)p Fq(!)p Fp(+)p Fq(1)1613 4840 y Ft(1)p 1608 4885 59 4 v 1608 4976 a Fs(n)1693 4908 y Ft(log)g Fo(k)p Fs(D)s(f)2029 4867 y Fr(n)2075 4908 y Ft(\()p Fs(s;)g(x)p Ft(\))p Fs(v)t Fo(k)27 b(\025)h Fs(\025)1900 5251 y Ft(35)p eop %%Page: 36 36 36 35 bop 118 548 a Ft(for)27 b(Leb)s(esgue)i(almost)c(ev)m(ery)k(\()p Fs(s;)17 b(x)p Ft(\))28 b Fo(2)g Fs(S)1649 512 y Fp(1)1700 548 y Fo(\002)11 b Fs(I)36 b Ft(and)27 b(ev)m(ery)j(non-zero)d Fs(v)k Fo(2)e Fs(T)2924 563 y Fp(\()p Fr(s;x)p Fp(\))3075 548 y Ft(\()p Fs(S)3179 512 y Fp(1)3229 548 y Fo(\002)11 b Fs(I)d Ft(\).)43 b(In)28 b(fact,)118 668 y(the)h(metho)s(d)f(used)i (for)e(sho)m(wing)h(this)g(result)f(also)g(giv)m(es)h(that)g Fs(f)39 b Ft(is)29 b(a)f(non-uniformly)e(expanding)118 789 y(map)k(and)h(its)f(orbits)g(ha)m(v)m(e)j(slo)m(w)d(appro)m (ximation)f(to)h(the)i(critical)c(set,)k(as)f(w)m(e)h(no)m(w)f (explain.)42 b(F)-8 b(or)118 909 y(the)33 b(sak)m(e)h(of)e(clearness,)i (w)m(e)f(start)g(b)m(y)g(assuming)f(that)h Fs(f)43 b Ft(has)33 b(the)g(sp)s(ecial)e(form)466 1129 y Fs(f)11 b Ft(\()p Fs(s;)17 b(x)p Ft(\))27 b(=)h(\()p Fs(g)t Ft(\()p Fs(s)p Ft(\))p Fs(;)17 b(q)t Ft(\()p Fs(s;)g(x)p Ft(\)\))p Fs(;)112 b Ft(with)98 b Fs(@)1916 1144 y Fr(x)1960 1129 y Fs(q)t Ft(\()p Fs(s;)17 b(x)p Ft(\))27 b(=)h(0)97 b(if)32 b(and)g(only)g(if)97 b Fs(x)28 b Ft(=)f(0)p Fs(;)191 b Ft(\(25\))118 1349 y(and)33 b(describ)s(e)g(ho)m(w)g(the)g (conclusions)g(in)f([Vi1)o(])g(are)h(obtained)f(for)g(eac)m(h)h 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Fo(\002)23 b Ft(\()p Fo(\000)1360 2319 y(p)p 1443 2319 63 4 v 72 x Fs(\013)q(;)1550 2319 y Fo(p)p 1633 2319 V 72 x Fs(\013)17 b Ft(\))32 b(of)g(the)h(critical)e(set)i Fo(C)h Ft(=)27 b Fo(f)p Ft(\()p Fs(s;)17 b(x)p Ft(\))28 b(:)g Fs(x)g Ft(=)f(0)p Fo(g)p Ft(.)43 b(W)-8 b(e)33 b(set)492 2611 y Fs(J)9 b Ft(\(0\))27 b(=)h Fs(I)i Fo(n)22 b Ft(\()p Fo(\000)1071 2534 y(p)p 1154 2534 V 77 x Fs(\013)q(;)1261 2534 y Fo(p)p 1344 2534 V 77 x Fs(\013)p Ft(\))98 b(and)g Fs(J)9 b Ft(\()p Fs(r)s Ft(\))27 b(=)h Fo(f)p Fs(x)f Fo(2)i Fs(I)35 b Ft(:)28 b Fo(j)p Fs(x)p Fo(j)f Fs(<)h(e)2761 2569 y Fq(\000)p Fr(r)2854 2611 y Fo(g)97 b Ft(for)32 b Fs(r)f Fo(\025)d Ft(0)p Fs(:)118 2830 y Ft(F)-8 b(rom)41 b(here)h(on)g(w)m(e)h(only)f(consider)g(p)s(oin)m(ts)g(\()p Fs(s;)17 b(x)p Ft(\))44 b Fo(2)g Fs(S)2259 2794 y Fp(1)2327 2830 y Fo(\002)29 b Fs(I)49 b Ft(whose)44 b(orbit)d(do)s(es)h(not)g (hit)f(the)118 2951 y(critical)28 b(set)i Fo(C)6 b Ft(.)43 b(This)30 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Fs(n)p Ft(\)\))28 b Fo(\024)g Ft(const)17 b Fs(e)2246 4925 y Fq(\000)2301 4877 y(p)p 2360 4877 43 3 v 48 x Fr(n=)p Fp(4)3606 4966 y Ft(\(27\))1900 5251 y(36)p eop %%Page: 37 37 37 36 bop 118 548 a Ft(and,)33 b(for)f(ev)m(ery)i(small)d Fs(c)c(>)h Ft(0)k(\(only)g(dep)s(ending)h(on)f(the)h(quadratic)g(map)e Fs(Q)p Ft(\),)651 844 y(log)794 719 y Fr(n)p Fq(\000)p Fp(1)797 749 y Fh(Y)799 959 y Fr(j)t Fp(=0)944 844 y Fo(j)p Fs(@)1023 859 y Fr(x)1067 844 y Fs(q)t Ft(\()p Fs(s)1198 859 y Fr(j)1234 844 y Fs(;)17 b(x)1333 859 y Fr(j)1369 844 y Ft(\))p Fo(j)28 b(\025)g Ft(2)p Fs(cn)22 b Fo(\000)1839 749 y Fh(X)1843 961 y Fr(j)t Fq(2)p Fr(G)1999 844 y Fs(r)2043 859 y Fr(j)2080 844 y Ft(\()p Fs(s;)17 b(x)p Ft(\))97 b(for)g(\()p Fs(s;)17 b(x)p Ft(\))39 b Fs(=)-60 b Fo(2)28 b Fs(B)3029 859 y Fp(2)3068 844 y Ft(\()p Fs(n)p Ft(\))p Fs(;)377 b Ft(\(28\))118 1152 y(see)34 b([Vi1)o(,)f(pp.)43 b(75)32 b(&)h(76].)43 b(Moreo)m(v)m(er,)34 b(if)e(w)m(e)h(de\014ne)h(for)e Fs(\015)h(>)27 b Ft(0)980 1415 y Fs(B)1054 1430 y Fp(1)1094 1415 y Ft(\()p Fs(n)p Ft(\))h(=)1359 1274 y Fh(\032)1434 1415 y Ft(\()p Fs(s;)17 b(x)p Ft(\))39 b Fs(=)-61 b Fo(2)29 b Fs(B)1851 1430 y Fp(2)1890 1415 y Ft(\()p Fs(n)p Ft(\))f(:)2107 1320 y Fh(X)2111 1531 y Fr(j)t Fq(2)p Fr(G)2267 1415 y Fs(r)2311 1430 y Fr(j)2348 1415 y Ft(\()p Fs(s;)17 b(x)p Ft(\))27 b Fo(\025)h Fs(\015)5 b(n)2815 1274 y Fh(\033)2890 1415 y Fs(;)118 1717 y Ft(then,)33 b(for)f(small)f Fs(\015)h(>)c Ft(0,)k(there)i(is)e(a)g(constan)m(t)h Fs(\030)g(>)27 b Ft(0)32 b(for)g(whic)m(h)1558 1923 y Fs(m)1643 1842 y Fh(\000)1689 1923 y Fs(B)1763 1938 y Fp(1)1803 1923 y Ft(\()p Fs(n)p Ft(\))1937 1842 y Fh(\001)2010 1923 y Fo(\024)c Fs(e)2160 1882 y Fq(\000)p Fr(\030)s(n)2296 1923 y Fs(;)1283 b Ft(\(29\))118 2128 y(see)35 b([Vi1)o(,)g(p.)47 b(77].)g(T)-8 b(aking)34 b(in)m(to)f(accoun)m(t)i(the)f(de\014nitions)f (of)h Fs(J)9 b Ft(\()p Fs(r)s Ft(\))33 b(and)h Fs(r)2939 2143 y Fr(j)2976 2128 y Ft(,)g(this)f(sho)m(ws)j(that)e(if)118 2249 y(w)m(e)g(tak)m(e)f 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b Ft(admits)42 b(a)g(unique)i(in)m(v)-5 b(arian)m(t)118 2489 y(cen)m(tral)37 b(foliation)c Fo(F)913 2452 y Fr(c)983 2489 y Ft(of)k Fs(S)1165 2452 y Fp(1)1229 2489 y Fo(\002)25 b Fs(I)44 b Ft(b)m(y)38 b(smo)s(oth)e(curv)m(es)i(uniformly)d(close)h(to)g(v)m (ertical)g(segmen)m(ts,)118 2609 y(see)d([Vi1)o(,)f(Section)f(2.5].)43 b(Actually)-8 b(,)31 b Fo(F)1568 2573 y Fr(c)1633 2609 y Ft(is)g(obtained)g(as)h(the)g(set)g(of)g(in)m(tegral)d(curv)m(es)34 b(of)d(a)g(v)m(ector)118 2729 y(\014eld)k(\()p Fs(\030)418 2693 y Fr(c)452 2729 y Fs(;)17 b Ft(1\))34 b(in)g Fs(S)799 2693 y Fp(1)862 2729 y Fo(\002)25 b Fs(I)42 b Ft(with)35 b Fs(\030)1322 2693 y Fr(c)1391 2729 y Ft(uniformly)e(close)i(to)f (zero.)51 b(The)36 b(previous)f(analysis)g(can)g(then)118 2850 y(b)s(e)42 b(carried)f(out)g(in)f(terms)i(of)f(the)g(expansion)h (of)f Fs(f)52 b Ft(along)40 b(this)h(cen)m(tral)g(foliation)d Fo(F)3429 2814 y Fr(c)3463 2850 y Ft(.)70 b(More)118 2970 y(precisely)-8 b(,)33 b Fo(j)p Fs(@)615 2985 y Fr(x)659 2970 y Fs(q)t Ft(\()p Fs(s;)17 b(x)p Ft(\))p Fo(j)32 b Ft(is)g(replaced)h(b)m(y)1303 3168 y Fo(j)p Fs(@)1382 3183 y Fr(c)1417 3168 y Fs(q)t Ft(\()p Fs(s;)17 b(x)p Ft(\))p Fo(j)27 b(\021)i(j)p Fs(D)s(f)11 b Ft(\()p Fs(s;)17 b(x)p Ft(\))p Fs(v)2285 3183 y Fr(c)2319 3168 y Ft(\()p Fs(s;)g(x)p Ft(\))p Fo(j)p Fs(;)118 3367 y Ft(where)33 b Fs(v)446 3382 y Fr(c)481 3367 y Ft(\()p Fs(s;)17 b(x)p Ft(\))32 b(is)g(a)f(unit)h(v)m(ector)h(tangen)m(t)f(to)g(the)h (foliation)28 b(at)k(\()p Fs(s;)17 b(x)p Ft(\).)43 b(The)33 b(previous)f(observ)-5 b(a-)118 3487 y(tions)38 b(imply)e(that)h Fs(v)905 3502 y Fr(c)978 3487 y Ft(is)h(uniformly)e(close)i(to)f(\(0)p Fs(;)17 b Ft(1\))37 b(if)g Fs(f)49 b Ft(is)37 b(close)h(to)2830 3461 y(^)2809 3487 y Fs(f)11 b Ft(.)59 b(Moreo)m(v)m(er,)41 b(cf.)60 b([Vi1)o(,)118 3607 y(Section)32 b(2.5],)g(it)f(is)g(no)h (restriction)f(to)g(supp)s(ose)i Fo(j)p Fs(@)2028 3622 y Fr(c)2063 3607 y Fs(q)t Ft(\()p Fs(s;)17 b Ft(0\))p Fo(j)27 b(\021)h Ft(0,)k(so)g(that)f Fs(@)2973 3622 y Fr(c)3009 3607 y Fs(q)t Ft(\()p Fs(s;)17 b(x)p Ft(\))27 b Fo(\031)h(j)p Fs(x)p Fo(j)p Ft(,)k(as)g(in)118 3728 y(the)h(unp)s(erturb)s(ed)h(case.)44 b(Indeed,)34 b(if)e(w)m(e)h (de\014ne)h(the)f Fi(critic)-5 b(al)35 b(set)d Ft(of)g Fs(f)44 b Ft(b)m(y)1172 3926 y Fo(C)34 b Ft(=)28 b Fo(f)p Ft(\()p Fs(s;)17 b(x)p Ft(\))27 b Fo(2)h Fs(S)1820 3885 y Fp(1)1882 3926 y Fo(\002)22 b Fs(I)36 b Ft(:)28 b Fs(@)2166 3941 y Fr(c)2201 3926 y Fs(q)t Ft(\()p Fs(s;)17 b(x)p Ft(\))27 b(=)h(0)p Fo(g)p Fs(:)118 4124 y Ft(b)m(y)40 b(an)e(easy)i(implicit)34 b(function)k(argumen)m(t)h(it)e(is)h(sho)m (wn)i(in)e([Vi1)o(,)i(Section)f(2.5])f(that)g Fo(C)45 b Ft(is)38 b(the)118 4245 y(graph)29 b(of)f(some)g Fs(C)815 4208 y Fp(2)884 4245 y Ft(map)f Fs(\021)32 b Ft(:)c Fs(S)1297 4208 y Fp(1)1364 4245 y Fo(!)f Fs(I)36 b Ft(arbitrarily)26 b Fs(C)2112 4208 y Fp(2)2152 4245 y Ft(-close)i(to)g(zero)h(if)f Fs(\013)h Ft(is)g(small.)40 b(This)28 b(means)118 4365 y(that)i(up)g(to)g(a)f(c)m(hange)i(of)f(co)s(ordinates)f Fs(C)1681 4329 y Fp(2)1720 4365 y Ft(-close)h(to)f(the)i(iden)m(tit)m (y)f(w)m(e)h(ma)m(y)e(supp)s(ose)i(that)f Fs(\021)i Fo(\021)c Ft(0)118 4485 y(and,)33 b(hence,)h(write)e(for)g Fs(\013)d(>)e Ft(0)32 b(small)936 4683 y Fs(@)987 4698 y Fr(c)1022 4683 y Fs(q)t Ft(\()p Fs(s;)17 b(x)p Ft(\))28 b(=)f Fs(x )t Ft(\()p Fs(s;)17 b(x)p Ft(\))98 b(with)32 b Fo(j)p Fs( )26 b Ft(+)c(2)p Fo(j)32 b Ft(close)h(to)f(zero)p Fs(:)118 4882 y Ft(This)44 b(pro)m(vides)g(an)f(analog)f(to)h(the)h(second)h (part)e(of)g(assumption)f(\(25\).)76 b(A)m(t)44 b(this)f(p)s(oin)m(t,)i (the)118 5002 y(argumen)m(ts)d(apply)f(with)g Fs(@)1161 5017 y Fr(x)1206 5002 y Fs(q)t Ft(\()p Fs(s;)17 b(x)p Ft(\))41 b(replaced)h(b)m(y)g Fs(@)2104 5017 y Fr(c)2140 5002 y Fs(q)t Ft(\()p Fs(s;)17 b(x)p Ft(\),)43 b(to)e(sho)m(w)i(that)f (orbits)f(ha)m(v)m(e)h(slo)m(w)1900 5251 y(38)p eop %%Page: 39 39 39 38 bop 118 548 a Ft(appro)m(ximation)34 b(to)i(the)h(critical)c(set) k Fo(C)43 b Ft(and)1839 473 y Fh(Q)1933 500 y Fr(n)p Fq(\000)p Fp(1)1933 577 y Fr(i)p Fp(=0)2087 548 y Fo(j)p Fs(@)2166 563 y Fr(c)2201 548 y Fs(q)t Ft(\()p Fs(s)2332 563 y Fr(i)2360 548 y Fs(;)17 b(x)2459 563 y Fr(i)2487 548 y Ft(\))p Fo(j)36 b Ft(gro)m(ws)h(exp)s(onen)m(tially)e(fast)h(for) 118 687 y(Leb)s(esgue)f(almost)c(ev)m(ery)k(\()p Fs(s;)17 b(x)p Ft(\))29 b Fo(2)g Fs(S)1525 650 y Fp(1)1587 687 y Fo(\002)23 b Fs(I)8 b Ft(.)46 b(A)33 b(matrix)f(form)m(ula)f(for)2743 606 y Fh(\000)2788 687 y Fs(D)s(f)2931 650 y Fr(n)2978 687 y Ft(\()p Fs(s;)17 b(x)p Ft(\))3199 606 y Fh(\001)3244 628 y Fq(\000)p Fp(1)3372 687 y Ft(similar)30 b(to)118 807 y(that)f(in)g(\(30\))g(can)h(b)s(e)f(obtained)g(if)g(w)m(e)h (replace)g(the)g(v)m(ector)g(\(0)p Fs(;)17 b Ft(1\))29 b(in)g(the)h(canonical)e(basis)h(of)g(the)118 941 y(space)k(tangen)m(t) f(to)f Fs(S)916 905 y Fp(1)976 941 y Fo(\002)20 b Fs(I)40 b Ft(at)31 b(\()p Fs(s;)17 b(x)p Ft(\))31 b(b)m(y)i Fs(v)1708 956 y Fr(c)1743 941 y Ft(\()p Fs(s;)17 b(x)p Ft(\),)32 b(and)f(consider)h(the)g(matrix)e(of)3184 861 y Fh(\000)3229 941 y Fs(D)s(f)3372 905 y Fr(n)3419 941 y Ft(\()p Fs(s;)17 b(x)p Ft(\))3640 861 y Fh(\001)3685 883 y Fq(\000)p Fp(1)118 1062 y Ft(with)32 b(resp)s(ect)i(to)e(the)h(new)h(basis.)264 1220 y(F)-8 b(or)40 b(future)h(reference,)j(let)c(us)i(mak)m(e)e(some)h (considerations)f(on)g(the)h(w)m(a)m(y)h(the)f(sets)h Fs(B)3606 1235 y Fp(1)3645 1220 y Ft(\()p Fs(n)p Ft(\))118 1340 y(and)36 b Fs(B)385 1355 y Fp(2)424 1340 y Ft(\()p Fs(n)p Ft(\))g(are)f(obtained.)52 b(Let)36 b Fs(X)k Ft(:)33 b Fs(S)1632 1304 y Fp(1)1704 1340 y Fo(!)f Fs(I)44 b Ft(b)s(e)35 b(a)h(smo)s(oth)e(map)h(whose)i(graph)e(in)g Fs(S)3463 1304 y Fp(1)3526 1340 y Fo(\002)25 b Fs(I)43 b Ft(is)118 1460 y(nearly)38 b(horizon)m(tal)f(\(see)i(the)f(notion)g (of)f(admissible)f(curv)m(e)k(in)e([Vi1)o(,)h(Section)f(2])g(for)g(a)g (precise)118 1591 y(de\014nition\).)65 b(Denote)1018 1565 y Fh(b)993 1591 y Fs(X)1074 1606 y Fr(n)1121 1591 y Ft(\()p Fs(s)p Ft(\))40 b(=)h Fs(f)1459 1555 y Fr(n)1505 1510 y Fh(\000)1551 1591 y Fs(s;)17 b(X)8 b Ft(\()p Fs(s)p Ft(\))1852 1510 y Fh(\001)1937 1591 y Ft(for)40 b Fs(n)h Fo(\025)g Ft(0)f(and)g Fs(s)h Fo(2)g Fs(S)2857 1555 y Fp(1)2896 1591 y Ft(.)66 b(T)-8 b(ak)m(e)42 b(some)e(leaf)f Fs(L)3740 1606 y Fp(0)118 1711 y Ft(of)e(the)h(foliation)d Fo(F)877 1675 y Fr(c)911 1711 y Ft(.)59 b(Letting)37 b Fs(L)1411 1726 y Fr(n)1494 1711 y Ft(=)g Fs(f)1666 1675 y Fr(n)1712 1711 y Ft(\()p Fs(L)1816 1726 y Fp(0)1856 1711 y Ft(\))h(for)f Fs(n)g Fo(\025)f Ft(1,)j(w)m(e)g(de\014ne)g(a)e (sequence)k(of)c(Mark)m(o)m(v)118 1831 y(partitions)31 b(\()p Fo(P)670 1846 y Fr(n)717 1831 y Ft(\))755 1846 y Fr(n)835 1831 y Ft(of)h Fs(S)1012 1795 y Fp(1)1084 1831 y Ft(in)f(the)i(follo)m(wing)d(w)m(a)m(y:)373 2036 y Fo(P)442 2051 y Fr(n)517 2036 y Ft(=)621 1925 y Fh(n)687 2036 y Ft([)p Fs(s)760 1995 y Fq(0)783 2036 y Fs(;)17 b(s)873 1995 y Fq(00)915 2036 y Ft(\))11 b(:)34 b(\()p Fs(s)1109 1995 y Fq(0)1132 2036 y Fs(;)17 b(s)1222 1995 y Fq(00)1264 2036 y Ft(\))32 b(is)h(a)f(connected)i(comp)s(onen)m(t)f (of)2604 2011 y Fh(b)2579 2036 y Fs(X)2668 1995 y Fq(\000)p Fp(1)2660 2060 y Fr(n)2762 1955 y Fh(\000)2808 2036 y Ft(\()p Fs(S)2912 1995 y Fp(1)2973 2036 y Fo(\002)22 b Fs(I)8 b Ft(\))22 b Fo(n)g Fs(L)3321 2051 y Fr(n)3369 1955 y Fh(\001)3414 1925 y(o)3497 2036 y Fs(:)118 2234 y Ft(It)33 b(is)f(easy)h(to)g(c)m(hec)m(k)i(that)d Fo(P)1199 2249 y Fr(n)p Fp(+1)1369 2234 y Ft(re\014nes)i Fo(P)1742 2249 y Fr(n)1822 2234 y Ft(for)e(eac)m(h)h Fs(n)28 b Fo(\025)g Ft(1)33 b(and)1105 2411 y(\()p Fs(d)22 b Ft(+)g(const)17 b Fs(\013)q Ft(\))1654 2370 y Fq(\000)p Fr(n)1783 2411 y Fo(\024)28 b(j)p Fs(!)t Fo(j)f(\024)h Ft(\()p Fs(d)21 b Fo(\000)i Ft(const)17 b Fs(\013)q Ft(\))2691 2370 y Fq(\000)p Fr(n)118 2588 y Ft(for)33 b(eac)m(h)i Fs(!)e Fo(2)d(P)748 2603 y Fr(n)796 2588 y Ft(.)47 b(Due)33 b(to)h(the)g(large)f(expansion)h(of)f Fs(f)45 b Ft(in)33 b(the)h(horizon)m(tal)e(direction,)h(w)m(e)i(ha)m(v)m(e)118 2709 y(that)d(if)g Fs(J)37 b Fo(\032)28 b Fs(I)40 b Ft(is)32 b(an)h(in)m(terv)-5 b(al)31 b(with)h Fo(j)p Fs(J)9 b Fo(j)27 b(\024)i Fs(\013)q Ft(,)j(then)h(for)f(eac)m(h)i Fs(!)d Fo(2)d(P)2726 2724 y Fr(n)909 2896 y Fs(m)994 2816 y Fh(\000)1040 2896 y Fo(f)p Fs(s)f Fo(2)h Fs(!)15 b Ft(:)1418 2871 y Fh(b)1393 2896 y Fs(X)1474 2911 y Fr(j)1511 2896 y Ft(\()p Fs(s)p Ft(\))27 b Fo(2)h Fs(S)1820 2855 y Fp(1)1882 2896 y Fo(\002)22 b Fs(J)9 b Fo(g)2111 2816 y Fh(\001)2184 2896 y Fo(\024)28 b Ft(const)2529 2806 y Fh(p)p 2628 2806 119 4 v 2628 2896 a Fo(j)p Fs(J)9 b Fo(j)16 b Fs(m)p Ft(\()p Fs(!)t Ft(\))617 b(\(33\))118 3073 y(see)37 b([Vi1)o(,)g(Corollary)d(2.3].)53 b(The)36 b(estimate)f(\(27\))h(on)f(the)i(Leb)s(esgue)g(measure)f(of)f Fs(B)3298 3088 y Fp(2)3338 3073 y Ft(\()p Fs(n)p Ft(\))g(is)h(no)m(w) 118 3194 y(an)31 b(easy)i(consequence)h(of)d(\(33\).)43 b(F)-8 b(or)31 b(that)g(w)m(e)h(only)f(ha)m(v)m(e)i(to)e(compute)h(the) f(Leb)s(esgue)i(measure)118 3314 y(of)41 b Fs(B)312 3329 y Fp(2)351 3314 y Ft(\()p Fs(n)p Ft(\))g(on)g(eac)m(h)h(horizon)m(tal)d (line)h(of)h Fs(S)1743 3278 y Fp(1)1810 3314 y Fo(\002)28 b Fs(I)49 b Ft(and)41 b(in)m(tegrate.)69 b(The)42 b(estimate)e(\(27\))g (on)h(the)118 3434 y(Leb)s(esgue)48 b(measure)e(of)g Fs(B)1146 3449 y Fp(1)1185 3434 y Ft(\()p Fs(n)p Ft(\))h(is)e(is)h (obtained)g(b)m(y)h(mean)f(of)f(a)h(large)f(deviations)h(argumen)m(t) 118 3555 y(applied)32 b(to)g(the)h(horizon)m(tal)e(curv)m(es)j(in)e Fs(S)1683 3519 y Fp(1)1744 3555 y Fo(\002)23 b Fs(I)8 b Ft(.)118 3721 y Fc(Remark)37 b(6.4.)49 b Fi(The)35 b(choic)-5 b(e)35 b(of)g(the)h(c)-5 b(onstants)35 b Fs(c;)17 b(\030)5 b(;)17 b(\015)40 b Fi(and)35 b Fs(\016)40 b Fi(only)35 b(dep)-5 b(ends)35 b(on)g(the)h(quadr)-5 b(atic)118 3842 y(map)31 b Fs(Q)h Fi(and)g Fs(\013)c(>)g Ft(0)p Fi(.)43 b(In)31 b(p)-5 b(articular)32 b(the)g(de)-5 b(c)g(ay)32 b(estimates)f(on)h(the)g(L)-5 b(eb)g(esgue)31 b(me)-5 b(asur)g(e)31 b(of)h Fs(B)3606 3857 y Fp(1)3645 3842 y Ft(\()p Fs(n)p Ft(\))118 3962 y Fi(and)i Fs(B)381 3977 y Fp(2)421 3962 y Ft(\()p Fs(n)p Ft(\))h Fi(dep)-5 b(end)34 b(only)g(on)h(the)g(quadr)-5 b(atic)34 b(map)h Fs(Q)g Fi(and)f Fs(\013)28 b(>)g Ft(0)p Fi(.)118 4215 y Fc(6.2.2)113 b(Random)37 b(p)s(erturbations)118 4400 y Ft(Let)f Fs(f)46 b Ft(b)s(e)35 b(close)g(to)905 4374 y(^)883 4400 y Fs(f)46 b Ft(in)35 b(the)h Fs(C)1342 4364 y Fp(3)1416 4400 y Ft(top)s(ology)-8 b(.)50 b(As)36 b(w)m(e)h(ha)m(v)m(e)f(seen)h(b)s (efore,)f(it)e(is)h(no)g(restriction)g(if)118 4521 y(w)m(e)i(assume)f (that)f Fo(C)40 b Ft(=)32 b Fo(f)p Ft(\()p Fs(s;)17 b(x)p Ft(\))33 b Fo(2)g Fs(S)1488 4484 y Fp(1)1552 4521 y Fo(\002)25 b Fs(I)19 b Ft(:)34 b Fs(x)f Ft(=)g(0)p Fo(g)i Ft(is)g(the)i(critical)c (set)j(of)f Fs(f)11 b Ft(.)53 b(Fix)35 b Fo(f)p Ft(\010)p Fs(;)17 b Ft(\()p Fs(\022)3542 4536 y Fr(\017)3575 4521 y Ft(\))3613 4536 y Fr(\017)3645 4521 y Fo(g)36 b Ft(a)118 4641 y(random)30 b(p)s(erturbation)h(of)f Fs(f)42 b Ft(for)31 b(whic)m(h)g(\(8\))g(holds.)43 b(Our)31 b(goal)f(no)m(w)h(is)g(to)g (pro)m(v)m(e)h(that)f(an)m(y)h(suc)m(h)118 4761 y Fs(f)46 b Ft(satis\014es)35 b(the)h(h)m(yp)s(otheses)h(of)d(Theorems)i(C)f(and) h(D)e(for)g Fs(\017)e(>)g Ft(0)i(su\016cien)m(tly)i(small,)d(and)i(th)m (us)118 4882 y(conclude)41 b(that)f Fs(f)50 b Ft(is)40 b(sto)s(c)m(hastically)f(stable.)65 b(So,)42 b(w)m(e)g(w)m(an)m(t)f(to) e(sho)m(w)i(that)f(if)f Fs(\017)i(>)f Ft(0)g(is)g(small)118 5002 y(enough)33 b(then)1900 5251 y(39)p eop %%Page: 40 40 40 39 bop 263 548 a Fo(\017)49 b Fs(f)43 b Ft(is)32 b(non-uniformly)e (expanding)j(for)f(random)g(orbits;)263 741 y Fo(\017)49 b Ft(random)32 b(orbits)g(ha)m(v)m(e)i(slo)m(w)e(appro)m(ximation)e(to) j(the)g(critical)d(set)j Fo(C)6 b Ft(;)263 934 y Fo(\017)49 b Ft(the)33 b(family)d(of)i(h)m(yp)s(erb)s(olic)g(time)f(maps)i(\()p Fs(h)1988 949 y Fr(\017)2020 934 y Ft(\))2058 949 y Fr(\017)2123 934 y Ft(has)g(uniform)e Fs(L)2729 898 y Fp(1)2769 934 y Ft(-tail.)264 1105 y(W)-8 b(e)46 b(remark)e(that)h(in)f(the)h (estimates)f(w)m(e)i(ha)m(v)m(e)g(obtained)f(for)f(log)16 b Fo(k)p Ft(\()p Fs(D)s(f)11 b Ft(\()p Fs(s)3186 1120 y Fr(j)3222 1105 y Fs(;)17 b(x)3321 1120 y Fr(j)3358 1105 y Ft(\)\))3434 1069 y Fq(\000)p Fp(1)3528 1105 y Fo(k)44 b Ft(and)118 1226 y(log)17 b(dist)418 1241 y Fr(\016)456 1226 y Ft(\()p Fs(x)549 1241 y Fr(j)586 1226 y Fs(;)g Fo(C)6 b Ft(\))41 b(o)m(v)m(er)g(the)g(orbit)e(of)h(a)h(giv)m (en)f(p)s(oin)m(t)g(\()p Fs(s;)17 b(x)p Ft(\))41 b Fo(2)g Fs(S)2574 1190 y Fp(1)2641 1226 y Fo(\002)28 b Fs(I)8 b Ft(,)42 b(w)m(e)g(can)f(easily)e(replace)118 1346 y(the)33 b(iterates)g(\()p Fs(s)718 1361 y Fr(j)754 1346 y Fs(;)17 b(x)853 1361 y Fr(j)890 1346 y Ft(\))32 b(b)m(y)i(random)d(iterates)i (\()p Fs(s)1885 1299 y Fr(j)1885 1369 y(t)p 1885 1381 30 3 v 1921 1346 a Fs(;)17 b(x)2020 1299 y Fr(j)2020 1369 y(t)p 2020 1381 V 2057 1346 a Ft(\))28 b(=)f Fs(f)2285 1299 y Fr(j)2274 1369 y(t)p 2274 1381 V 2322 1346 a Ft(\()p Fs(s;)17 b(x)p Ft(\).)43 b(Actually)-8 b(,)32 b(the)h(metho)s(ds)f (used)118 1467 y(for)42 b(obtaining)f(estimate)h(\(28\))f(rely)i(on)f (a)h(delicate)e(decomp)s(osition)g(of)h(the)h(orbit)f(of)g(a)g(giv)m (en)118 1587 y(p)s(oin)m(t)36 b(\()p Fs(s;)17 b(x)p Ft(\))36 b(from)f(time)g(0)h(un)m(til)f(time)g Fs(n)i Ft(in)m(to)f(\014nite)g (pieces)h(according)f(to)g(its)g(returns)h(to)f(the)118 1707 y(neigh)m(b)s(orho)s(o)s(d)27 b Fs(S)792 1671 y Fp(1)843 1707 y Fo(\002)12 b Ft(\()p Fo(\000)1047 1636 y(p)p 1132 1636 63 4 v 1132 1707 a Fs(\013;)1238 1636 y Fo(p)p 1321 1636 V 71 x Fs(\013)q Ft(\))28 b(of)f(the)h(critical)e (set.)42 b(The)29 b(main)d(to)s(ols)h(are)g([Vi1,)h(Lemma)f(2.4])118 1828 y(and)h([Vi1)o(,)h(Lemma)d(2.5])h(whose)i(pro)s(ofs)e(ma)m(y)h (easily)f(b)s(e)h(mimic)m(k)m(ed)e(for)h(random)g(orbits.)41 b(Indeed,)118 1948 y(the)d(imp)s(ortan)m(t)d(fact)i(in)f(the)h(pro)s (of)g(of)f(the)i(referred)f(lemmas)f(is)g(that)h(orbits)g(of)f(p)s(oin) m(ts)h(in)f(the)118 2069 y(cen)m(tral)27 b(direction)e(sta)m(y)j(close) e(to)h(orbits)f(of)g(the)h(quadratic)f(map)g Fs(Q)h Ft(for)f(long)g(p)s (erio)s(ds,)h(as)g(long)e(as)118 2189 y Fs(\013)j(>)g Ft(0)f(is)g(tak)m(en)i(su\016cien)m(tly)f(small.)39 b(Hence,)30 b(suc)m(h)f(results)f(can)g(easily)f(b)s(e)g(obtained)g(for)g(random) 118 2309 y(orbits)32 b(as)h(long)e(as)i(w)m(e)h(tak)m(e)f Fs(\017)28 b(>)g Ft(0)k(with)g Fs(\017)c Fo(\034)f Fs(\013)34 b Ft(and)e(p)s(erturbation)g(v)m(ectors)i Fs(t)p 3057 2325 36 4 v 28 w Fo(2)28 b Ft(supp)18 b(\()p Fs(\022)3515 2324 y Fr(\017)3548 2309 y Ft(\).)264 2430 y(Th)m(us,)33 b(the)d(pro)s(cedure)h(of)f([Vi1)o(])g(describ)s(ed)h(in)e(Subsection)i (6.2.1)e(applies)h(to)f(this)h(situation,)118 2550 y(and)j(w)m(e)g(are) g(able)f(to)g(pro)m(v)m(e)i(that)e(there)i(is)e Fs(c)27 b(>)h Ft(0,)k(and)h(for)f Fs(\015)h(>)27 b Ft(0)33 b(there)g(is)f Fs(\016)g(>)27 b Ft(0,)33 b(suc)m(h)h(that)574 2693 y Fr(n)p Fq(\000)p Fp(1)569 2723 y Fh(X)579 2933 y Fr(j)t Fp(=0)729 2818 y Ft(log)17 b Fo(k)p Fs(D)s(f)11 b Ft(\()p Fs(s)1149 2771 y Fr(j)1149 2840 y(t)p 1149 2852 30 3 v 1184 2818 a Fs(;)17 b(x)1283 2771 y Fr(j)1283 2840 y(t)p 1283 2852 V 1320 2818 a Ft(\)\))1396 2777 y Fq(\000)p Fp(1)1490 2818 y Fo(k)28 b(\024)g(\000)p Fs(cn)98 b Ft(and)2225 2693 y Fr(n)p Fq(\000)p Fp(1)2219 2723 y Fh(X)2230 2933 y Fr(j)t Fp(=0)2380 2818 y Fo(\000)17 b Ft(log)g(dist)2774 2833 y Fr(\016)2812 2818 y Ft(\()p Fs(x)2905 2771 y Fr(j)2905 2840 y(t)p 2905 2852 V 2942 2818 a Fs(;)g Fo(C)6 b Ft(\))28 b Fo(\024)g Fs(\015)5 b(n)118 3106 y Ft(for)32 b(\()p Fs(s;)17 b(x)p Ft(\))39 b Fs(=)-60 b Fo(2)28 b Fs(B)684 3121 y Fp(1)723 3106 y Ft(\()p Fs(n)p Ft(\))23 b Fo([)f Fs(B)1042 3121 y Fp(2)1082 3106 y Ft(\()p Fs(n)p Ft(\),)32 b(where)i Fs(B)1631 3121 y Fp(1)1671 3106 y Ft(\()p Fs(n)p Ft(\))e(and)h Fs(B)2101 3121 y Fp(2)2141 3106 y Ft(\()p Fs(n)p Ft(\))f(are)h(subsets)h Fs(S)2873 3070 y Fp(1)2935 3106 y Fo(\002)22 b Fs(I)41 b Ft(with)875 3289 y Fs(m)960 3208 y Fh(\000)1006 3289 y Fs(B)1080 3304 y Fp(1)1120 3289 y Ft(\()p Fs(n)p Ft(\))1254 3208 y Fh(\001)1327 3289 y Fo(\024)28 b Fs(e)1477 3248 y Fq(\000)p Fr(\030)s(n)1710 3289 y Ft(and)98 b Fs(m)p Ft(\()p Fs(B)2162 3304 y Fp(2)2202 3289 y Ft(\()p Fs(n)p Ft(\)\))28 b Fo(\024)g Ft(const)17 b Fs(e)2791 3248 y Fq(\000)2846 3200 y(p)p 2905 3200 43 3 v 48 x Fr(n=)p Fp(4)118 3472 y Ft(for)36 b(some)g(constan)m(t)h Fs(\030)h(>)c Ft(0)i(dep)s(ending)h(only)e(on)i Fs(\015)5 b Ft(.)54 b(This)36 b(giv)m(es)h(the)g(non-uniform)d(expansion)118 3593 y(and)29 b(slo)m(w)f(appro)m(ximation)f(to)h(the)h(critical)d(set) j(for)f(random)g(orbits.)41 b(Moreo)m(v)m(er,)31 b(the)e(argumen)m(ts) 118 3713 y(sho)m(w)34 b(that)e(w)m(e)i(ma)m(y)e(tak)m(e)i(the)f(map)e Fs(N)1600 3728 y Fr(\017)1666 3713 y Ft(with)676 3896 y(\()p Fs(\022)762 3855 y Fe(N)759 3921 y Fr(\017)836 3896 y Fo(\002)23 b Fs(m)p Ft(\))1076 3815 y Fh(\000)o(\010)1179 3896 y Ft(\()p Fs(t)p 1217 3912 36 4 v(;)17 b(x)p Ft(\))28 b Fo(2)g Fs(T)1582 3855 y Fe(N)1656 3896 y Fo(\002)23 b Fs(M)f Ft(:)33 b Fs(N)2010 3911 y Fr(\017)2043 3896 y Ft(\()p Fs(t)p 2081 3912 V(;)17 b(x)p Ft(\))28 b Fs(>)f(n)2442 3815 y Fh(\011)q(\001)2574 3896 y Fo(\024)h Ft(const)17 b Fs(e)2963 3855 y Fq(\000)3018 3807 y(p)p 3077 3807 43 3 v 48 x Fr(n=)p Fp(4)3195 3896 y Fs(;)118 4079 y Ft(th)m(us)36 b(giving)d(that)i(the)g(family)d(of)i(\014rst)h(h)m(yp)s (erb)s(olic)f(time)g(maps)g(has)h(uniform)e Fs(L)3198 4043 y Fp(1)3238 4079 y Ft(-tail)f(\(see)k(the)118 4200 y(considerations)c(at)g(the)h(b)s(eginning)f(of)g(Section)g(6\).)264 4359 y(F)-8 b(or)40 b(the)g(sak)m(e)i(of)d(completeness,)k(an)d (explanation)f(is)h(required)h(on)f(the)g(w)m(a)m(y)i(the)e(Mark)m(o)m (v)118 4480 y(partitions)g Fo(P)641 4495 y Fr(n)729 4480 y Ft(of)h Fs(S)915 4443 y Fp(1)996 4480 y Ft(can)g(b)s(e)h(de\014ned)h (in)d(this)h(case,)k(in)40 b(order)i(to)f(obtain)f(the)i(estimates)f (on)118 4600 y(the)d(Leb)s(esgue)h(measure)f(of)e Fs(B)1292 4615 y Fp(1)1332 4600 y Ft(\()p Fs(n)p Ft(\))i(and)f Fs(B)1772 4615 y Fp(2)1812 4600 y Ft(\()p Fs(n)p Ft(\).)58 b(W)-8 b(e)38 b(consider)f Fs(M)47 b Ft(=)36 b Fs(S)2907 4564 y Fp(1)2972 4600 y Fo(\002)26 b Fs(I)45 b Ft(and)37 b(de\014ne)i(the)118 4720 y(sk)m(ew-pro)s(duct)34 b(map)1232 4895 y Fs(F)41 b Ft(:)84 b Fs(T)1518 4859 y Fe(N)1592 4895 y Fo(\002)22 b Fs(M)94 b Fo(\000)-16 b(!)169 b Fs(T)2280 4859 y Fe(N)2353 4895 y Fo(\002)23 b Fs(M)5 b(;)1519 5016 y Ft(\()p Fs(t)p 1557 5032 36 4 v(;)17 b(z)t Ft(\))156 b Fo(7\000)-16 b(!)2122 4935 y Fh(\000)2168 5016 y Fs(\033)t Ft(\()p Fs(t)p 2265 5032 V Ft(\))p Fs(;)17 b(f)2430 5031 y Fr(t)2455 5040 y Fd(1)2494 5016 y Ft(\()p Fs(z)t Ft(\))2619 4935 y Fh(\001)1900 5251 y Ft(40)p eop %%Page: 41 41 41 40 bop 118 548 a Ft(where)40 b Fs(\033)i Ft(is)c(the)h(left)f(shift) g(map.)60 b(W)-8 b(riting)36 b Fs(f)1867 563 y Fr(t)1897 548 y Ft(\()p Fs(z)t Ft(\))i(=)2174 467 y Fh(\000)2220 548 y Fs(g)2267 563 y Fr(t)2296 548 y Ft(\()p Fs(z)t Ft(\))p Fs(;)17 b(q)2508 563 y Fr(t)2538 548 y Ft(\()p Fs(z)t Ft(\))2663 467 y Fh(\001)2748 548 y Ft(for)38 b Fs(z)k Ft(=)c(\()p Fs(s;)17 b(x)p Ft(\))37 b Fo(2)i Fs(S)3533 512 y Fp(1)3598 548 y Fo(\002)27 b Fs(I)8 b Ft(,)118 668 y(w)m(e)33 b(ha)m(v)m(e)h(that)e Fs(q)740 683 y Fr(t)770 668 y Ft(\()p Fs(s;)17 b Fo(\001)p Ft(\))31 b(is)h(a)f(unimo)s(dal)f(map)h(close)h(to)39 b(^)-56 b Fs(q)36 b Ft(for)c(all)e Fs(s)d Fo(2)i Fs(S)2769 632 y Fp(1)2840 668 y Ft(and)j Fs(t)c Fo(2)g Ft(supp)18 b(\()p Fs(\022)3487 683 y Fr(\017)3520 668 y Ft(\))32 b(with)118 789 y Fs(\017)c(>)g Ft(0)k(small.)118 992 y Fc(Prop)s(osition)k(6.5.)49 b Fi(Given)35 b Fs(t)p 1232 1008 36 4 v 30 w Fo(2)30 b Fs(T)1464 956 y Fe(N)1552 992 y Fi(ther)-5 b(e)35 b(is)h(a)g Fs(C)2064 956 y Fp(1)2139 992 y Fi(foliation)f Fo(F)2608 956 y Fr(c)2598 1017 y(t)p 2598 1029 30 3 v 2678 992 a Fi(of)g Fs(M)47 b Fi(such)36 b(that)g(if)f Fs(L)3518 1007 y Fr(t)p 3518 1019 V 3548 992 a Ft(\()p Fs(z)t Ft(\))i Fi(is)118 1112 y(the)e(le)-5 b(af)34 b(of)h Fo(F)656 1076 y Fr(c)646 1137 y(t)p 646 1149 V 725 1112 a Fi(thr)-5 b(ough)35 b(a)g(p)-5 b(oint)34 b Fs(z)f Fo(2)28 b Fs(M)10 b Fi(,)35 b(then)234 1316 y(1.)48 b Fs(L)428 1331 y Fr(t)p 428 1343 V 458 1316 a Ft(\()p Fs(z)t Ft(\))35 b Fi(is)g(a)g Fs(C)885 1280 y Fp(1)959 1316 y Fi(submanifold)f(of)g Fs(M)46 b Fi(close)34 b(to)h(a)g(vertic)-5 b(al)34 b(line)g(in)h(the)g Fs(C)3085 1280 y Fp(1)3159 1316 y Fi(top)-5 b(olo)g(gy;)234 1519 y(2.)48 b Fs(f)410 1534 y Fr(t)435 1543 y Fd(1)474 1439 y Fh(\000)520 1519 y Fs(L)586 1534 y Fr(t)p 586 1546 V 616 1519 a Ft(\()p Fs(z)t Ft(\))741 1439 y Fh(\001)822 1519 y Fi(is)35 b(c)-5 b(ontaine)g(d)34 b(in)g Fs(L)1548 1534 y Fr(\033)r(t)p 1590 1546 26 3 v 1621 1439 a Fh(\000)1667 1519 y Fs(f)1715 1534 y Fr(t)1740 1543 y Fd(1)1779 1519 y Ft(\()p Fs(z)t Ft(\))1904 1439 y Fh(\001)1950 1519 y Fi(.)118 1723 y(Pr)-5 b(o)g(of.)49 b Ft(This)33 b(will)e(b)s(e)j (obtained)f(as)g(a)g(consequence)k(of)c(the)g(fact)h(that)f(the)h(set)g (of)f(v)m(ertical)f(lines)118 1843 y(constitutes)47 b(a)f(normally)e (expanding)j(in)m(v)-5 b(arian)m(t)44 b(foliation)f(for)2629 1817 y(^)2608 1843 y Fs(f)11 b Ft(.)84 b(Let)47 b Fo(H)g Ft(b)s(e)g(the)g(space)g(of)118 1963 y(con)m(tin)m(uous)40 b(maps)f Fs(\030)44 b Ft(:)39 b Fs(T)1099 1927 y Fe(N)1178 1963 y Fo(\002)27 b Fs(M)50 b Fo(!)38 b Ft([)p Fo(\000)p Ft(1)p Fs(;)17 b Ft(1])40 b(endo)m(w)m(ed)h(with)e(the)h(sup)g(norm,)g (and)f(de\014ne)i(the)118 2084 y(map)32 b Fs(A)c Ft(:)f Fo(H)i(!)f(H)33 b Ft(b)m(y)263 2353 y Fs(A\030)5 b Ft(\()p Fs(t)p 422 2369 36 4 v(;)17 b(z)t Ft(\))28 b(=)760 2286 y Fs(@)811 2301 y Fr(x)856 2286 y Fs(q)899 2301 y Fr(t)924 2310 y Fd(1)963 2286 y Ft(\()p Fs(z)t Ft(\))p Fs(\030)5 b Ft(\()p Fs(F)14 b Ft(\()p Fs(t)p 1289 2302 V(;)j(z)t Ft(\)\))23 b Fo(\000)f Fs(@)1666 2301 y Fr(x)1711 2286 y Fs(g)1758 2301 y Fr(t)1783 2310 y Fd(1)1822 2286 y Ft(\()p Fs(z)t Ft(\))p 730 2330 1249 4 v 730 2421 a Fo(\000)p Fs(@)858 2436 y Fr(s)895 2421 y Fs(q)938 2436 y Fr(t)963 2445 y Fd(1)1003 2421 y Ft(\()p Fs(z)t Ft(\))p Fs(\030)5 b Ft(\()p Fs(F)14 b Ft(\()p Fs(t)p 1329 2437 36 4 v(;)j(z)t Ft(\)\))22 b(+)g Fs(@)1704 2436 y Fr(s)1741 2421 y Fs(g)1788 2436 y Fr(t)1813 2445 y Fd(1)1852 2421 y Ft(\()p Fs(z)t Ft(\))1988 2353 y Fs(;)114 b(t)p 2129 2369 V 28 w Ft(=)28 b(\()p Fs(t)2369 2368 y Fp(1)2408 2353 y Fs(;)17 b(t)2487 2368 y Fp(2)2527 2353 y Fs(;)g(:)g(:)g(:)e Ft(\))28 b Fo(2)g Fs(T)2932 2312 y Fe(N)3082 2353 y Ft(and)97 b Fs(z)33 b Fo(2)28 b Fs(M)5 b(:)118 2623 y Ft(Note)33 b(that)f Fs(A)h Ft(is)f(w)m(ell-de\014ned,)h(since)981 2893 y Fo(j)p Fs(A\030)5 b Ft(\()p Fs(t)p 1168 2909 V -1 w(;)17 b(z)t Ft(\))p Fo(j)28 b(\024)1711 2825 y Ft(\(4)22 b(+)g Fs(\013)g Ft(+)g Fs(\017)p Ft(\))h(+)f Fs(\013)h Ft(+)f Fs(\017)p 1504 2870 1224 4 v 1504 2961 a Fo(\000)p Ft(\(const)34 b Fs(\013)22 b Ft(+)g Fs(\017)p Ft(\))h(+)f(\()p Fs(d)g Fo(\000)g Fs(\013)h Fo(\000)g Fs(\017)p Ft(\))2765 2893 y Fs(<)k Ft(1)118 3163 y(for)38 b(small)f Fs(\013)h(>)g Ft(0)g(and)h Fs(\017)g(>)e Ft(0.)62 b(Moreo)m(v)m(er,)42 b Fs(A)c Ft(is)g(a)h(con)m(traction)f(on)h Fo(H)q Ft(:)55 b(giv)m(en)39 b Fs(\030)5 b(;)17 b(\020)44 b Fo(2)39 b(H)g Ft(and)118 3284 y(\()p Fs(t)p 156 3299 36 4 v(;)17 b(z)t Ft(\))28 b Fo(2)g Fs(T)515 3247 y Fe(N)589 3284 y Fo(\002)23 b Fs(M)43 b Ft(then)143 3503 y Fo(j)p Fs(A\030)5 b Ft(\()p Fs(t)p 330 3519 V -1 w(;)17 b(z)t Ft(\))23 b Fo(\000)f Fs(A\020)8 b Ft(\()p Fs(t)p 779 3519 V(;)17 b(z)t Ft(\))p Fo(j)698 3704 y(\024)1673 3636 y(j)p Ft(det)p Fs(D)s(f)1968 3651 y Fr(t)1993 3660 y Fd(1)2032 3636 y Ft(\()p Fs(z)t Ft(\))p Fo(j)22 b(\001)g(j)p Fs(\030)5 b Ft(\()p Fs(t)p 2371 3652 V -1 w(;)17 b(z)t Ft(\))23 b Fo(\000)g Fs(\020)8 b Ft(\()p Fs(t)p 2748 3652 V -1 w(;)17 b(z)t Ft(\))p Fo(j)p 869 3681 2876 4 v 869 3693 a Fh(\014)869 3753 y(\014)902 3697 y(\000)953 3777 y Fo(\000)23 b Fs(@)1104 3792 y Fr(s)1141 3777 y Fs(q)1184 3792 y Fr(t)1209 3801 y Fd(1)1249 3777 y Ft(\()p Fs(z)t Ft(\))p Fs(\030)5 b Ft(\()p Fs(F)14 b Ft(\()p Fs(t)p 1575 3793 36 4 v -1 w(;)j(z)t Ft(\)\))23 b(+)f Fs(@)1950 3792 y Fr(s)1987 3777 y Fs(g)2034 3792 y Fr(t)2059 3801 y Fd(1)2098 3777 y Ft(\()p Fs(z)t Ft(\))2223 3697 y Fh(\001)2292 3777 y Fo(\001)2341 3697 y Fh(\000)2393 3777 y Fo(\000)g Fs(@)2543 3792 y Fr(s)2581 3777 y Fs(q)2624 3792 y Fr(t)2649 3801 y Fd(1)2688 3777 y Ft(\()p Fs(z)t Ft(\))p Fs(\020)8 b Ft(\()p Fs(F)14 b Ft(\()p Fs(t)p 3017 3793 V(;)j(z)t Ft(\)\))22 b(+)g Fs(@)3392 3792 y Fr(s)3429 3777 y Fs(g)3476 3792 y Fr(t)3501 3801 y Fd(1)3540 3777 y Ft(\()p Fs(z)t Ft(\))3665 3697 y Fh(\001)3712 3693 y(\014)3712 3753 y(\014)698 4004 y Fo(\024)869 3850 y Fh(\000)915 3930 y Ft(\()p Fs(d)f Ft(+)h Fs(\013)h Ft(+)f Fs(\017)p Ft(\)\(4)g(+)g Fs(\013)h Ft(+)f Fs(\017)p Ft(\))h(+)f Fs(\013)g Ft(+)g Fs(\017)2192 3850 y Fh(\001)2261 3930 y Fo(\001)2310 3846 y Fh(\014)2310 3905 y(\014)2344 3930 y Fs(\030)5 b Ft(\()p Fs(t)p 2430 3946 V -1 w(;)17 b(z)t Ft(\))23 b Fo(\000)f Fs(\020)8 b Ft(\()p Fs(t)p 2806 3946 V(;)17 b(z)t Ft(\))2972 3846 y Fh(\014)2972 3905 y(\014)p 869 3981 2137 4 v 1570 4072 a Ft(\()p Fs(d)22 b Fo(\000)g Ft(const)q Fs(\013)h Fo(\000)f Fs(\017)p Ft(\))2264 4043 y Fp(2)3015 4004 y Fs(:)118 4280 y Ft(This)30 b(last)f(quan)m(tit)m(y)i (can)f(b)s(e)g(made)f(smaller)f(than)i Fo(j)p Fs(\030)5 b Ft(\()p Fs(t)p 2144 4296 36 4 v -1 w(;)17 b(z)t Ft(\))f Fo(\000)g Fs(\021)t Ft(\()p Fs(t)p 2508 4296 V 1 w(;)h(z)t Ft(\))p Fo(j)p Fs(=)p Ft(2,)30 b(as)g(long)f(as)h Fs(\013)g Ft(and)g Fs(\017)g Ft(are)118 4400 y(c)m(hosen)36 b(su\016cien)m(tly)g (small.)47 b(This)34 b(sho)m(ws)j(that)d Fs(A)h Ft(is)f(a)g(con)m (traction)g(on)h(the)g(Banac)m(h)g(space)g Fo(H)q Ft(,)118 4521 y(and)e(so)g(it)e(has)i(a)f(unique)h(\014xed)h(p)s(oin)m(t)e Fs(\030)1636 4484 y Fr(c)1698 4521 y Fo(2)c(H)q Ft(.)264 4641 y(It)33 b(is)f(no)g(restriction)g(for)f(our)i(purp)s(oses)g(if)f (w)m(e)h(think)f(of)g Fs(T)46 b Ft(as)33 b(b)s(eing)f(equal)g(to)g (supp)18 b(\()p Fs(\022)3560 4656 y Fr(\017)3593 4641 y Ft(\))32 b(for)118 4761 y(some)38 b(small)e Fs(\017)p Ft(.)60 b(Note)38 b(that)g(the)h(map)e Fs(A)h Ft(dep)s(ends)i(con)m (tin)m(uously)e(on)g Fs(F)52 b Ft(and)38 b(for)g Fs(\017)f(>)g Ft(0)h(small)118 4882 y(enough)25 b(the)g(\014xed)g(p)s(oin)m(t)e(of)h Fs(A)h Ft(is)e(close)i(to)f(the)g(zero)h(constan)m(t)g(map.)40 b(This)25 b(holds)e(b)s(ecause)j(w)m(e)f(are)118 5002 y(c)m(ho)s(osing)h(supp)17 b(\()p Fs(\022)808 5017 y Fr(\017)841 5002 y Ft(\))26 b(close)g(to)f Fo(f)p Fs(t)1329 4966 y Fq(\003)1369 5002 y Fo(g)p Ft(,)i Fs(f)1521 5017 y Fr(t)1546 4998 y Ff(\003)1614 5002 y Ft(=)h Fs(f)36 b Ft(and)26 b Fs(f)37 b Ft(close)26 b(to)2431 4976 y(^)2409 5002 y Fs(f)11 b Ft(.)41 b(Then,)29 b(for)c Fs(\017)j(>)g Ft(0)d(small)f(enough,)1900 5251 y(41)p eop %%Page: 42 42 42 41 bop 118 548 a Ft(w)m(e)31 b(ha)m(v)m(e)h Fs(\030)530 512 y Fr(c)564 548 y Ft(\()p Fs(t)p 602 564 36 4 v(;)17 b Fo(\001)p Ft(\))30 b(uniformly)e(close)i(to)g Fs(\030)1615 512 y Fr(c)1649 548 y Ft(\()p Fs(t)p 1687 564 V -36 x Fq(\003)1762 548 y Fs(;)17 b Fo(\001)p Ft(\))29 b(and)h(it)g(is)f(not)h (hard)h(to)f(c)m(hec)m(k)i(that)e Fs(\030)3310 512 y Fr(c)3305 573 y Fp(0)3372 548 y Ft(=)d Fs(\030)3523 512 y Fr(c)3558 548 y Ft(\()p Fs(t)p 3596 564 V -36 x Fq(\003)3670 548 y Fs(;)17 b Fo(\001)p Ft(\))118 668 y(is)35 b(precisely)i(the)f (map)f(whose)i(in)m(tegral)d(lea)m(v)m(es)j(of)f(the)g(v)m(ector)h (\014eld)e(\()p Fs(\030)2827 632 y Fr(c)2822 693 y Fp(0)2862 668 y Fs(;)17 b Ft(1\))35 b(giv)m(e)h(the)g(in)m(v)-5 b(arian)m(t)118 789 y(foliation)37 b Fo(F)590 753 y Fr(c)665 789 y Ft(asso)s(ciated)j(to)g Fs(f)1312 804 y Fr(t)1337 785 y Ff(\003)1419 789 y Ft(=)i Fs(f)11 b Ft(.)67 b(Since)40 b(this)h(foliation)c(dep)s(ends)42 b(con)m(tin)m(uously)f(on)f(the)118 909 y(dynamics)31 b(and)g(for)g Fs(f)38 b Ft(=)1095 883 y(^)1074 909 y Fs(f)k Ft(w)m(e)32 b(ha)m(v)m(e)g Fs(\030)1577 873 y Fr(c)1572 934 y Fp(0)1639 909 y Fo(\021)c Ft(0)j(\(see)h([Vi1)o (,)g(Section)f(2.5]\),)g(w)m(e)h(\014nally)e(deduce)j(that)118 1029 y Fs(\030)166 993 y Fr(c)200 1029 y Ft(\()p Fs(t)p 238 1045 V(;)17 b Fo(\001)p Ft(\))32 b(is)g(uniformly)f(close)h(to)h (zero)f(for)h(small)d Fs(\017)e(>)f Ft(0.)264 1150 y(W)-8 b(e)31 b(ha)m(v)m(e)g(de\014ned)g Fs(A)f Ft(in)f(suc)m(h)i(a)e(w)m(a)m (y)i(that)f(if)e(w)m(e)j(tak)m(e)g Fs(E)2412 1114 y Fr(c)2446 1150 y Ft(\()p Fs(t)p 2484 1166 V 1 w(;)17 b(z)t Ft(\))28 b(=)f(span)q Fo(f)p Ft(\()p Fs(\030)3114 1114 y Fr(c)3148 1150 y Ft(\()p Fs(t)p 3186 1166 V(;)17 b(z)t Ft(\))p Fs(;)g Ft(1\))p Fo(g)p Ft(,)30 b(then)118 1270 y(for)i(ev)m(ery)j Fs(t)p 525 1286 V 27 w Fo(2)29 b Fs(T)753 1234 y Fe(N)837 1270 y Ft(and)k Fs(z)f Fo(2)c Fs(S)1264 1234 y Fp(1)1325 1270 y Fo(\002)23 b Fs(I)1314 1490 y(D)s(f)1446 1505 y Fr(t)1471 1514 y Fd(1)1510 1490 y Ft(\()p Fs(z)t Ft(\))p Fs(E)1713 1449 y Fr(c)1749 1490 y Ft(\()p Fs(t)p 1787 1506 V(;)17 b(z)t Ft(\))28 b Fo(\032)g Fs(E)2164 1449 y Fr(c)2199 1490 y Ft(\()p Fs(F)14 b Ft(\()p Fs(t)p 2352 1506 V(;)j(z)t Ft(\)\))p Fs(:)1023 b Ft(\(34\))118 1710 y(No)m(w,)39 b(for)e(\014xed)h Fs(t)p 767 1726 V 36 w Fo(2)e Fs(T)1011 1674 y Fe(N)1063 1710 y Ft(,)i(w)m(e)g(tak)m(e)g Fo(F)1574 1674 y Fr(c)1564 1735 y(t)p 1564 1747 30 3 v 1646 1710 a Ft(to)f(b)s(e)g(the)h(set)g(of)e(in)m(tegral)g(curv)m(es) j(of)e(the)g(v)m(ector)i(\014eld)118 1831 y Fs(z)e Fo(!)32 b Ft(\()p Fs(\030)418 1794 y Fr(c)452 1831 y Ft(\()p Fs(t)p 490 1847 36 4 v(;)17 b(z)t Ft(\))p Fs(;)g Ft(1\))36 b(de\014ned)g(on)g Fs(S)1366 1794 y Fp(1)1429 1831 y Fo(\002)24 b Fs(I)8 b Ft(.)52 b(Since)35 b(the)h(v)m(ector)h(\014eld)e (is)g(tak)m(en)h(of)f(class)g Fs(C)3389 1794 y Fp(0)3429 1831 y Ft(,)h(it)e(do)s(es)118 1951 y(not)45 b(follo)m(w)e(immediately) e(that)k(through)g(eac)m(h)g(p)s(oin)m(t)f(in)g Fs(S)2460 1915 y Fp(1)2530 1951 y Fo(\002)31 b Fs(I)52 b Ft(passes)47 b(only)d(one)h(in)m(tegral)118 2071 y(curv)m(e.)58 b(W)-8 b(e)37 b(will)e(pro)m(v)m(e)j(uniqueness)h(of)d(solutions)g(b)m(y)i (using)e(the)h(fact)g(that)g(the)g(map)f Fs(f)48 b Ft(has)37 b(a)118 2192 y(big)32 b(expansion)h(in)f(the)h(horizon)m(tal)e (direction.)264 2312 y(Assume,)46 b(b)m(y)e(con)m(tradiction,)g(that)e (there)i(are)e(t)m(w)m(o)h(distinct)f(in)m(tegral)f(curv)m(es)k Fs(Y)5 b(;)17 b(Z)51 b Fo(2)45 b(F)3745 2276 y Fr(c)3735 2337 y(t)p 3735 2349 30 3 v 118 2433 a Ft(with)33 b(a)g(common)g(p)s (oin)m(t.)45 b(So)33 b(w)m(e)i(ma)m(y)e(tak)m(e)h(three)g(distinct)f (nearb)m(y)i(p)s(oin)m(ts)e Fs(z)3079 2448 y Fp(0)3119 2433 y Fs(;)17 b(z)3208 2448 y Fp(1)3247 2433 y Fs(;)g(z)3336 2448 y Fp(2)3405 2433 y Fo(2)29 b Fs(S)3566 2396 y Fp(1)3628 2433 y Fo(\002)24 b Fs(I)118 2553 y Ft(suc)m(h)38 b(that)e Fs(z)602 2568 y Fp(0)676 2553 y Fo(2)f Fs(Y)46 b Fo(\\)25 b Fs(Z)7 b Ft(,)38 b Fs(z)1155 2568 y Fp(1)1229 2553 y Fo(2)d Fs(Y)21 b Ft(,)37 b Fs(z)1517 2568 y Fp(2)1591 2553 y Fo(2)e Fs(Z)44 b Ft(and)36 b Fs(z)2041 2568 y Fp(1)2081 2553 y Fs(;)17 b(z)2170 2568 y Fp(2)2246 2553 y Ft(ha)m(v)m(e)38 b(the)f(same)f Fs(x)p Ft(-co)s(ordinate.)54 b(Let)37 b Fs(X)118 2673 y Ft(b)s(e)e(the)g(horizon)m(tal)d(curv)m(e)k (joining)d Fs(z)1514 2688 y Fp(1)1588 2673 y Ft(to)h Fs(z)1754 2688 y Fp(2)1794 2673 y Ft(.)49 b(If)34 b(w)m(e)i(consider)f Fs(X)2578 2688 y Fr(n)2655 2673 y Ft(=)c Fs(\031)2817 2688 y Fp(2)2880 2673 y Fo(\016)24 b Fs(F)3031 2637 y Fr(n)3077 2673 y Ft(\()p Fs(t)p 3115 2689 36 4 v(;)17 b(X)8 b Ft(\))34 b(for)g Fs(n)d Fo(\025)h Ft(1,)118 2794 y(where)c Fs(\031)449 2809 y Fp(2)516 2794 y Ft(is)e(the)h(pro)5 b(jection)27 b(from)e Fs(T)1523 2758 y Fe(N)1585 2794 y Fo(\002)10 b Fs(S)1738 2758 y Fp(1)1789 2794 y Fo(\002)g Fs(I)35 b Ft(on)m(to)27 b Fs(S)2234 2758 y Fp(1)2284 2794 y Fo(\002)10 b Fs(I)e Ft(,)28 b(w)m(e)g(ha)m(v)m(e)g(that)f(the)g (curv)m(es)i Fs(X)3576 2809 y Fr(n)3649 2794 y Ft(are)118 2914 y(nearly)36 b(horizon)m(tal)e(and)i(gro)m(w)g(in)g(the)g(horizon)m (tal)e(direction)h(\(when)i Fs(n)f Ft(increases\))h(b)m(y)g(a)f(factor) 118 3034 y(close)c(to)f Fs(d)g Ft(for)g(small)e Fs(\013)j Ft(and)g Fs(\017)p Ft(,)g(see)g([Vi1,)f(Section)h(2.1].)42 b(Hence,)34 b(for)d(large)f Fs(n)p Ft(,)i Fs(X)3187 3049 y Fr(n)3265 3034 y Ft(wraps)g(man)m(y)118 3155 y(times)38 b(around)g(the)i(cylinder)e Fs(S)1339 3119 y Fp(1)1404 3155 y Fo(\002)27 b Fs(I)8 b Ft(.)62 b(On)38 b(the)h(other)g(hand,)i (since)e Fs(Y)2832 3170 y Fr(n)2916 3155 y Ft(=)f Fs(\031)3085 3170 y Fp(2)3151 3155 y Fo(\016)26 b Fs(F)3304 3119 y Fr(n)3351 3155 y Ft(\()p Fs(t)p 3389 3171 V(;)17 b(Y)k Ft(\))38 b(and)118 3275 y Fs(Z)185 3290 y Fr(n)260 3275 y Ft(=)27 b Fs(\031)418 3290 y Fp(2)476 3275 y Fo(\016)19 b Fs(F)622 3239 y Fr(n)668 3275 y Ft(\()p Fs(t)p 706 3291 V(;)e(Z)7 b Ft(\))31 b(are)g(alw)m(a)m(ys)g(tangen)m(t)g(to)g(the) g(v)m(ector)h(\014eld)e Fs(z)j Fo(!)2741 3195 y Fh(\000)2787 3275 y Fs(\030)2835 3239 y Fr(c)2869 3275 y Ft(\()p Fs(\033)2966 3239 y Fr(n)3013 3275 y Fs(t)p 3013 3291 V(;)17 b(z)t Ft(\))p Fs(;)g Ft(1)3272 3195 y Fh(\001)3348 3275 y Ft(on)31 b Fs(S)3548 3239 y Fp(1)3606 3275 y Fo(\002)19 b Fs(I)8 b Ft(,)118 3396 y(it)38 b(follo)m(ws)g(that)h(all)d(the)k(iterates)f (of)f Fs(Y)1611 3411 y Fr(n)1697 3396 y Ft(and)h Fs(Z)1960 3411 y Fr(n)2045 3396 y Ft(ha)m(v)m(e)i(small)36 b(amplitude)h(in)i (the)g Fs(s)p Ft(-direction.)118 3516 y(This)d(giv)m(es)g(a)g(con)m (tradiction,)f(since)h(the)h(closed)f(curv)m(e)h(made)e(b)m(y)i Fs(Y)21 b Ft(,)37 b Fs(Z)42 b Ft(and)36 b Fs(X)44 b Ft(is)35 b(homotopic)118 3636 y(to)29 b(zero)h(in)f Fs(S)614 3600 y Fp(1)669 3636 y Fo(\002)16 b Fs(I)37 b Ft(and)30 b(the)g(closed)f (curv)m(e)j(made)d(b)m(y)h Fs(Y)2182 3651 y Fr(n)2228 3636 y Ft(,)h Fs(Z)2353 3651 y Fr(n)2429 3636 y Ft(and)e Fs(X)2696 3651 y Fr(n)2773 3636 y Ft(cannot)g(b)s(e)h(homotopic)e(to) 118 3757 y(zero)35 b(for)g(large)f Fs(n)p Ft(.)51 b(Th)m(us,)37 b(for)d(\014xed)j Fs(t)p 1522 3773 V 32 w Fo(2)32 b Fs(T)1758 3721 y Fe(N)1845 3757 y Ft(w)m(e)k(ha)m(v)m(e)g(uniqueness)h(of)d (solutions)g(of)h(the)g(v)m(ector)118 3877 y(\014eld)c Fs(z)h Fo(!)c Ft(\()p Fs(\030)619 3841 y Fr(c)653 3877 y Ft(\()p Fs(t)p 691 3893 V(;)17 b(z)t Ft(\))p Fs(;)g Ft(1\),)31 b(and)g(from)e(\(34\))i(it)f(follo)m(ws)f(that)i Fo(F)2373 3841 y Fr(c)2363 3902 y(t)p 2363 3914 30 3 v 2438 3877 a Ft(is)f(an)h Fs(F)14 b Ft(-in)m(v)-5 b(arian)m(t)29 b(foliation)e(of)k Fs(M)118 3998 y Ft(b)m(y)j(nearly)e(v)m(ertical)g (lea)m(v)m(es.)p 3709 3998 4 66 v 3713 3935 59 4 v 3713 3998 V 3771 3998 4 66 v 264 4193 a(No)m(w,)27 b(using)d(the)h (foliations)c(giv)m(en)k(b)m(y)g(the)g(previous)g(prop)s(osition)d(w)m (e)k(are)e(also)f(able)h(to)g(de\014ne)118 4313 y(the)32 b(Mark)m(o)m(v)g(partitions)e(of)h Fs(S)1258 4277 y Fp(1)1328 4313 y Ft(in)g(this)g(setting.)43 b(Giv)m(en)31 b(an)m(y)h(smo)s(oth)e (map)h Fs(X)k Ft(:)28 b Fs(S)3248 4277 y Fp(1)3315 4313 y Fo(!)f Fs(I)39 b Ft(whose)118 4433 y(graph)f(is)f(nearly)g(horizon)m (tal,)h(denote)1639 4408 y Fh(b)1614 4433 y Fs(X)1703 4397 y Fr(n)1695 4458 y(t)p 1695 4470 30 3 v 1750 4433 a Ft(\()p Fs(s)p Ft(\))e(=)h Fs(f)2080 4397 y Fr(n)2069 4458 y(t)p 2069 4470 V 2126 4353 a Fh(\000)2172 4433 y Fs(s;)17 b(X)8 b Ft(\()p Fs(s)p Ft(\))2473 4353 y Fh(\001)2556 4433 y Ft(for)37 b Fs(n)g Fo(\025)g Ft(0)g(and)h Fs(s)e Fo(2)h Fs(S)3451 4397 y Fp(1)3490 4433 y Ft(.)59 b(T)-8 b(ak)m(e)118 4554 y(some)28 b(leaf)f Fs(L)601 4518 y Fp(0)601 4578 y Fr(t)p 601 4590 V 668 4554 a Ft(of)h(the)g(foliation)c Fo(F)1397 4518 y Fr(c)1387 4578 y(t)p 1387 4590 V 1432 4554 a Ft(.)41 b(Letting)27 b Fs(L)1904 4518 y Fr(n)1904 4578 y(t)p 1904 4590 V 1980 4554 a Ft(=)g Fs(f)2142 4518 y Fr(n)2131 4578 y(t)p 2131 4590 V 2189 4554 a Ft(\()p Fs(L)2293 4569 y Fr(t)p 2293 4581 V 2323 4554 a Ft(\))g(for)h Fs(n)g Fo(\025)g Ft(1,)g(w)m(e)h(de\014ne)g(the)g(sequence)118 4674 y(of)j(Mark)m(o)m(v)i(partitions)d(\()p Fo(P)1144 4638 y Fr(n)1136 4699 y(t)p 1136 4711 V 1191 4674 a Ft(\))1229 4689 y Fr(n)1309 4674 y Ft(of)h Fs(S)1486 4638 y Fp(1)1558 4674 y Ft(as)308 4936 y Fo(P)385 4895 y Fr(n)377 4961 y(t)p 377 4973 V 460 4936 a Ft(=)563 4826 y Fh(n)630 4936 y Ft([)p Fs(s)703 4895 y Fq(0)726 4936 y Fs(;)17 b(s)816 4895 y Fq(00)858 4936 y Ft(\))11 b(:)33 b(\()p Fs(s)1051 4895 y Fq(0)1075 4936 y Fs(;)17 b(s)1165 4895 y Fq(0)o(0)1207 4936 y Ft(\))32 b(is)g(a)h(connected)h(comp)s(onen)m(t) e(of)g(\()2584 4911 y Fh(b)2559 4936 y Fs(X)2648 4895 y Fr(n)2640 4961 y(t)p 2640 4973 V 2695 4936 a Ft(\))2733 4895 y Fq(\000)p Fp(1)2827 4856 y Fh(\000)2873 4936 y Ft(\()p Fs(S)2977 4895 y Fp(1)3038 4936 y Fo(\002)23 b Fs(I)8 b Ft(\))22 b Fo(n)g Fs(L)3387 4895 y Fr(n)3387 4961 y(t)p 3387 4973 V 3434 4856 a Fh(\001)3480 4826 y(o)3563 4936 y Fs(:)1900 5251 y Ft(42)p eop %%Page: 43 43 43 42 bop 118 548 a Ft(It)33 b(is)f(easy)h(to)g(c)m(hec)m(k)i(that)d Fo(P)1207 507 y Fr(n)p Fp(+1)1199 570 y Fr(t)p 1199 582 30 3 v 1377 548 a Ft(re\014nes)i Fo(P)1758 512 y Fr(n)1750 573 y(t)p 1750 585 V 1838 548 a Ft(for)e(eac)m(h)i Fs(n)27 b Fo(\025)i Ft(1)j(and,)h(taking)e Fs(\017)e Fo(\034)e Fs(\013)q Ft(,)1105 777 y(\()p Fs(d)22 b Ft(+)g(const)17 b Fs(\013)q Ft(\))1654 736 y Fq(\000)p Fr(n)1783 777 y Fo(\024)28 b(j)p Fs(!)t Fo(j)f(\024)h Ft(\()p Fs(d)21 b Fo(\000)i Ft(const)17 b Fs(\013)q Ft(\))2691 736 y Fq(\000)p Fr(n)118 997 y Ft(for)38 b(eac)m(h)i Fs(!)i Fo(2)d(P)784 961 y Fr(n)776 1021 y(t)p 776 1033 V 831 997 a Ft(.)63 b(This)39 b(p)s(ermits)f(to)g(obtain)g(estimates)h(for)f (the)h(Leb)s(esgue)h(measure)g(of)e(the)118 1117 y(sets)f Fs(B)386 1132 y Fp(1)426 1117 y Ft(\()p Fs(n)p Ft(\))f(and)f Fs(B)862 1132 y Fp(2)902 1117 y Ft(\()p Fs(n)p Ft(\))h(exactly)g(in)f (the)h(same)g(w)m(a)m(y)h(as)f(b)s(efore)g(also)f(with)h(the)g(constan) m(ts)h(only)118 1237 y(dep)s(ending)c(on)f(the)h(quadratic)g(map)e Fs(Q)i Ft(\(cf.)44 b(Remark)32 b(6.4\).)118 1570 y Fu(References)118 1789 y Ft([Al])163 b(J.)42 b(F.)e(Alv)m(es,)k Fi(SRB)e(me)-5 b(asur)g(es)42 b(for)g(non-hyp)-5 b(erb)g(olic)41 b(systems)i(with)f (multidimensional)436 1910 y(exp)-5 b(ansion)p Ft(,)31 b(Ann.)j(Scien)m(t.)1476 1885 y(\023)1468 1910 y(Ec.)f(Norm.)f(Sup.,)h (4)2236 1874 y Fr(e)2305 1910 y Ft(s)m(\023)-46 b(erie,)33 b(33)f(\(2000\),)g(1-32.)118 2113 y([ABV])49 b(J.)22 b(F.)g(Alv)m(es,)j(C.)d(Bonatti,)h(M.)g(Viana,)g Fi(SRB)h(me)-5 b(asur)g(es)25 b(for)f(p)-5 b(artial)5 b(ly)25 b(hyp)-5 b(erb)g(olic)25 b(systems)436 2233 y(with)35 b(mostly)g(exp)-5 b(anding)33 b(c)-5 b(entr)g(al)35 b(dir)-5 b(e)g(ction)p Ft(,)32 b(In)m(v)m(en)m(t.)j(Math.)e(\(to)f(app)s(ear\).)118 2437 y([A)-11 b(V])129 b(J.)33 b(F.)g(Alv)m(es,)h(M.)f(Viana,)f Fi(Statistic)-5 b(al)35 b(stability)g(for)g(r)-5 b(obust)35 b(classes)f(of)h(maps)f(with)h(non-)436 2557 y(uniform)g(exp)-5 b(ansion)p Ft(,)31 b(preprin)m(t)h(CMUP)i(1999.)118 2761 y([Ar1])104 b(V.)38 b(Ara)s(\023)-51 b(ujo,)36 b Fi(A)n(ttr)-5 b(actors)40 b(and)e(time)h(aver)-5 b(ages)38 b(for)h(r)-5 b(andom)38 b(maps)p Ft(,)f(Ann.)h(de)f(l'Inst.)g(H.)436 2881 y(P)m(oincar)m(\023)-46 b(e)33 b(-)f(Anal.)g(non-Lin.)f(\(to)i (app)s(ear\).)118 3084 y([Ar2])104 b(V.)23 b(Ara)s(\023)-51 b(ujo,)23 b Fi(In\014nitely)i(many)g(sto)-5 b(chastic)g(al)5 b(ly)24 b(stable)i(attr)-5 b(actors)p Ft(,)24 b(preprin)m(t)f(CMUP)g (2000.)118 3288 y([BaV])73 b(V.)27 b(Baladi,)f(M.)h(Viana,)g Fi(Str)-5 b(ong)29 b(sto)-5 b(chastic)29 b(stability)h(and)f(r)-5 b(ate)30 b(of)f(mixing)f(for)h(unimo)-5 b(dal)436 3408 y(maps)p Ft(,)32 b(Ann.)h(Scien)m(t.)1282 3383 y(\023)1273 3408 y(Ec.)h(Norm.)d(Sup.,)j(4)2042 3372 y Fr(e)2111 3408 y Ft(s)m(\023)-46 b(erie,)33 b(29)f(\(1996\),)f(483-517.)118 3612 y([BC1])76 b(M.)30 b(Benedic)m(ks)g(and)f(L.)g(Carleson,)h Fi(On)h(iter)-5 b(ations)31 b(of)g Ft(1)15 b Fo(\000)g Fs(ax)2726 3575 y Fp(2)2797 3612 y Fi(on)31 b Ft(\()p Fo(\000)p Ft(1)p Fs(;)17 b Ft(1\),)30 b(Ann.)f(Math.)436 3732 y(122)j(\(1985\),)g(1-25.)118 3935 y([BC2])76 b(M.)37 b(Benedic)m(ks)h(and)f(L.)f(Carleson,)h Fi(The)h(dynamics)f(of)h(the)h (H)n(\023)-47 b(enon)37 b(map)p Ft(,)f(Ann.)h(Math.)436 4056 y(133)32 b(\(1991\),)g(73-169.)118 4259 y([BY])122 b(M.)32 b(Benedic)m(ks,)h(L.-S.)e(Y)-8 b(oung,)31 b Fi(A)n(bsolutely)j (c)-5 b(ontinuous)33 b(invariant)f(me)-5 b(asur)g(es)33 b(and)g(r)-5 b(an-)436 4380 y(dom)32 b(p)-5 b(erturb)g(ations)33 b(for)g(c)-5 b(ertain)32 b(one-dimensional)e(maps)p Ft(,)g(Erg.)g(Th.)h (&)f(Dyn.)g(Sys.)62 b(12)436 4500 y(\(1992\),)32 b(13-37.)118 4703 y([BY])122 b(M.)30 b(Benedic)m(ks)h(and)f(L.-S.)f(Y)-8 b(oung,)30 b Fi(SRB-me)-5 b(asur)g(es)30 b(for)i(c)-5 b(ertain)31 b(H)n(\023)-47 b(enon)31 b(maps)p Ft(,)e(In)m(v)m(en)m(t.) 436 4824 y(Math.)66 b(112)32 b(\(1993\),)f(541-576.)1900 5251 y(43)p eop %%Page: 44 44 44 43 bop 118 548 a Ft([BV])122 b(C.)53 b(Bonatti,)j(M.)c(Viana,)k Fi(SRB)c(me)-5 b(asur)g(es)52 b(for)h(p)-5 b(artial)5 b(ly)52 b(hyp)-5 b(erb)g(olic)52 b(systems)h(with)436 668 y(mostly)35 b(c)-5 b(ontr)g(acting)35 b(c)-5 b(entr)g(al)34 b(dir)-5 b(e)g(ction)p Ft(,)32 b(Israel)h(J.)f(Math.)h(\(to)f(app)s (ear\).)118 865 y([BR])123 b(R.)35 b(Bo)m(w)m(en)i(and)e(D.)g(Ruelle,)g Fi(The)h(er)-5 b(go)g(dic)37 b(the)-5 b(ory)37 b(of)g(Axiom)g(A)h (\015ows)p Ft(,)e(In)m(v)m(en)m(t.)h(Math.)436 985 y(29)32 b(\(1975\),)g(181-202.)118 1182 y([HPS])71 b(M.)36 b(Hirsc)m(h,)g(C.)g (Pugh,)g(M.)g(Sh)m(ub,)g Fi(Invariant)h(manifolds)p Ft(,)d(Lect.)i (Notes)f(in)g(Math.)g(583,)436 1303 y(Springer)e(V)-8 b(erlag,)31 b(1977.)118 1499 y([Ja])165 b(M.)27 b(Jak)m(obson,)i Fi(A)n(bsolutely)g(c)-5 b(ontinuous)29 b(invariant)f(me)-5 b(asur)g(es)29 b(for)f(one-p)-5 b(ar)g(ameter)28 b(fam-)436 1620 y(ilies)35 b(of)f(one-dimensional)f(maps)p Ft(,)e(Comm.)h(Math.)h (Ph)m(ys.)67 b(81)32 b(\(1981\),)f(39-88.)118 1816 y([Ki1])111 b(Y)-8 b(u.)39 b(Kifer,)g Fi(Er)-5 b(go)g(dic)40 b(the)-5 b(ory)41 b(of)f(r)-5 b(andom)39 b(p)-5 b(erturb)g(ations)p Ft(,)40 b(Birkh\177)-49 b(auser,)41 b(Boston)e(Basel,)436 1937 y(1986.)118 2133 y([Ki2])111 b(Y)-8 b(u.)50 b(Kifer,)i Fi(R)-5 b(andom)49 b(p)-5 b(erturb)g(ations)50 b(of)g(dynamic)-5 b(al)49 b(systems)p Ft(,)k(Birkh\177)-49 b(auser,)54 b(Boston)436 2254 y(Basel,)33 b(1988.)118 2451 y([Ma])126 b(R.)33 b(Ma)s(~)-51 b(n)m(\023)-46 b(e,)32 b Fi(Er)-5 b(go)g(dic)34 b(the)-5 b(ory)36 b(and)e(di\013er)-5 b(entiable)33 b(dynamics)p Ft(,)f(Springer-V)-8 b(erlag,)31 b(1987.)118 2647 y([Pl])170 b(V.)33 b(Pliss,)f Fi(On)j(a)f(c)-5 b(onje)g(ctur)g(e) 35 b(due)g(to)g(Smale)p Ft(,)c(Di\013.)h(Ura)m(vnenija,)65 b(8)32 b(\(1972\),)g(262-268.)118 2844 y([Ru])138 b(D.)36 b(Ruelle,)g Fi(A)j(me)-5 b(asur)g(e)38 b(asso)-5 b(ciate)g(d)37 b(with)h(Axiom)g(A)g(attr)-5 b(actors)p Ft(,)38 b(Amer.)e(Jour.)g (Math.)436 2964 y(98)c(\(1976\),)g(619-654.)118 3161 y([Si])182 b(Y)-8 b(a.)27 b(Sinai,)f Fi(Gibbs)j(me)-5 b(asur)g(es)29 b(in)g(er)-5 b(go)g(dic)28 b(the)-5 b(ory)p Ft(,)28 b(Russ.)g(Math.)f(Surv.)54 b(27,)27 b(n.)g(4,)g(\(1972\),)436 3281 y(21-69.)118 3478 y([Vi1])114 b(M.)39 b(Viana,)g Fi(Multidimensional)h(non-hyp)-5 b(erb)g(olic)38 b(attr)-5 b(actors)p Ft(,)41 b(Publ.)d(Math.)h(IHES)78 b(85)436 3599 y(\(1997\),)32 b(63-96.)118 3795 y([Vi2])114 b(M.)28 b(Viana,)f Fi(Sto)-5 b(chastic)29 b(dynamics)f(of)i(deterministic)e (systems)p Ft(,)g(Lect.)g(Notes)f(XXI)g(Braz.)436 3916 y(Math)33 b(Collo)s(q.,)e(IMP)-8 b(A,)34 b(1997.)118 4112 y([Y)-8 b(o])150 b(L.-S.)33 b(Y)-8 b(oung,)33 b Fi(Sto)-5 b(chastic)34 b(stability)i(of)f(hyp)-5 b(erb)g(olic)34 b(attr)-5 b(actors)p Ft(,)33 b(Erg.)g(Th.)h(&)e(Dyn.)h(Sys.)436 4233 y(6)g(\(1986\),)e(311-319.)118 4521 y(Jos)m(\023)-46 b(e)33 b(F)-8 b(erreira)32 b(Alv)m(es)h(\()p Fa(jfalves@fc.up.pt)p Ft(\))118 4641 y(V)-11 b(\023)-38 b(\020tor)31 b(Ara)s(\023)-51 b(ujo)32 b(\()p Fa(vdaraujo@fc.up.pt)p Ft(\))118 4761 y(Cen)m(tro)i(de)f(Matem\023)-49 b(atica)31 b(da)h(Univ)m(ersidade)h (do)g(P)m(orto)118 4882 y(Pra\030)-43 b(ca)33 b(Gomes)f(T)-8 b(eixeira,)32 b(4099-002)e(P)m(orto,)j(P)m(ortugal)1900 5251 y(44)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0011211027497--