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Fg(Z)3039 5388 y Fo(1)p Fm(=)p Fo(\()p Fm(z)d Fn(\000)c Fm(w)h Fo(+)f Fm(n)p Fo(\))22 b(is)g(a)g(sum)-47 5479 y(of)h(translates)h(of)g(the)g(standard)h(Cauc)n(h)n(y)g(k)n (ernel.)1822 5777 y FA(7)p eop %%Page: 8 8 8 7 bop 77 71 a FA(W)-7 b(e)29 b(ma)n(y)e(de\014ne)h(a)g(holomorphic)f (extension)g(of)h(the)h(P)n(erron)d(F)-7 b(rob)r(enius)27 b(op)r(erator)g Fz(L)g FA(acting)h(on)g(the)g(space)f Fz(E)3659 83 y Fy(\003)3709 71 y FA(:)-47 170 y(F)-7 b(or)32 b Fz(\036)h Ft(2)f Fz(E)337 182 y Fy(\003)420 170 y FA(and)h(an)n(y)f Ft(j)p FA(\003)p Ft(j)p FA(-c)n(hain)g Fz(\033)37 b FA(in)c Fz(A)1353 182 y Fy(\003)1436 170 y FA(w)n(e)g(mak)n(e)f(the)h(follo)n(wing)f(iden)n(ti\014cation)h(\(as) g(already)e(men)n(tioned,)k Ft(\006)-47 270 y FA(signi\014es)27 b(the)h(orien)n(tation)e(of)i(the)f(map)h Fz(F)1303 282 y Fy(\003)1381 270 y FA(restricted)f(to)g(the)h(real)f(torus\))g(:)1342 375 y Fx(Z)1389 564 y Fw(\033)1447 488 y Fz(\026)1497 500 y Fy(\003)1570 488 y Fz(L)1627 500 y Fy(\003)1676 488 y Fz(\036)c Ft(\021)g(\006)1915 375 y Fx(Z)1961 564 y Fw(F)2012 537 y Ff(\000)p Fk(1)2003 583 y(\003)2089 564 y Fw(\033)2147 488 y Fz(\026)2197 500 y Fy(\003)2270 488 y Fz(\036:)1178 b FA(\(3.13\))-47 725 y(A)28 b(c)n(hange)e(of)i(co) r(ordinates)e(yields)h(the)h(follo)n(wing)f(form)n(ula)f(for)h(the)h (op)r(erator:)1320 939 y Fz(L)1377 951 y Fy(\003)1426 939 y Fz(\036)1475 951 y Fw(w)1552 939 y FA(=)23 b Ft(\006)1794 860 y Fx(X)1719 1038 y Fw(z)r Fy(:)p Fw(F)1814 1046 y Fk(\003)1855 1038 y Fw(z)r Fy(=)p Fw(w)2129 883 y Fz(\036)2178 895 y Fw(z)p 2013 920 320 4 v 2013 996 a FA(det)15 b Fz(F)2196 1008 y Fy(\003)k Fv(\003)p Fw(z)2342 939 y Fz(:)1155 b FA(\(3.14\))77 1202 y(T)-7 b(o)31 b(see)f(that)i(\(3.13\))e (is)h(w)n(ell-de\014ned,)g(note)g(that)g(b)n(y)g(Section)g(E,)g Fz(F)2283 1214 y Fy(\003)2364 1202 y FA(has)g(lo)r(cal)f(in)n(v)n (erses)f(whic)n(h)i(are)f(holomor-)-47 1301 y(phic.)54 b(Summing)34 b(the)f(pull-bac)n(ks)g(of)g Fz(\026)1254 1313 y Fy(\003)1303 1301 y Fz(\036)h 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y Fz(\036)h FA(=)e Ft(\006)1824 1900 y Fx(Z)1870 2088 y Fw(F)1921 2061 y Ff(\000)p Fk(1)1912 2108 y(\003)1998 2088 y Fw(\033)2057 2013 y Fz(\026)2107 2025 y Fy(\003)2179 2013 y Fz(a)d Ft(\016)f Fz(F)2355 2025 y Fy(\003)2428 2013 y Fz(\036;)1020 b FA(\(3.15\))-47 2254 y(v)-5 b(alid)31 b(for)f(an)n(y)g Ft(j)p FA(\003)p Ft(j)p FA(-c)n(hain)g Fz(\033)k FA(and)d(an)n(y)f(con)n(tin)n(uous)g (function)i Fz(a)c Ft(2)h Fz(C)6 b FA(\()p Fz(\033)s FA(\))32 b(on)f(the)g(c)n(hain.)46 b(If)31 b(w)n(e)g(here)f(let)i Fz(\033)g FA(=)c Fz(S)3590 2266 y Fy(\003)3668 2254 y FA(=)-47 2354 y Fz(F)18 2318 y Fv(\000)p Fy(1)6 2378 y(\003)107 2354 y Fz(S)158 2366 y Fy(\003)241 2354 y FA(the)34 b(form)n(ula)f(extends)g(to)h Fz(a)f Ft(2)g Fz(L)1339 2324 y Fv(1)1409 2354 y FA(\()p Fz(S)1492 2366 y Fy(\003)1542 2354 y FA(\))g(and)h(w)n(e)f(see)g(that)h(\(3.13\))f (extends)h(the)g(real)e(P)n(erron)g(F)-7 b(rob)r(enius)-47 2453 y(op)r(erator)27 b(when)j(acting)f(up)r(on)h(an)f(elemen)n(t)g(of) h Fz(E)1554 2465 y Fy(\003)1604 2453 y FA(.)42 b(W)-7 b(e)30 b(shall)f(not)h(mak)n(e)e(use)i(of)f(the)h(explicit)g(form)n (ula)e(\(3.14\))h(but)-47 2553 y(rather)d(the)i(follo)n(wing)f (represen)n(tation)f(theorem:)-47 2729 y Fr(Theorem)k(3.2)41 b Fs(Under)e(the)g(assumption)g Fz(\024)h(<)f FA(\()p Fz(\025)26 b Ft(\000)f FA(1\))p Fz(\032)p Fs(,)41 b(the)e(holomorphic)j (lo)l(c)l(al)e(Perr)l(on-F)-6 b(r)l(ob)l(enius)39 b(op)l(er)l(ator)-47 2828 y Fz(L)10 2840 y Fy(\003)82 2828 y FA(:)23 b Fz(E)189 2840 y Fy(\003)262 2828 y Ft(!)g Fz(E)429 2840 y Fy(\003)479 2828 y Fs(,)30 b FA(\003)23 b Ft(2)g(F)k(n)18 b(;)29 b Fs(has)h(the)g(fol)t(lowing)i(inte)l(gr)l(al)e(r)l(epr)l(esentation)g (:)957 3046 y Fz(L)1014 3058 y Fy(\003)1063 3046 y Fz(\036)1112 3058 y Fw(w)1189 3046 y FA(=)23 b Ft(\006)1356 2933 y Fx(Z)1401 3122 y Fy(\000)1442 3130 y Fk(\003)1502 3046 y Fz(\026)1552 3058 y Fy(\003)1625 3046 y Fz(k)1668 3058 y Fw(w)1740 3046 y Ft(\016)18 b Fz(F)1853 3058 y Fy(\003)1926 3046 y Fz(\036;)104 b(w)26 b Ft(2)d 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Fw(p)674 5414 y FA(\000[)p Fz(\025\032)p FA(])h(to)f(\000[)p Fz(\032)p FA(])g(whic)n(h)g(sta)n(ys) g(in)g Fz(A)p FA([)p Fz(\032)p FA(])19 b Ft(\000)f Fz(U)36 b FA(and)27 b(preserv)n(es)e(the)j(orien)n(tation.)35 b(Let)27 b(us)g(do)g(so)g(for)-47 5528 y(eac)n(h)33 b Fz(p)h Ft(2)g FA(\003.)56 b(Their)33 b(direct)h(pro)r(duct)g(then)h (giv)n(es)e(a)g(smo)r(oth)h(homotop)n(y)f(b)r(et)n(w)n(een)h(\000)2794 5540 y Fy(\003)2844 5528 y FA([)p Fz(\032)p FA(])g(and)g Fz(f)3185 5492 y Fv(\000)p Fy(1)3176 5552 y(\003)3273 5528 y FA(\000[)p Fz(\025\032)p FA(])h(during)1822 5777 y(8)p eop %%Page: 9 9 9 8 bop -47 71 a FA(whic)n(h)27 b Ft(j)p FA(Im)14 b Fz(f)367 83 y Fw(p)405 71 y Ft(j)24 b(\025)e Fz(\025\032)28 b FA(for)f(eac)n(h)g Fz(p)c Ft(2)g FA(\003.)37 b(Hence,)28 b Fz(k)1546 83 y Fw(w)1618 71 y Ft(\016)18 b Fz(F)1731 83 y Fy(\003)1809 71 y FA(is)27 b(non-singular)f(under)h(this)h (homotop)n(y)f(as)g(w)n(ell.)77 270 y(Comp)r(osing)c(the)h(ab)r(o)n(v)n 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Fw(n)2252 5442 y Ft(\001)g FA(\()p Fz(g)2365 5454 y Fw(p)2404 5442 y FA(\))2436 5408 y Fw(n)3520 5442 y FA(\(3.35\))1801 5777 y(10)p eop %%Page: 11 11 11 10 bop -47 71 a FA(and)29 b(since)f Ft(j)p Fz(g)383 83 y Fw(p)421 71 y Ft(j)e(\024)f(j)p Fz(g)623 83 y Fw(p)661 71 y Ft(j)684 83 y Fw(\022)746 71 y Ft(\024)g Fz(\024)g(<)g FA(\()p Fz(\025)20 b Ft(\000)f FA(1\))p Fz(\032)29 b FA(this)g(sum)g(con)n(v)n(erges)e(uniformly)h(in)i(the)f(Banac)n(h)f (algebra)f Fz(E)3316 83 y Fw(p;)p Fy(\012)3422 71 y FA(.)41 b(This)28 b(is)-47 170 y(not)j(y)n(et)g(in)h(the)f(form)g(w)n(e)g(w)n (an)n(t)g(since)g(the)g(righ)n(t)g(hand)g(side)g(in)n(v)n(olv)n(es)f (`in)n(teractions')g(with,)i(in)g(general,)f(in\014nitely)-47 270 y(man)n(y)c(v)-5 b(ariables.)77 370 y(Using)31 b(that)h Fz(H)567 382 y Fw(\022)635 370 y FA(is)g(a)e(Banac)n(h)g(algebra,)h(w)n (e)f(ha)n(v)n(e)g Ft(j)p FA(\()p Fz(g)1829 382 y Fw(p)1868 370 y FA(\))1900 340 y Fw(n)1945 370 y Ft(j)1968 382 y Fw(\022)2035 370 y Ft(\024)e(j)p Fz(g)2191 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y Fw(I)1689 5528 y FA(=)d(0)k(and)g Fz(\031)2054 5540 y Fw(\013;K)2182 5528 y Fz(L)2239 5540 y Fw(K)2302 5528 y FA([)p Fz(C)6 b FA(])28 b(v)-5 b(anishes)27 b(as)g(claimed.)1801 5777 y(14)p eop %%Page: 15 15 15 14 bop 912 -59 1861 4 v 912 1270 4 1329 v 1412 896 4 748 v 1744 896 V 2076 896 V 2408 896 V 2078 647 a Fe(\032)2161 585 y(\032)2244 522 y(\032)2327 460 y(\032)2078 647 y Fd(i)1414 896 y(r)1414 647 y(r)1414 398 y(r)1414 149 y(r)1746 896 y(r)1746 647 y(r)1746 398 y(r)1746 149 y(r)2078 896 y(r)2078 647 y(r)2078 398 y(r)2078 149 y(r)2410 896 y(r)2410 647 y(r)2410 398 y(r)2410 149 y(r)1081 174 y Fz(t)24 b FA(=)e(0)1081 921 y Fz(T)35 b FA(=)22 b(3)1414 1000 y Ft(j)h( )1873 1007 y Fz(K)2281 1000 y Ft(!)g(j)1746 87 y(j)g( )2056 94 y Fz(I)2281 87 y Ft(!)g(j)1356 1158 y FA(1a\))214 b(1b\))g(2a\))g(2b\))p 2770 1270 4 1329 v 912 1273 1861 4 v -47 1452 a(Figure)20 b(2:)34 b(A)22 b(sample)e(con\014guration)g(whic)n(h)h(illustrates)g(the)h(four)f (di\013eren)n(t)g(cases)g(\(b)n(y)g(remo)n(ving)f(the)h(corresp)r (onding)-47 1552 y(p)r(oin)n(t\))29 b(in)f(the)h(pro)r(of)f(of)g(Prop)r (osition)f(4.5.)39 b(Here,)28 b Ft(j)p Fz(K)6 b Ft(j)24 b FA(=)g(4,)29 b Ft(j)p Fz(I)7 b Ft(j)24 b FA(=)g(3)k(and)h(there)f(is) g(one)g(branc)n(hing)f(\(the)i(big)g(circle\))-47 1652 y(in)n(to)e(a)g(set)h(of)f(t)n(w)n(o)g(p)r(oin)n(ts.)77 2122 y(When)h Fz(p)g FA(is)f(ev)n(en)n(tually)g(included)h(in)g(the)g (expansion)f(of)g Fz(\013)p FA(,)h(then)g Fz(\034)33 b(>)23 b FA(0)k(and)g(the)h(dic)n(hotom)n(y)f(is)g(as)g(follo)n(ws:)3 2295 y(2a\))41 b(Either)32 b Fz(p)f Ft(2)h Fz(S)631 2307 y Fw(\034)7 b Fv(\000)p Fy(1)789 2295 y FA(and)33 b(\()p Fz(S)1039 2307 y Fw(\034)7 b Fv(\000)p Fy(1)1165 2295 y Fz(;)14 b(V)1250 2307 y Fw(\034)7 b Fv(\000)p Fy(1)1377 2295 y FA(\))33 b(can)f Fs(not)g FA(b)r(e)h(a)f(branc)n(hing)f(o)n(v)n (er)g Fz(\013)2566 2307 y Fw(\034)2639 2295 y FA(=)f Fz(K)2805 2307 y Fw(\034)2868 2295 y Ft(\000)21 b(f)p Fz(p)p Ft(g)p FA(.)51 b(Hence,)33 b Fz(C)39 b FA(is)33 b Fs(not)160 2395 y FA(a)e(con\014guration)e(o)n(v)n(er)g Fz(\013)g FA(=)f Fz(K)e Ft(\000)20 b(f)p Fz(p)p Ft(g)p FA(.)46 b(Indeed,)31 b(Lemma)g(4.2,)g(\(case)f(1\),)i(sho)n(ws)d(that)j Fz(\031)3041 2407 y Fw(\013)3084 2415 y Fl(\034)3121 2407 y Fw(;K)3197 2415 y Fl(\034)3238 2395 y Fz(L)3295 2407 y Fw(K)3351 2415 y Fl(\034)3388 2407 y Fw(;S)3449 2415 y Fl(\034)5 b Ff(\000)p Fk(1)3559 2407 y Fw(;V)3618 2415 y Fl(\034)g Ff(\000)p Fk(1)160 2494 y FA(v)-5 b(anishes.)-1 2657 y(2b\))41 b(Or,)31 b Fz(p)38 b(=)-51 b Ft(2)29 b Fz(S)518 2669 y Fw(\034)7 b Fv(\000)p Fy(1)676 2657 y FA(and)31 b Fz(p)f FA(m)n(ust)i(then)f(b)r(elong)g(to)g(the)g(set)g Fz(V)2016 2669 y Fw(\034)7 b Fv(\000)p Fy(1)2143 2657 y FA([)p Fz(S)2217 2669 y Fw(\034)g Fv(\000)p Fy(1)2344 2657 y FA(])20 b Ft(\000)h Fz(S)2524 2669 y Fw(\034)7 b Fv(\000)p Fy(1)2650 2657 y FA(.)47 b(Lemma)31 b(4.2,)g(\(case)g(2\),) g(sho)n(ws)160 2756 y(that)e Fz(\031)388 2768 y Fw(\013)431 2776 y Fl(\034)473 2756 y Fz(L)530 2768 y Fw(K)586 2776 y Fl(\034)622 2768 y Fw(;S)683 2776 y Fl(\034)5 b Ff(\000)p Fk(1)793 2768 y Fw(;V)852 2776 y Fl(\034)g Ff(\000)p Fk(1)992 2756 y FA(=)24 b Fz(L)1138 2768 y Fw(\013)1181 2776 y Fl(\034)1218 2768 y Fw(;S)1279 2776 y Fl(\034)5 b Ff(\000)p Fk(1)1388 2768 y Fw(;V)1447 2776 y Fl(\034)g Ff(\000)p Fk(1)1562 2756 y FA(.)40 b(When)29 b Fz(t)24 b(<)g(\034)38 b FA(w)n(e)28 b(ha)n(v)n(e)g Fz(\013)2454 2768 y Fw(t)2508 2756 y FA(=)c Fz(K)2668 2768 y Fw(t)2696 2756 y FA(,)29 b Fz(C)35 b FA(is)28 b(a)g(con\014guration)f(o)n(v)n(er) g Fz(\013)160 2856 y FA(and)h(w)n(e)f(end)h(up)g(with)g Fz(L)967 2868 y Fw(\013)1014 2856 y FA([)p Fz(C)6 b FA(])28 b(as)f(claimed.)p 1619 2802 33 4 v 1619 2870 4 68 v 1648 2870 V 1619 2873 33 4 v -47 3129 a([)p Fs(In)33 b(terms)h(of)g FA(c)n(hains)f Fs(to)h(b)l(e)g(de\014ne)l(d)g(b)l(elow,)i(c)l(ase)e (1a\))h(and)f(2b\))h(c)l(orr)l(esp)l(onds)f(to)g(the)g(r)l(emoval)h(of) g(end-chains.)52 b(In)-47 3228 y(c)l(ase)33 b(1b\))g(we)g(ar)l(e)g(pr)l (oje)l(cting)g(away)h(an)f(initial-chain)i(\(yields)f(zer)l(o\))e(and)h (in)g(c)l(ase)g(2a\))g(the)g(same)g(is)g(happ)l(ening)h(to)-47 3328 y(an)c(ap)l(ex-chain)p FA(].)-47 3559 y Fq(4.3)112 b(Chains)38 b(and)g(trees)-47 3712 y FA(Let)f Fz(C)46 b Ft(2)39 b(C)5 b FA([)p Fz(K)q(;)14 b(T)e FA(])36 b(b)r(e)h(a)g (\014xed)g(time-T)h(con\014guration)d(o)n(v)n(er)h Fz(K)44 b Ft(2)c(F)45 b FA(and)37 b(let)h Fz(K)44 b FA(=)39 b Fz(K)2915 3724 y Fw(T)3006 3712 y Ft(\032)g Fz(:)14 b(:)g(:)39 b Ft(\032)f Fz(K)3420 3724 y Fy(0)3494 3712 y FA(b)r(e)g(the)-47 3812 y(expansion)24 b(of)h Fz(K)30 b FA(along)24 b Fz(C)6 b FA(.)36 b(By)25 b(\()p Fz(S)1083 3824 y Fw(t)1113 3812 y Fz(;)14 b(V)1198 3824 y Fw(t)1227 3812 y FA(\),)26 b Fz(t)d FA(=)g(0)p Fz(;)14 b(:)g(:)g(:)f(;)h(T)24 b Ft(\000)13 b FA(1)25 b(w)n(e)f(denote)h(the)h(branc)n(hing)d(pairs)i (of)g(the)g(con\014guration.)-47 3985 y Fr(De\014nition)31 b(4.6)40 b Fs(The)g(set)e(of)i(p)l(oints)f FA(\()p Fz(q)s(;)14 b(t)p FA(\))39 b Ft(2)h FA(\012)25 b 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Fz(\015)g FA(=)-47 4756 y(\()p Fz(q)s(;)14 b(t)p FA(\))124 4768 y Fw(t)149 4776 y Fk(1)182 4768 y Fv(\024)p Fw(t)p Fv(\024)p Fw(t)336 4776 y Fk(2)372 4756 y Fs(,)32 b(for)g(which)h FA(0)25 b Ft(\024)h Fz(t)986 4768 y Fy(1)1048 4756 y Fz(<)g(t)1169 4768 y Fy(2)1232 4756 y Ft(\024)f Fz(T)12 b Fs(,)31 b Fz(q)38 b(=)-51 b Ft(2)26 b Fz(S)1637 4768 y Fw(t)1662 4776 y Fk(2)1694 4768 y Fv(\000)p Fy(1)1814 4756 y Fs(and)32 b(every)g(interme)l(diate)g (p)l(oint,)g FA(\()p Fz(q)s(;)14 b(t)p FA(\))p Fs(,)33 b Fz(t)3176 4768 y Fy(1)3239 4756 y Fz(<)25 b(t)h(<)f(t)3505 4768 y Fy(2)3542 4756 y Fs(,)32 b(is)g(a)-47 4856 y(fr)l(e)l(e)g(p)l (oint.)46 b(We)32 b(say)h(that)f(the)g(chain)h(is)f(p)l(ositione)l(d)i FA(o)n(v)n(er)c Fz(q)s Fs(,)j FA(starts)e Fs(at)h(time)g Fz(t)2526 4868 y Fy(1)2564 4856 y Fs(,)h FA(stops)e Fs(at)h(time)g Fz(t)3162 4868 y Fy(2)3231 4856 y Fs(and)h(we)f(de\014ne)-47 4955 y(the)e FA(length)f Fs(of)i(the)f(chain)h Fz(\015)j Fs(to)c(b)l(e:)39 b Ft(j)p Fz(\015)5 b Ft(j)22 b(\021)h Fz(t)1354 4967 y Fy(2)1410 4955 y Ft(\000)18 b Fz(t)1523 4967 y Fy(1)1583 4955 y Ft(\025)23 b FA(1)p Fs(.)-47 5129 y Fr(De\014nition)31 b(4.8)40 b Fs(The)31 b(chain)g(\(of)f(the)g (pr)l(evious)h(De\014nition\))e(is)h(c)l(al)t(le)l(d:)51 5287 y(1.)42 b(an)30 b FA(ap)r(ex-c)n(hain)p Fs(,)f(if)i FA(\()p Fz(q)s(;)14 b(t)944 5299 y Fy(1)981 5287 y FA(\))30 b Fs(is)g(an)g(ap)l(ex)g(p)l(oint.)51 5407 y(2.)42 b(an)30 b FA(initial-c)n(hain)p Fs(,)g(if)g Fz(t)876 5419 y Fy(1)936 5407 y FA(=)23 b(0)p Fs(,)30 b FA(\()p Fz(q)s(;)14 b FA(0\))29 b Fs(is)i(a)f(fr)l(e)l(e)g(p)l(oint)g(and)g Fz(q)j Fs(b)l(elongs)d(to)g(the)g(initial)h(set,)p Fz(I)7 b Fs(.)51 5528 y(3.)42 b(an)30 b FA(end-c)n(hain)p Fs(,)g(otherwise)h FA(\()p Fs(i.e.)g Fz(q)26 b Ft(2)e Fz(V)1417 5540 y Fw(t)1442 5548 y Fk(1)1479 5528 y FA([)p Fz(S)1553 5540 y Fw(t)1578 5548 y Fk(1)1614 5528 y FA(])30 b Fs(or)g Fz(t)1804 5540 y Fy(1)1865 5528 y FA(=)22 b(0)29 b Fs(and)i Fz(q)26 b Ft(2)d FA(\()p Fz(K)2429 5540 y Fy(0)2485 5528 y Ft(\000)18 b Fz(I)7 b FA(\))18 b Ft([)h Fz(V)2783 5540 y Fy(0)2821 5528 y FA([)p Fz(S)2895 5540 y Fy(0)2932 5528 y FA(]\))1801 5777 y(15)p eop %%Page: 16 16 16 15 bop 414 -59 2858 4 v 414 938 4 997 v 998 772 a Fd(i)979 789 y Fz(e)p 1329 772 4 623 v 1331 772 a Fd(r)1331 605 y(r)1331 439 y(r)1331 273 y(r)1331 107 y(i)1316 135 y Fz(i)p 2076 772 4 167 v 2408 772 4 333 v 2740 772 4 499 v 1744 772 4 125 v 1746 772 a Fe(\000)1797 720 y(\000)1663 772 y(@)1611 720 y(@)1912 605 y Fd(i)-249 b(i)g(i)p 2076 605 V 415 w Fe(\000)2129 554 y(\000)1995 605 y(@)1943 554 y(@)2244 439 y Fd(i)g(i)g(i)p 2408 439 V 415 w Fe(\000)2461 388 y(\000)2327 439 y(@)2275 388 y(@)2576 273 y Fd(i)g(i)g(i)p 2740 273 V 415 w Fe(\000)2793 222 y(\000)2659 273 y(@)2607 222 y(@)2908 107 y Fd(i)g(i)g(i)p 874 109 374 4 v 1414 109 1080 4 v 2991 109 125 4 v 1746 772 a(r)307 b(r)2078 605 y(r)2410 772 y(r)2410 605 y(r)2410 439 y(r)2742 772 y(r)2742 605 y(r)2742 439 y(r)2742 273 y(r)583 124 y Fz(t)23 b FA(=)g(0)583 796 y Fz(T)34 b FA(=)23 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Fy(\003)688 1475 y FA(\(\()p Fz(j)786 1487 y Fy(\003)p Fw(;K)915 1475 y Fz(b)951 1487 y Fw(K)1015 1475 y FA(\))5 b Ft(\016)g Fz(F)1164 1440 y Fw(T)1152 1495 y Fy(\003)1234 1475 y Ft(\001)18 b Fz(\036)1324 1487 y Fy(\003)1374 1475 y FA(\))24 b(=)e Fz(m)1590 1487 y Fy(\003)1640 1475 y FA(\(\()p Fz(j)1738 1487 y Fy(\003)p Fw(;K)1868 1475 y Fz(b)1904 1487 y Fw(K)1967 1475 y FA(\))p Fz(L)2056 1440 y Fw(T)2056 1495 y Fy(\003)2108 1475 y Fz(\036)2157 1487 y Fy(\003)2207 1475 y FA(\))i(=)e Fz(m)2423 1487 y Fw(K)2487 1475 y FA(\()p Fz(b)2555 1487 y Fw(K)2619 1475 y Fz(\031)2666 1487 y Fw(K;)p Fy(\003)2792 1475 y FA(\()p Fz(L)2881 1440 y Fw(T)2881 1495 y Fy(\003)2933 1475 y Fz(\036)2982 1487 y Fy(\003)3032 1475 y FA(\)\))p Fz(:)360 b FA(\(4.105\))-47 1651 y(W)-7 b(e)34 b(wish)g(to)f(tak)n(e)g (the)h(limit)h(of)e(this)h(equalit)n(y)f(as)g(\003)g Ft(!)h FA(\012.)55 b(On)33 b(the)h(left)h(hand)f(side)f(w)n(e)h(ma)n(y) f(do)g(so)g(as)g(long)g(as)-47 1751 y Fz(\036)23 b Ft(2)h(M)204 1721 y Fw(m)292 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Fs(then)1255 2303 y Fz(\027)1296 2315 y Fw(\036)1340 2303 y FA(\()p Fz(b)19 b Ft(\016)f Fz(F)1552 2269 y Fw(T)1604 2303 y FA(\))23 b(=)g Fz(\027)1788 2322 y Fw(L)1834 2306 y Fk(\()p Fl(T)7 b Fk(\))1924 2322 y Fw(\036)1968 2303 y FA(\()p Fz(b)p FA(\))p Fz(;)104 b(T)34 b Ft(\025)22 b FA(0)p Fz(:)1049 b FA(\(4.106\))-47 2480 y(Pro)r(of:)40 b(Let)30 b Fz(b)404 2492 y Fw(K)495 2480 y Ft(2)d Fz(C)6 b FA(\()p Fz(S)725 2492 y Fw(K)789 2480 y FA(\),)31 b Fz(K)i Ft(2)27 b(F)38 b FA(b)r(e)30 b(a)g(net)g(suc)n(h)g (that)g Fz(j)1897 2492 y Fw(K)1961 2480 y Fz(b)1997 2492 y Fw(K)2091 2480 y FA(con)n(v)n(erges)d(to)j Fz(b)c Ft(2)i Fz(C)6 b FA(\()p Fz(S)2862 2492 y Fy(\012)2914 2480 y FA(\).)44 b(First,)30 b(let)g Fz(K)j Ft(2)27 b(F)38 b FA(b)r(e)-47 2579 y(\014xed.)k(The)30 b(net)g Fz(j)543 2591 y Fy(\003)612 2579 y Ft(\016)19 b FA(\(\()p Fz(j)771 2591 y Fy(\003)p Fw(;K)901 2579 y Fz(b)937 2591 y Fw(K)1001 2579 y FA(\))h Ft(\016)f Fz(F)1179 2549 y Fw(T)1167 2602 y Fy(\003)1231 2579 y FA(\))30 b(con)n(v)n(erges)d(to)i(\()p Fz(j)1836 2591 y Fw(K)1901 2579 y Fz(b)1937 2591 y Fw(K)2000 2579 y FA(\))20 b Ft(\016)g Fz(F)2179 2549 y Fw(T)2260 2579 y FA(as)29 b(\003)d Ft(!)g FA(\012.)43 b(Since)29 b Fz(\036)e Ft(2)f(M)3158 2549 y Fw(m)3221 2579 y FA(,)k(w)n(e)f(ma)n (y)g(tak)n(e)-47 2679 y(the)k(limit)g(on)f(the)h(left)h(hand)e(side)h (\(LHS\))g(in)g(\(4.105\))e(to)i(obtain)f Fz(\027)2163 2691 y Fw(\036)2207 2679 y FA(\(\()p Fz(j)2305 2691 y Fw(K)2370 2679 y Fz(b)2406 2691 y Fw(K)2469 2679 y FA(\))22 b Ft(\016)g Fz(F)2652 2649 y Fw(T)2704 2679 y FA(\))33 b(whic)n(h)f(tends)h(to)g Fz(\027)3386 2691 y Fw(\036)3430 2679 y FA(\()p Fz(b)21 b Ft(\016)h Fz(F)3648 2649 y Fw(T)3700 2679 y FA(\))-47 2779 y(as)29 b Fz(K)j Ft(!)26 b FA(\012.)43 b(Again)30 b(for)f(\014xed)g Fz(K)j Ft(2)27 b(F)38 b FA(w)n(e)29 b(ma)n(y)g(b)n(y)h(Lemma)f(4.26)f(and)i(the)g(assumption)f Fz(\036)e Ft(2)g(M)3161 2791 y Fw(#)3235 2779 y FA(tak)n(e)i(the)h (limit)-47 2878 y(\003)22 b Ft(!)i FA(\012)j(on)g(the)h(righ)n(t)f (hand)h(side)f(\(RHS\))i(of)f(the)g(same)f(equation)g(to)g(get)1128 3055 y Fz(\027)1169 3067 y Fw(\036)1213 3055 y FA(\(\()p Fz(j)1311 3067 y Fw(K)1376 3055 y Fz(b)1412 3067 y Fw(K)1476 3055 y FA(\))18 b Ft(\016)g Fz(F)1651 3020 y Fw(T)1704 3055 y FA(\))23 b(=)g Fz(m)1920 3067 y Fw(K)1983 3055 y FA(\()p Fz(b)2051 3067 y Fw(K)2115 3055 y Fz(\031)2162 3067 y Fw(K)2227 3055 y FA(\()p Fz(L)2316 3020 y Fy(\()p Fw(T)9 b Fy(\))2420 3055 y Fz(\036)p FA(\)\))p Fz(:)923 b FA(\(4.107\))-47 3231 y(F)-7 b(rom)29 b(this)i(last)f(iden)n(tit)n(y) g(w)n(e)g(ma)n(y)f(dra)n(w)g(the)i(w)n(an)n(ted)e(conclusions.)44 b(First,)30 b(the)h(LHS)f(is)g(b)r(ounded)h(b)n(y)f Ft(j)p Fz(b)3400 3243 y Fw(K)3463 3231 y Ft(j)e(k)p Fz(\036)p Ft(k)3647 3243 y Fw(m)3709 3231 y FA(,)-47 3331 y(whence)j(so)g(is)h (the)g(RHS.)g(W)-7 b(e)32 b(m)n(ust)g(therefore)f(ha)n(v)n(e)g Fz(L)1775 3301 y Fy(\()p Fw(T)9 b Fy(\))1878 3331 y Fz(\036)31 b Ft(2)f(M)2143 3301 y Fw(m)2237 3331 y FA(with)j Ft(k)p Fz(L)2530 3301 y Fy(\()p Fw(T)9 b Fy(\))2633 3331 y Fz(\036)p Ft(k)2724 3343 y Fw(m)2817 3331 y Ft(\024)29 b(k)p Fz(\036)p Ft(k)3044 3343 y Fw(m)3107 3331 y FA(.)49 b(Hence,)33 b(w)n(e)e(ma)n(y)-47 3431 y(write)c(the)h(RHS)g(as)f Fz(\027)648 3450 y Fw(L)694 3433 y Fk(\()p Fl(T)7 b Fk(\))785 3450 y Fw(\036)829 3431 y FA(\()p Fz(j)895 3443 y Fw(K)959 3431 y Fz(b)995 3443 y Fw(K)1059 3431 y FA(\))28 b(and)f(tak)n(e)g(the) h(limit)g(as)f Fz(K)i Ft(!)23 b FA(\012)28 b(to)f(obtain)g(\(4.106\))p 2865 3377 V 2865 3445 4 68 v 2894 3445 V 2865 3448 33 4 v -47 3707 a Fr(Remarks)j(4.29)41 b Fs(The)31 b(L)l(emma)f(also)h (shows)g(that)g(the)f(op)l(er)l(ators)h Fz(L)2164 3677 y Fy(\()p Fw(T)9 b Fy(\))2298 3707 y Fs(ar)l(e)31 b(p)l(ositive)g(and)g (sinc)l(e)f Fz(\031)3156 3722 y Fv(;)3195 3707 y Fz(L)3252 3677 y Fy(\()p Fw(T)9 b Fy(\))3380 3707 y FA(=)23 b Fz(\031)3515 3722 y Fv(;)3584 3707 y Fs(they)-47 3806 y(even)30 b(pr)l(eserve)g(the) g(a\016ne)g(subsp)l(ac)l(e)h(of)f(pr)l(ob)l(ability)i(me)l(asur)l(es)d (in)h Ft(M)2224 3771 y Fy(+)2224 3831 y Fw(#)2309 3806 y Fs(\(when)g Fz(T)k Ft(\025)23 b Fz(T)2775 3818 y Fy(0)2811 3806 y Fs(\).)-47 4080 y FB(5)134 b(Pro)t(of)45 b(of)h(Theorem)f(2.1) -47 4262 y FA(Recall)32 b(that)h(w)n(e)f(only)h(consider)e(the)j (time-indep)r(enden)n(t)f(case)f(here)g(\(lea)n(ving)g(the)h(`easy')f (extension)g(to)h(the)g(time-)-47 4361 y(dep)r(enden)n(t)j(case)f(to)g (the)h(reader\).)60 b(Let)36 b Fz(h)1354 4331 y Fy(0)1428 4361 y Ft(\021)g FA(\(1)1603 4373 y Fy(\003)1652 4361 y FA(\))1684 4373 y Fy(\003)p Fv(2F)1869 4361 y Ft(2)g Fz(M)2050 4326 y Fy(+)2041 4386 y Fw(#)2141 4361 y FA(denote)f(the)i (pro)5 b(jectiv)n(e)34 b(family)i(in)g(whic)n(h)f(eac)n(h)-47 4461 y(\003-comp)r(onen)n(t)26 b(is)i(iden)n(tically)f(1.)77 4561 y(F)-7 b(or)31 b Fz(T)40 b Ft(\025)29 b Fz(T)462 4573 y Fy(0)530 4561 y FA(the)i(sequence,)h Fz(h)1096 4530 y Fw(T)1177 4561 y Ft(\021)d Fz(L)1328 4530 y Fy(\()p Fw(T)9 b Fy(\))1431 4561 y Fz(h)1479 4530 y Fy(0)1516 4561 y FA(,)33 b(is)e(in)g Fz(M)1849 4525 y Fy(+)1840 4586 y Fw(#)1935 4561 y FA(and)g(b)n(y)g(Lemma)g(4.25)f(w)n(e)h(ha)n(v) n(e)f Fz(\031)3066 4576 y Fv(;)3104 4561 y Fz(h)3152 4530 y Fw(T)3233 4561 y FA(=)f(1)i(for)f(all)h Fz(T)12 b FA(.)-47 4660 y(The)27 b(same)g(Lemma)h(com)n(bined)f(with)h(Lemma)g (\(4.24\))e(no)n(w)h(implies:)742 4837 y Ft(k)p Fz(h)832 4802 y Fw(t)p Fy(+)p Fw(T)978 4837 y Ft(\000)18 b Fz(h)1109 4802 y Fw(\034)7 b Fy(+)p Fw(T)1250 4837 y Ft(k)1292 4849 y Fw(#)1359 4837 y FA(=)22 b Ft(k)p Fz(L)1545 4802 y Fy(\()p Fw(T)9 b Fy(\))1649 4837 y FA(\()p Fz(h)1729 4802 y Fw(t)1776 4837 y Ft(\000)18 b Fz(h)1907 4802 y Fw(\034)1949 4837 y FA(\))p Ft(k)2023 4849 y Fw(#)2090 4837 y Ft(\024)k FA(2)p Fz(\015)2267 4802 y Fv(\000)p Fw(T)2371 4837 y Fz(;)124 b(T)7 b(;)14 b(t;)g(\034)32 b Ft(\025)23 b Fz(T)2883 4849 y Fy(0)2920 4837 y Fz(;)536 b FA(\(5.108\))-47 5013 y(whic)n(h)35 b(sho)n(ws)e(that)i(the)h (sequence)e(is)h(Cauc)n(h)n(y)f(in)h Fz(M)1722 4978 y Fy(+)1713 5038 y Fw(#)1811 5013 y FA(\(hence)g(also)f(in)h Fz(M)2449 4978 y Fy(+)2440 5038 y Fw(\022)2504 5013 y FA(\).)59 b(The)35 b(limit,)i Fz(h)e Ft(2)h(M)3300 5025 y Fw(#)3344 5013 y FA(,)h(is)e(clearly)-47 5113 y(p)r(ositiv)n(e)25 b(and)g Fz(\031)464 5128 y Fv(;)503 5113 y Fz(h)e FA(=)f(1.)36 b(Hence,)26 b(the)g(asso)r(ciated)e(measure)g Fz(\027)1926 5125 y Fw(h)1995 5113 y FA(is)h(a)h(probabilit)n(y)e(measure)g(on)i Fz(S)3052 5125 y Fy(\012)3103 5113 y FA(.)36 b(By)25 b(Lemma)h(4.28)-47 5213 y(and)g(Lemma)h(4.24,)f Fz(\027)648 5225 y Fw(h)691 5213 y FA(\()p Fz(b)17 b Ft(\016)f Fz(F)899 5182 y Fw(T)952 5213 y FA(\))23 b(=)g Fz(\027)1136 5234 y Fw(L)1182 5215 y Fk(\()p Fl(T)1239 5227 y Fk(0)1271 5215 y(\))1297 5234 y Fw(h)1340 5213 y FA(\()p Fz(b)17 b Ft(\016)g Fz(F)1549 5182 y Fw(T)1601 5213 y FA(\))23 b(=)g Fz(\027)1785 5234 y Fw(L)1831 5215 y Fk(\()p Fl(T)7 b Fk(+)p Fl(T)1973 5227 y Fk(0)2004 5215 y(\))2031 5234 y Fw(h)2074 5213 y FA(\()p Fz(b)p FA(\))23 b(=)g Fz(\027)2326 5225 y Fw(h)2369 5213 y FA(\()p Fz(b)p FA(\))k(for)f(all)h Fz(T)34 b Ft(\025)23 b FA(0,)j Fz(b)d Ft(2)g Fz(C)6 b FA(\()p Fz(S)3284 5225 y Fy(\012)3336 5213 y FA(\).)37 b(No)n(w,)27 b(let)-47 5312 y Fz(a)c Ft(2)g Fz(H)167 5324 y Fw(#)211 5312 y FA(,)k Fz(b)22 b Ft(2)i Fz(C)457 5324 y Fw(\022)521 5312 y FA(and)h(set)h Fz(c)d FA(=)g Fz(\027)996 5324 y Fw(h)1039 5312 y FA(\()p Fz(a)p FA(\))g(=)g Fz(\031)1305 5327 y Fv(;)1343 5312 y FA(\()p Fz(a)16 b(?)e(h)p FA(\).)37 b(W)-7 b(e)26 b(note)g(that)g Fz(\027)2174 5324 y Fw(h)2217 5312 y FA(\()p Fz(b)15 b Ft(\016)f Fz(F)2421 5282 y Fw(T)2488 5312 y Ft(\001)h Fz(a)p FA(\))23 b(=)g Fz(\027)2754 5324 y Fw(a?h)2867 5312 y FA(\()p Fz(b)15 b Ft(\016)f Fz(F)3071 5282 y Fw(T)3124 5312 y FA(\))23 b(=)g Fz(\027)3308 5331 y Fw(L)3354 5315 y Fk(\()p Fl(T)6 b Fk(\))3444 5331 y Fy(\()p Fw(a?h)p Fy(\))3609 5312 y FA(\()p Fz(b)p FA(\).)-47 5428 y(Therefore,)30 b Fz(\027)397 5440 y Fw(h)440 5428 y FA(\()p Fz(b)20 b Ft(\016)g Fz(F)655 5398 y Fw(T)728 5428 y Ft(\001)g Fz(a)p FA(\))h Ft(\000)e Fz(\027)993 5440 y Fw(h)1036 5428 y FA(\()p Fz(b)p FA(\))p Fz(\027)1177 5440 y Fw(h)1221 5428 y FA(\()p Fz(a)p FA(\))28 b(=)f Fz(\027)1490 5440 y Fw( )1541 5428 y FA(\()p Fz(b)p FA(\))j(where)g Fz( )h FA(=)d Fz(L)2149 5398 y Fy(\()p Fw(T)9 b Fy(\))2252 5428 y FA(\()p Fz(a)21 b(?)f(h)p FA(\))g Ft(\000)g Fz(c)28 b(h)g FA(=)f Fz(L)2885 5398 y Fy(\()p Fw(T)9 b Fy(\))2989 5428 y FA(\(\()p Fz(a)21 b Ft(\000)f Fz(c)p FA(1\))g Fz(?)f(h)p FA(\).)46 b(Since)-47 5528 y Fz(\031)0 5543 y Fv(;)38 5528 y FA(\(\()p Fz(a)18 b Ft(\000)f Fz(c)p FA(1\))g Fz(?)g(h)p FA(\))23 b(=)g Fz(\031)670 5543 y Fv(;)708 5528 y FA(\()p Fz(a)18 b(?)f(h)p FA(\))h Ft(\000)e Fz(c)23 b FA(=)g(0)k(our)f(k)n(ernel)g(lemma)h (implies)h(that)f Ft(k)p Fz( )s Ft(k)2516 5540 y Fw(\022)2576 5528 y Ft(\024)22 b FA(2)p Fz(\015)2753 5497 y Fv(\000)p Fw(T)2857 5528 y Ft(j)p Fz(a)p Ft(j)2947 5540 y Fw(#)2991 5528 y FA(.)37 b(Finally)-7 b(,)27 b(b)n(y)g(Lemma)1801 5777 y(25)p eop %%Page: 26 26 26 25 bop -47 71 a FA(4.27)26 b(w)n(e)h(obtain)h Ft(j)p Fz(\027)573 83 y Fw( )623 71 y FA(\()p Fz(b)p FA(\))p Ft(j)23 b(\024)g FA(2)p Fz(\015)947 41 y Fv(\000)p Fw(T)1073 71 y Ft(j)p Fz(b)p Ft(j)1155 83 y Fw(C)1203 92 y Fl(\022)1263 71 y Ft(j)p Fz(a)p Ft(j)1353 83 y Fw(#)1425 71 y FA(whic)n(h)28 b(giv)n(es)e(equation)h(\(2.5\))h(of)f(the)h(Theorem.)77 270 y(In)22 b(order)e(to)h(see)g(that)h Fz(\027)824 282 y Fw(h)888 270 y FA(is)f(a)g(natural)g(measure)f(it)i(su\016ces)f(to)g (sho)n(w)f(that)i(for)f(a)g(con)n(tin)n(uous)f(function)i Fz(b)h Ft(2)g Fz(C)6 b FA(\()p Fz(S)3648 282 y Fy(\012)3700 270 y FA(\))-47 370 y(the)30 b(`time-shifted')h(Birkho\013)e(a)n(v)n (erage)e(1)p Fz(=n)1378 307 y Fx(P)1464 328 y Fw(n)1464 395 y(k)q Fy(=1)1603 370 y Fz(b)19 b Ft(\016)h Fz(F)1785 340 y Fw(T)1824 348 y Fk(0)1856 340 y Fy(+)p Fw(k)1948 370 y FA(\()p Fz(x)p FA(\))31 b(con)n(v)n(erges)c(Leb)r(esgue)i(almost) g(surely)g(to)h Fz(\027)3485 382 y 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b(k)n(er)13 b Fz(`)p FA(.)37 b(Using)27 b(e.g.)g(F)-7 b(ourier)27 b(expansion)f(of)i(functions)g(in)g Fz(E)5 b FA(\()p Fz(A)p FA([)p Fz(\032)p FA(]\))28 b(w)n(e)g(\014nd)g(for)f Fz(T)34 b Ft(\025)22 b FA(1:)828 1994 y Ft(k)p Fz(L)927 1960 y Fw(T)927 2017 y Fv(j)p Fw(Z)999 1994 y Ft(k)h(\024)1526 1938 y FA(1)p 1161 1975 771 4 v 1161 2051 a(exp\(2)p Fz(\031)s FA(\(2)1486 2027 y Fw(T)1557 2051 y Ft(\000)18 b FA(1\))p Fz(\032)p FA(\))g Ft(\000)g FA(1)1960 1994 y(+)2418 1938 y(1)p 2053 1975 V 2053 2051 a(exp\(2)p Fz(\031)s FA(\(2)2378 2027 y Fw(T)2449 2051 y FA(+)g(1\))p Fz(\032)p FA(\))g Ft(\000)g FA(1)2834 1994 y Fz(:)622 b FA(\(6.118\))77 2231 y(T)-7 b(o)31 b(c)n(hec)n(k)g(Condition)g(TR)h (w)n(e)f(ma)n(y)f(instead)i(of)f(\(sums)h(of)6 b(\))32 b Fz(c)2091 2243 y Fw(r)2127 2231 y Fz(\021)2171 2201 y Fw(n)2248 2231 y FA(use)f(the)h(more)f(precise)f(v)-5 b(alues,)33 b Fz(\013)3359 2201 y Fy(1)3426 2231 y FA(=)c Ft(k)p Fz(L)3619 2246 y Fv(j)p Fw(Z)3691 2231 y Ft(k)-47 2339 y FA(and)36 b 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FA(as)e(ab)r(o)n(v)n(e\).)53 b(Theorem)32 b(2.5)h(no)n(w)f(sho)n(ws)g(that)-47 4182 y(indeed)c(the)g(densities)f(of)h(the)g(natural)f(measure)f(exhibit)i (a)f(p)r(olynomial)g(spatial)g(deca)n(y)-7 b(,)773 4381 y Fz(#)822 4347 y Fv(j)p Fw(K)t Fv(j)p Fy(+)p Fv(j)p Fy(\003)p Fv(j)1061 4381 y Ft(j)p Fz(h)1132 4393 y Fw(K)t Fv([)p Fy(\003)1305 4381 y Ft(\000)18 b Fz(h)1436 4393 y Fw(K)1499 4381 y Fz(h)1547 4393 y Fy(\003)1597 4381 y Ft(j)23 b(\024)g FA(2)p Fz(\030)1813 4347 y Fv(\000)p Fw(d)p Fy(\()p Fw(K;)p Fy(\003\))2099 4381 y FA(=)g(2\(1)18 b(+)g(dist\()p Fz(K)q(;)c FA(\003\)\))2801 4347 y Fv(\000)p Fw(\016)2889 4381 y Fz(:)567 b FA(\(6.121\))1801 5777 y(28)p eop %%Page: 29 29 29 28 bop -47 71 a FB(A)134 b(Expanding)45 b(analytic)h(circle)f(maps) -47 253 y FA(In)33 b(the)f(follo)n(wing)g(let)h Fz(\032)e(>)g FA(0)h(and)h Fz(\025)e(>)g FA(1)i(b)r(e)g(\014xed.)51 b(W)-7 b(e)33 b(will)g(establish)f(uniform)h(b)r(ounds)g(for)f(in)n(v) -5 b(arian)n(t)31 b(natural)-47 352 y(measures)24 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b(=)e(max)p Ft(f)p Fz(d)p FA(\()p Fz(x;)14 b(y)s FA(\))23 b(:)g Fz(x)h Ft(2)f Fz(S)3199 2014 y Fy(1)3236 2044 y Fz(;)14 b(y)26 b Ft(2)d Fz(A)p FA([)p Fz(\032)p FA(])p Ft(g)k FA(w)n(e)-47 2144 y(denote)22 b(the)g(maximal)g(distance) f(b)r(et)n(w)n(een)h(p)r(oin)n(ts)g(on)g(the)h(circle)e(and)h(p)r(oin)n (ts)g(in)g(the)h(closed)e(ann)n(ulus.)35 b(One)22 b(ma)n(y)f(ev)n(en) -47 2244 y(calculate)29 b(this)h(quan)n(tit)n(y)f(explicitly)h(through) f(a)h(mapping)f(to)h(the)g(unit)h(disc.)43 b(With)31 b Fz(\013)c FA(=)f Ft(j)14 b FA(tanh)3172 2211 y Fw(\031)p 3139 2225 107 4 v 3139 2272 a Fy(4)p Fw(\025\032)3256 2244 y FA(\()3298 2211 y Fy(1)p 3298 2225 34 4 v 3298 2272 a(2)3361 2244 y FA(+)20 b Fz(i\032)p FA(\))p Ft(j)26 b Fz(<)h FA(1)-47 2362 y(one)g(has)g(\(w)n(e)h(omit)f(the)h(details\),) g Fz(D)d FA(=)e(log)14 b Ft(j)1399 2329 y Fy(1+)p Fw(\013)p 1399 2343 129 4 v 1399 2391 a Fy(1)p Fv(\000)p Fw(\013)1537 2362 y Ft(j)p FA(.)77 2561 y(Let)28 b Fz(E)292 2531 y Fb(R)361 2561 y Ft(\032)23 b Fz(E)32 b FA(denote)27 b(the)h(subset)f (of)g(real-analytic)e(functions)j(of)f Fz(E)h FA(=)23 b Fz(E)5 b FA(\()p Fz(A)p FA([)p Fz(\032)p FA(]\).)38 b(F)-7 b(or)26 b Fz(\033)h(>)22 b FA(0)27 b(w)n(e)g(in)n(tro)r(duce)g (the)-47 2661 y(con)n(v)n(ex)f(cone)580 2760 y Fz(C)639 2772 y Fw(\033)708 2760 y FA(=)c Ft(f)p Fz(\036)h Ft(2)h Fz(E)1054 2726 y Fb(R)1118 2760 y Ft(n)18 b(f)p FA(0)p Ft(g)k FA(:)h Ft(j)p Fz(\036)1444 2772 y Fw(y)1503 2760 y Ft(\000)18 b Fz(\036)1635 2772 y Fw(x)1677 2760 y Ft(j)23 b(\024)g Fz(\036)1860 2772 y Fw(x)1925 2760 y FA(\()p Fz(e)1996 2726 y Fw(\033)r(d)p Fy(\()p Fw(x;y)r Fy(\))2239 2760 y Ft(\000)18 b FA(1\))p Fz(;)41 b(x)24 b Ft(2)f Fz(S)2665 2726 y Fy(1)2702 2760 y Fz(;)42 b(y)25 b Ft(2)f Fz(A)p FA([)p Fz(\032)p FA(])p Ft(g)353 b FA(\(A.129\))-47 2910 y(and)27 b(w)n(e)g(de\014ne)h(for)f Fz(\036)652 2922 y Fy(1)690 2910 y Fz(;)14 b(\036)776 2922 y Fy(2)837 2910 y Ft(2)23 b Fz(C)974 2922 y Fw(\033)1047 2910 y FA(a)k(functional)h Fz(\014)1553 2922 y Fw(\033)1625 2910 y FA(and)g(then)g(Hilb)r(ert's)g(pro)5 b(jectiv)n(e)26 b(metric)i Fz(\022)3001 2922 y Fw(\033)3073 2910 y FA(b)n(y)927 3109 y Fz(\014)974 3121 y Fw(\033)1018 3109 y FA(\()p Fz(\036)1099 3121 y Fy(1)1137 3109 y Fz(;)14 b(\036)1223 3121 y Fy(2)1261 3109 y FA(\))23 b(=)g(inf)7 b Ft(f)p Fz(\026)22 b Ft(2)i Fu(R)1758 3121 y Fy(+)1831 3109 y Ft([)19 b(f1g)j FA(:)h Fz(\026)g(\036)2262 3121 y Fy(1)2319 3109 y Ft(\000)18 b Fz(\036)2451 3121 y Fy(2)2511 3109 y Ft(2)24 b Fz(C)2649 3121 y Fw(\033)2694 3109 y Ft(g)p Fz(;)699 b FA(\(A.130\))1144 3258 y Fz(\022)1183 3270 y Fw(\033)1228 3258 y FA(\()p Fz(\036)1309 3270 y Fy(1)1347 3258 y Fz(;)14 b(\036)1433 3270 y Fy(2)1470 3258 y FA(\))24 b(=)e(log\()p Fz(\014)1799 3270 y Fw(\033)1844 3258 y FA(\()p Fz(\036)1925 3270 y Fy(1)1963 3258 y Fz(;)14 b(\036)2049 3270 y Fy(2)2087 3258 y FA(\))p Fz(\014)2166 3270 y Fw(\033)2211 3258 y FA(\()p Fz(\036)2292 3270 y Fy(2)2330 3258 y Fz(;)g(\036)2416 3270 y Fy(1)2454 3258 y FA(\)\))p Fz(:)917 b FA(\(A.131\))-47 3408 y(W)-7 b(e)28 b(ma)n(y)f(relate)f(the)i(pro)5 b(jectiv)n(e)27 b(metric)g(and)h(the)g(standard)e(Banac)n(h)h(norm)g(on)g Fz(E)2639 3377 y Fb(R)2713 3408 y FA(through)g(the)h(follo)n(wing)-47 3573 y Fr(Lemma)h(A.3)51 3739 y Fs(1.)42 b(F)-6 b(or)30 b Fz(\036)364 3751 y Fy(1)402 3739 y Fz(;)14 b(\036)488 3751 y Fy(2)549 3739 y Ft(2)23 b Fz(C)686 3751 y Fw(\033)750 3739 y Ft(\\)18 b(f)p Fz(`)23 b FA(=)f(1)p Ft(g)29 b Fs(we)h(have)h Ft(j)p Fz(\036)1509 3751 y Fy(1)1565 3739 y Ft(\000)18 b Fz(\036)1697 3751 y Fy(2)1735 3739 y Ft(j)23 b(\024)g FA(2)p Fz(e)1950 3709 y Fw(\033)r(D)2050 3739 y FA(\()p Fz(e)2121 3709 y Fw(\022)2153 3717 y Fl(\033)2192 3709 y Fy(\()p Fw(\036)2258 3717 y Fk(1)2290 3709 y Fw(;\036)2350 3717 y Fk(2)2382 3709 y Fy(\))2430 3739 y Ft(\000)18 b FA(1\))p Fs(.)51 3905 y(2.)42 b(F)-6 b(or)30 b Fz(u)23 b Ft(2)g Fz(E)530 3875 y Fb(R)606 3905 y Fs(one)30 b(has:)40 b Fz(\022)985 3917 y Fw(\033)1029 3905 y FA(\(1)19 b(+)f Fz(u;)c FA(1\))22 b Ft(\024)h FA(\(2)18 b(+)1663 3872 y Fy(8)p Fw(\025)p 1659 3886 82 4 v 1659 3934 a(\031)r(\033)1750 3905 y FA(\))p Ft(j)p Fz(u)p Ft(j)h FA(+)f Fz(o)p FA(\()p Ft(j)p Fz(u)p Ft(j)p FA(\))p Fs(.)-47 4071 y(Pr)l(o)l(of.)38 b FA(F)-7 b(or)27 b Fz(\036)d Ft(2)f Fz(C)567 4083 y Fw(\033)640 4071 y FA(w)n(e)k(ha)n(v)n(e)g(the)h(b)r(ounds)1215 4291 y Ft(j)p Fz(\036)p Ft(j)c(\025)e Fz(`)p FA(\()p Fz(\036)p FA(\))i(=)1681 4178 y Fx(I)1727 4367 y Fw(S)1771 4350 y Fk(1)1821 4291 y Fz(\036)1870 4303 y Fw(x)1936 4291 y Fz(dx)f Ft(\025)g Fz(e)2176 4257 y Fv(\000)p Fw(\033)r(D)2351 4291 y Ft(j)p Fz(\036)p Ft(j)p Fz(:)989 b FA(\(A.132\))-47 4520 y(Let)29 b Fz(\036)152 4532 y Fy(1)190 4520 y Fz(;)14 b(\036)276 4532 y Fy(2)340 4520 y Ft(2)26 b Fz(C)480 4532 y Fw(\033)545 4520 y Ft(\\)20 b(f)p Fz(`)25 b FA(=)g(1)p Ft(g)k FA(and)g(set)g Fz(\026)d FA(=)g Fz(\014)t FA(\()p Fz(\036)1518 4532 y Fy(1)1556 4520 y Fz(;)14 b(\036)1642 4532 y Fy(2)1680 4520 y FA(\).)42 b(Then)30 b(0)25 b Ft(\024)h Fz(`)p FA(\()p Fz(\026\036)2320 4532 y Fy(1)2377 4520 y Ft(\000)19 b Fz(\036)2510 4532 y Fy(2)2548 4520 y FA(\))26 b(=)g Fz(\026)20 b Ft(\000)f FA(1)29 b(sho)n(ws)f(that)i Fz(\026)c Ft(\025)f FA(1.)42 b(W)-7 b(e)-47 4620 y(then)27 b(obtain)f Ft(j)p Fz(\036)470 4632 y Fy(1)524 4620 y Ft(\000)17 b Fz(\036)655 4632 y Fy(2)692 4620 y Ft(j)23 b(\024)g(j)p Fz(\026\036)948 4632 y Fy(1)1002 4620 y Ft(\000)16 b Fz(\036)1132 4632 y Fy(2)1170 4620 y Ft(j)h FA(+)f(\()p Fz(\026)g Ft(\000)g FA(1\))p Ft(j)p Fz(\036)1616 4632 y Fy(1)1654 4620 y Ft(j)23 b(\024)g FA(2)p Fz(e)1869 4590 y Fw(\033)r(D)1969 4620 y FA(\()p Fz(\026)17 b Ft(\000)f FA(1\))22 b Ft(\024)h FA(2)p Fz(e)2414 4590 y Fw(\033)r(D)2514 4620 y FA(\()p Fz(e)2585 4590 y Fw(\022)2617 4598 y Fl(\033)2656 4590 y Fy(\()p Fw(\036)2722 4598 y Fk(1)2754 4590 y Fw(;phi)2870 4598 y Fk(2)2903 4590 y Fy(\))2950 4620 y Ft(\000)16 b FA(1\))26 b(sho)n(wing)g(the)h(\014rst)-47 4719 y(claim.)77 4819 y(First,)i(let)f Fz(M)33 b(<)24 b FA(1.)39 b(Giv)n(en)28 b Fz(u)c Ft(2)h Fz(E)1190 4789 y Fb(R)1236 4819 y FA(,)j Ft(j)p Fz(u)p Ft(j)c(\024)g Fz(M)37 b FA(w)n(e)28 b(lift)h(it)g(to)f(a) g(function)35 b(^)-48 b Fz(u)28 b FA(on)g(the)h(co)n(v)n(er)3020 4798 y Fx(b)3000 4819 y Fz(A)q FA([)p Fz(\032)p FA(].)39 b(Then)29 b(for)e(\014xed)-47 4919 y Fz(x)c Ft(2)h Fu(R)6 b FA(,)1544 5005 y Ft(j)f FA(^)-47 b Fz(u)1615 5017 y Fw(y)1673 5005 y Ft(\000)23 b FA(^)-47 b Fz(u)1804 5017 y Fw(x)1845 5005 y Ft(j)p 1544 5042 325 4 v 1567 5119 a(j)p FA(1)18 b(+)23 b(^)-47 b Fz(u)1781 5131 y Fw(x)1822 5119 y Ft(j)1897 5062 y(\001)2024 5005 y FA(1)p 1948 5042 193 4 v 1948 5119 a Fz(y)21 b Ft(\000)d Fz(x)3458 5062 y FA(\(A.133\))-47 5278 y(is)27 b(holomorphic)g(in)g Fz(y)f Ft(2)773 5257 y Fx(b)754 5278 y Fz(A)p FA([)p Fz(\032)p FA(])i(and)f(b)r(ounded)h(b)n(y)f Fz(\026)c FA(=)g(2)p Fz(M)t(=)p FA(\()p Fz(\032)p FA(\(1)17 b Ft(\000)h Fz(M)9 b FA(\)\).)38 b(Hence,)898 5504 y Ft(j)5 b FA(^)-47 b Fz(u)969 5516 y Fw(y)1026 5504 y Ft(\000)24 b FA(^)-47 b Fz(u)1158 5516 y Fw(x)1199 5504 y Ft(j)23 b(\024)g FA(\(1)18 b(+)23 b(^)-47 b Fz(u)1556 5516 y Fw(x)1597 5504 y FA(\))p Fz(\026)p Ft(j)p Fz(y)22 b Ft(\000)c Fz(x)p Ft(j)23 b(\024)g FA(\(1)18 b(+)24 b(^)-48 b Fz(u)2252 5516 y Fw(x)2294 5504 y FA(\))p Fz(\026)2386 5448 y FA(2)p Fz(\025\032)p 2386 5485 133 4 v 2427 5561 a(\031)2543 5482 y FA(^)2529 5504 y Fz(d)p FA(\()p Fz(x;)14 b(y)s FA(\))p Fz(;)671 b FA(\(A.134\))1801 5777 y(30)p eop %%Page: 31 31 31 30 bop -47 71 a FA(whic)n(h)27 b(on)h(the)g(ann)n(ulus)f(implies:) 1176 264 y Ft(j)p Fz(u)1247 276 y Fw(y)1305 264 y Ft(\000)18 b Fz(u)1436 276 y Fw(x)1477 264 y Ft(j)23 b(\024)g FA(\(1)18 b(+)g Fz(u)1834 276 y Fw(x)1875 264 y FA(\)\()p Fz(e)1988 203 y Fk(2)p Fl(\025\032\026)p 1989 217 129 3 v 2035 250 a(\031)2127 230 y Fw(d)p Fy(\()p Fw(x;y)r Fy(\))2329 264 y Ft(\000)g FA(1\))p Fz(:)949 b FA(\(A.135\))-47 443 y(In)34 b(order)e(for)h(1)22 b(+)g Fz(u)34 b FA(to)f(b)r(e)h(in)g (the)h(cone)e(w)n(e)g(need)h(that)g(2)p Fz(\025\032\026=\031)i Ft(\024)d Fz(\033)k FA(and)d(this)g(is)f(assured)g(when)h(w)n(e)f(set)h Fz(M)42 b FA(=)-47 542 y Fz(\031)s(\033)s(=)p FA(\(4)p Fz(\025)19 b FA(+)g Fz(\031)s(\033)s FA(\).)40 b(Th)n(us,)28 b(the)g(ball)h(cen)n(tered)e(at)h(1)g(and)g(ha)n(ving)g(radius)f Fz(M)37 b FA(b)r(elongs)28 b(to)g(the)g(cone.)39 b(F)-7 b(or)28 b Ft(j)p Fz(u)p Ft(j)23 b Fz(<)h(M)37 b FA(this)-47 642 y(implies:)1107 785 y(\002)1172 797 y Fw(\033)1216 785 y FA(\(1)18 b(+)g Fz(u;)c FA(1\))22 b Ft(\024)1670 729 y Fz(M)27 b FA(+)18 b Ft(j)p Fz(u)p Ft(j)p 1670 766 285 4 v 1670 842 a Fz(M)27 b Ft(\000)18 b(j)p Fz(u)p Ft(j)1988 785 y(\024)2109 729 y FA(2)p 2085 766 90 4 v 2085 842 a Fz(M)2185 785 y Ft(j)p Fz(u)p Ft(j)g FA(+)g Fz(o)p FA(\()p Ft(j)p Fz(u)p Ft(j)p FA(\))880 b(\(A.136\))-47 976 y(whic)n(h)27 b(yields)h(the)g(desired)f(b)r(ound)p 1132 922 33 4 v 1132 990 4 68 v 1161 990 V 1132 993 33 4 v 77 1175 a(F)-7 b(or)27 b Fz(z)g Ft(2)d FA(In)n(t)496 1154 y Fx(b)477 1175 y Fz(A)p FA([)p Fz(\025\032)p FA(])k(the)g (function)1187 1153 y Fx(b)1173 1175 y Fz( )i FA(is)e(univ)-5 b(alen)n(t)28 b(in)g(a)f(disk)h(of)f(radius)g(\(at)h(least\))g Fz(r)e FA(=)d Fz(\025\032)c Ft(\000)f(j)p FA(Im)23 b Fz(z)t Ft(j)k FA(cen)n(tered)g(at)-47 1286 y Fz(z)t FA(.)36 b(Ko)r(eb)r(e)27 b(distortion)g([6,)g(Theorem)g(1.5])g(then)h(implies)g Ft(j)1824 1264 y Fx(b)1809 1286 y Fz( )1866 1256 y Fv(00)1863 1307 y Fw(z)1909 1286 y Fz(=)1965 1264 y Fx(b)1951 1286 y Fz( )2008 1256 y Fv(0)2005 1307 y Fw(z)2042 1286 y Ft(j)c(\024)e FA(4)p Fz(=r)30 b FA(and)d(a)g(calculation)g(no)n(w)g (sho)n(ws)f(that)1342 1532 y Ft(j)1394 1476 y Fz(d)p 1375 1513 83 4 v 1375 1589 a(ds)1481 1532 y FA(log)1616 1510 y Fx(b)1602 1532 y Fz( )1659 1498 y Fv(0)1656 1553 y Fw(z)1694 1532 y Ft(j)d FA(=)g Ft(j)1876 1454 y Fx(b)1861 1476 y Fz( )1918 1446 y Fv(00)1915 1497 y Fw(z)p 1861 1513 100 4 v 1879 1583 a Fx(b)1865 1605 y Fz( )1922 1581 y Fv(0)1919 1625 y Fw(z)1970 1532 y Ft(j)2026 1476 y(j)p Fz(dz)t Ft(j)p 2026 1513 132 4 v 2051 1589 a Fz(ds)2191 1532 y Ft(\024)g FA(4)p Fz(:)1114 b FA(\(A.137\))-47 1786 y(Since)31 b Ft(j)p Fz(e)235 1756 y Fw(z)293 1786 y Ft(\000)20 b FA(1)p Ft(j)28 b(\024)g Fz(e)603 1756 y Fv(j)p Fw(z)r Fv(j)701 1786 y Ft(\000)20 b FA(1)30 b(for)h Fz(z)g Ft(2)e Fu(C)52 b FA(w)n(e)30 b(ma)n(y)g(in)n(tegrate)g (the)h(ab)r(o)n(v)n(e)e(inequalit)n(y)i(and)f(obtain)h(for)f Fz(u;)14 b(f)9 b FA(\()p Fz(u)p FA(\))27 b Ft(2)i Fz(A)p FA([)p Fz(\032)p FA(])-47 1886 y(and)e Fz(v)f Ft(2)e Fz(S)315 1856 y Fy(1)379 1886 y FA(:)1288 2016 y Ft(j)1323 1960 y Fz(f)1373 1930 y Fv(0)1396 1960 y FA(\()p Fz(v)s FA(\))p 1321 1997 185 4 v 1321 2073 a Fz(f)1371 2049 y Fv(0)1394 2073 y FA(\()p Fz(u)p FA(\))1534 2016 y Ft(\000)18 b FA(1)p Ft(j)23 b(\024)g Fz(e)1832 1982 y Fy(4)p Fw(d)p Fy(\()p Fw(f)7 b Fy(\()p Fw(u)p Fy(\))p Fw(;f)g Fy(\()p Fw(v)r Fy(\)\))2249 2016 y Ft(\000)18 b FA(1)p Fz(:)1061 b FA(\(A.138\))-47 2249 y Fr(Lemma)29 b(A.4)41 b Fs(L)l(et)e FA(~)-47 b Fz(\033)35 b FA(=)c(4)21 b(+)h Fz(\025)1003 2219 y Fv(\000)p Fy(1)1092 2249 y Fz(\033)38 b Fs(and)d(supp)l(ose)g (that)k FA(~)-47 b Fz(\033)35 b(<)c(\033)s Fs(.)53 b(Then)35 b Fz(L)c FA(:)h Fz(C)2554 2261 y Fw(\033)2630 2249 y Ft(!)g Fz(C)2808 2261 y Fy(~)-37 b Fw(\033)2883 2249 y Fs(and)35 b(ther)l(e)f(is)h(a)g(c)l(onstant)-47 2348 y Fz(\021)26 b FA(=)d Fz(\021)s FA(\()p Fz(\032;)14 b(\025;)g(\033)s FA(\))24 b Fz(<)f FA(1)29 b Fs(such)h(that)994 2527 y Fz(\022)1033 2539 y Fw(\033)1078 2527 y FA(\()p Fz(L\036)1216 2539 y Fy(1)1254 2527 y Fz(;)14 b(L\036)1397 2539 y Fy(2)1434 2527 y FA(\))23 b Ft(\024)g Fz(\021)17 b(\022)1674 2539 y Fw(\033)1718 2527 y FA(\()p Fz(\036)1799 2539 y Fy(1)1837 2527 y Fz(;)d(\036)1923 2539 y Fy(2)1961 2527 y FA(\))p Fz(;)74 b Fs(for)60 b Fz(\036)2301 2539 y Fy(1)2339 2527 y Fz(;)14 b(\036)2425 2539 y Fy(2)2485 2527 y Ft(2)24 b Fz(C)2623 2539 y Fw(\033)2668 2527 y Fz(:)767 b FA(\(A.139\))-47 2706 y Fs(Pr)l(o)l(of.)44 b FA(Let)30 b Fz(x)d Ft(2)g Fz(S)577 2675 y Fy(1)644 2706 y FA(and)i Fz(y)g Ft(2)e Fz(A)p FA([)p Fz(\032)p FA(].)44 b(W)-7 b(e)30 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FA(\()p Fz(L)439 1722 y Fy(\()p Fw(n)p Fy(\))536 1752 y FA(1)p Fz(;)14 b(L)672 1722 y Fy(\()p Fw(n)p Fy(\))768 1752 y FA(\(1)k(+)g Fz(u)p FA(\)\))23 b Ft(\024)g Fz(\021)1210 1722 y Fw(n)1255 1752 y Fz(c)p Ft(j)p Fz(u)p Ft(j)18 b FA(+)g Fz(o)p FA(\()p Ft(j)p Fz(u)p Ft(j)p FA(\).)38 b(By)27 b(Lemma)h(A.3,)517 1932 y Ft(j)p Fz(L)597 1898 y Fy(\()p Fw(n)p Fy(\))694 1932 y Fz(u)p Ft(j)22 b FA(=)h Ft(j)p Fz(L)955 1898 y Fy(\()p Fw(n)p Fy(\))1052 1932 y FA(\(1)18 b(+)g Fz(u)p FA(\))g Ft(\000)g Fz(L)1465 1898 y Fy(\()p Fw(n)p Fy(\))1562 1932 y FA(1)p Ft(j)k(\024)h FA(2)14 b Fz(e)1832 1898 y Fw(\033)r(D)1946 1932 y Fz(\021)1990 1898 y Fw(n)2049 1932 y Fz(c)p Ft(j)p Fz(u)p Ft(j)k FA(+)g Fz(o)p FA(\()p Ft(j)p Fz(u)p Ft(j)p FA(\))24 b Ft(\021)e Fz(c)2625 1944 y Fw(r)2662 1932 y Fz(\021)2706 1898 y Fw(n)2751 1932 y Ft(j)p Fz(u)p Ft(j)c FA(+)g Fz(o)p FA(\()p Ft(j)p Fz(u)p Ft(j)p FA(\))p Fz(;)291 b FA(\(A.150\))-47 2112 y(and)27 b(b)n(y)g(linearit)n(y)-7 b(,)27 b(the)h(term)g Fz(o)p FA(\()p Ft(j)p 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Fy(\003)3536 5081 y FA(is)g(the)-47 5180 y(closure)h(of)i(its)g(o)n(wn)e(in)n(terior.)36 b(More)27 b(generally)-7 b(,)26 b(one)h(has)g(an)h(ob)n(vious)e (inclusion,)1186 5380 y Fz(E)5 b FA(\()p Fz(F)12 b FA(;)i Fz(Y)19 b FA(\))k Ft(\032)g Fz(C)1661 5345 y Fy(0)1698 5380 y FA(\()p Fz(F)12 b FA(;)i Fz(Y)19 b FA(\))g Ft(\\)g Fz(C)2089 5345 y Fw(!)2137 5380 y FA(\(In)n(t)p Fz(F)12 b FA(;)i Fz(Y)19 b FA(\))p Fz(:)962 b FA(\(B.152\))-47 5528 y(\(The)28 b(reader)e(ma)n(y)h(try)g(to)g(\014nd)i(general)d (conditions)h(under)g(whic)n(h)h(equalit)n(y)f(holds.\))1801 5777 y(32)p eop %%Page: 33 33 33 32 bop -47 71 a Fr(Example)30 b(B.2)41 b Fs(F)-6 b(or)35 b Fz(\032)e(>)f FA(0)i Fs(let)h Fz(F)1115 83 y Fy(1)1186 71 y FA(=)d Fz(A)p FA([)p Fz(\032)p FA(])h Ft(\032)f(C)39 b Fs(b)l(e)d(a)f(close)l(d)h(annulus)e(and)i Fz(F)2614 83 y Fy(2)2684 71 y FA(=)c Fz(@)5 b(A)p FA([)p Fz(\032)p FA(])35 b Fs(its)g(b)l(oundary.)56 b(Then)-47 170 y Fz(E)5 b FA(\()p Fz(F)104 182 y Fy(1)160 170 y Ft(\002)18 b Fz(F)296 182 y Fy(2)334 170 y FA(\))30 b Fs(ar)l(e)g(the)g Fu(C)21 b Fs(-value)l(d)30 b(c)l(ontinuous)f(functions)h Fz(\036)p FA(\()p Fz(z)1916 182 y Fy(1)1953 170 y Fz(;)14 b(z)2029 182 y Fy(2)2066 170 y FA(\))30 b Fs(in)g FA(\()p Fz(z)2301 182 y Fy(1)2338 170 y Fz(;)14 b(z)2414 182 y Fy(2)2451 170 y FA(\))23 b Ft(2)g Fz(F)2637 182 y Fy(1)2694 170 y Ft(\002)18 b Fz(F)2830 182 y Fy(2)2897 170 y Fs(for)31 b(which)g(the)f(map)1329 353 y Fz(z)1368 365 y Fy(1)1427 353 y Ft(2)24 b FA(In)n(t)p Fz(A)p FA([)p Fz(\032)p FA(])f Ft(7!)g Fz(\036)p FA(\()p Fz(z)2012 365 y Fy(1)2050 353 y Fz(;)14 b(z)2126 365 y Fy(2)2163 353 y FA(\))23 b Ft(2)h Fu(C)1125 b FA(\(B.153\))-47 536 y Fs(is)30 b(holomorphic)j(for)e(any)f (\014xe)l(d)f Fz(z)1036 548 y Fy(2)1096 536 y Ft(2)23 b Fz(@)5 b(A)p FA([)p Fz(\032)p FA(])p Fs(.)39 b(\(This)31 b(is)f(the)g(pr)l(ototyp)l(e)g(of)h(example)g(which)g(is)f(use)l(d)g (in)g(se)l(ction)f(3.3\).)-47 718 y Fr(Theorem)h(B.3)41 b Fs(L)l(et)30 b Fz(F)43 b Fs(and)32 b Fz(K)k 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3721 y Fw(z)456 3709 y Fz(\036)23 b Ft(2)h Fz(E)5 b FA(\()p Fz(K)h FA(;)14 b Fz(Y)k FA(\).)36 b(Similarly)21 b(if)i Fz( )1451 3679 y Fw(F)1529 3709 y Ft(2)g Fz(C)1672 3679 y Fw(!)1721 3709 y FA(\()p Fu(D)d FA(;)14 b Fz(F)e FA(\))22 b(then)h Fz(r)2190 3726 y Fw( )2236 3709 y Fl(F)2283 3726 y Fy(\()p Fw(u)p Fy(\))2379 3709 y Fz(\036)f FA(has)g(a)f(con)n(v) n(ergen)n(t)f(series)h(expansion)g(in)-47 3809 y Fz(E)5 b FA(\()p Fz(K)h FA(;)14 b Fz(Y)k FA(\),)27 b(hence)e(so)g(has)g Fz(M)9 b(r)914 3825 y Fw( )960 3809 y Fl(F)1007 3825 y Fy(\()p Fw(u)p Fy(\))1103 3809 y Fz(\036)p FA(.)36 b(By)26 b(a)f(sligh)n(t)g(abuse)g(of)h(notation)f(w)n(e)g(write)g Fz(M)9 b(r)r(\036)27 b FA(for)e(the)h(resulting)f Fz(E)5 b FA(\()p Fz(K)h FA(;)14 b Fz(Y)k FA(\))-47 3909 y(v)-5 b(alued)31 b(function)h(of)f Fz(F)44 b FA(and)31 b(w)n(e)g(conclude)g (that)g Fz(M)9 b(r)r(\036)31 b Ft(2)e Fz(E)5 b FA(\()p Fz(F)12 b FA(;)i Fz(E)5 b FA(\()p Fz(K)h FA(;)14 b Fz(Y)19 b FA(\)\).)49 b(Comp)r(osing)30 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5496 y Ff(\000)1540 5412 y Fz(k)1583 5424 y Fw(w)1637 5412 y FA(\()p Fz(z)t FA(\)\010\()p Fz(z)t FA(\))14 b Fz(dz)26 b FA(=)d(\010)2181 5424 y Fy(+)2236 5412 y FA(\()p Fz(w)r FA(\))c(+)f(\010)2523 5424 y Fv(\000)2579 5412 y FA(\()p Fz(w)r FA(\))p Fz(;)734 b FA(\(B.161\))1801 5777 y(33)p eop %%Page: 34 34 34 33 bop -47 71 a FA(where)21 b(\000)239 83 y Fy(+)316 71 y FA(and)h(\000)524 83 y Fv(\000)602 71 y FA(are)f(the)i(t)n(w)n(o)e (connected)h(parts)f(of)h Fz(@)5 b(A)p FA([)p Fz(\032)p FA(].)36 b(Clearly)-7 b(,)22 b(\010)2321 83 y Fy(+)2398 71 y FA(extends)g(holomorphically)e(for)i(Im)p Fz(w)k(<)c(\032)-47 170 y FA(and)28 b(\010)175 182 y Fv(\000)260 170 y FA(similarly)g(for)g (Im)p Fz(w)g(>)c Ft(\000)p Fz(\032)p FA(.)40 b(Then)29 b(\010)1453 182 y Fy(+)1508 170 y FA(\()p Fz(w)r FA(\))d(=)e(\010\()p Fz(w)r FA(\))d Ft(\000)e FA(\010)2098 182 y Fv(\000)2154 170 y FA(\()p Fz(w)r FA(\))30 b(is)e(the)h(di\013erence)g(of)f(t)n(w)n (o)g(functions)h(whic)n(h)-47 270 y(extends)i(con)n(tin)n(uously)f(to)h (Im)p Fz(w)h FA(=)c Fz(\032)p FA(,)k(whence)f(\010)1585 282 y Fy(+)1641 270 y FA(\()p Fz(w)r FA(\))h(itself)g(extends)f(con)n (tin)n(uously)f(to)h(Im)p Fz(w)h Ft(\024)c Fz(\032)j FA(\(similarly)g(for)-47 370 y(\010)13 382 y Fv(\000)69 370 y FA(\).)37 b(By)27 b(uniform)h(con)n(tin)n(uit)n(y)f(it)h(follo)n (ws)f(that)1071 552 y(\010)1131 564 y Fw(\017)1163 552 y FA(\()p Fz(w)r FA(\))d(=)f(\010)1460 564 y Fy(+)1515 552 y FA(\()p Fz(w)e Ft(\000)d Fz(i\017)p FA(\))h(+)f(\010)1967 564 y Fv(\000)2023 552 y FA(\()p Fz(w)j FA(+)d Fz(i\017)p FA(\))p Fz(;)69 b(\017)23 b(>)f FA(0)p Fz(;)847 b FA(\(B.162\))-47 735 y(con)n(v)n(erges)28 b(uniformly)i(to)h(\010)g(in)f(the)h Fz(E)5 b FA(\()p Fz(A)p FA([)p Fz(\032)p FA(];)14 b Fz(Y)20 b FA(\)-norm)30 b(as)g Fz(\017)e Ft(!)g FA(0)2104 705 y Fy(+)2158 735 y FA(.)46 b(By)31 b(construction)f(\010)2900 747 y Fw(\017)2960 735 y Ft(2)e Fz(E)5 b FA(\()p Fz(A)p FA([)p Fz(\032)21 b FA(+)f Fz(\017)p FA(];)14 b Fz(Y)19 b FA(\))31 b(and)-47 835 y(its)f(F)-7 b(ourier)30 b(series)f(con)n(v)n (erges)f(uniformly)i(\(in)h(fact)g(exp)r(onen)n(tially)f(fast\))h(in)f (the)h Fz(E)5 b FA(\()p Fz(A)p FA([)p Fz(\032)p FA(];)14 b Fz(Y)19 b FA(\)-norm.)45 b(T)-7 b(aking)30 b(\014nite)-47 934 y(truncations)36 b(of)h(these)g(F)-7 b(ourier)36 b(series)g(w)n(e)h(obtain)g(an)g(elemen)n(t)g(of)g Fz(E)5 b FA(\()p Fz(A)p FA([)p Fz(\032)p FA(]\))26 b Ft(\012)e Fz(Y)56 b FA(whic)n(h)37 b(is)g(arbitrarily)e(close)h(to)-47 1034 y(\010)23 b Ft(2)g Fz(E)5 b FA(\()p Fz(A)p FA([)p Fz(\032)p FA(];)14 b Fz(Y)19 b FA(\))p 555 980 33 4 v 555 1048 4 68 v 584 1048 V 555 1051 33 4 v -47 1316 a Fr(Corollary)31 b(B.6)41 b Fs(F)-6 b(or)30 b FA(\003)g Fs(an)f(index)h(set,)g Ft(\012)1363 1328 y Fw(p)p Fv(2)p Fy(\003)1491 1316 y Fz(E)5 b FA(\()p Fz(A)p FA([)p Fz(\032)p FA(]\))31 b Fs(is)f(dense)g(in)g Fz(E)5 b FA(\()2321 1254 y Fx(Q)2400 1341 y Fw(p)p Fv(2)p Fy(\003)2542 1316 y Fz(A)p FA([)p Fz(\032)p FA(]\))p Fs(.)-47 1499 y FA(Pro)r(of:)35 b(When)27 b(\003)f(is)g(\014nite)h(this)g(follo)n(ws)e(from)i (induction.)36 b(Denoting)27 b Fz(A)2278 1511 y Fy(\003)2351 1499 y FA(=)2438 1437 y Fx(Q)2517 1524 y Fw(p)p Fv(2)p Fy(\003)2659 1499 y Fz(A)p FA([)p Fz(\032)p FA(],)g(Theorem)f(B.3)g (and)g(Theo-)-47 1598 y(rem)f(B.5)g(implies)h(that)f Fz(E)5 b FA(\()p Fz(A)883 1610 y Fw(p)922 1598 y FA(\))15 b Ft(\012)f Fz(E)5 b FA(\()p Fz(A)1208 1610 y Fy(\003)p Fv(\000)p Fw(p)1344 1598 y FA(\))23 b Ft(\030)g Fz(E)5 b FA(\()p Fz(A)1647 1610 y Fw(p)1686 1598 y FA(;)14 b Fz(E)5 b FA(\()p Fz(A)1883 1610 y Fy(\003)p Fv(\000)p Fw(p)2019 1598 y FA(\)\))26 b(is)g(dense)f(in)h Fz(E)5 b FA(\()p Fz(A)2670 1610 y Fy(\003)2720 1598 y FA(\).)37 b(Also)25 b Fz(E)5 b FA(\()p Fz(A)3157 1610 y Fw(q)3194 1598 y FA(\))14 b Ft(\012)g Fz(E)5 b FA(\()p Fz(A)3479 1610 y Fy(\003)p Fv(\000)p Fw(p)p Fv(\000)p Fw(q)3700 1598 y FA(\))-47 1698 y(is)31 b(dense)g(in)g Fz(E)5 b FA(\()p Fz(A)530 1710 y Fy(\003)p Fv(\000)p Fw(p)666 1698 y FA(\))32 b(and)f(inductiv)n(ely)g(w)n(e)f(see)h(that)g Fz(E)5 b FA(\()p Fz(A)1931 1710 y Fw(p)1970 1698 y FA(\))21 b Ft(\012)g Fz(E)5 b FA(\()p Fz(A)2269 1710 y Fw(q)2306 1698 y FA(\))21 b Ft(\012)f Fz(E)5 b FA(\()p Fz(A)2604 1710 y Fy(\003)p Fv(\000)p Fw(p)p Fv(\000)p Fw(q)2825 1698 y FA(\))31 b(is)g(dense)g(in)h Fz(E)5 b FA(\()p Fz(A)3466 1710 y Fy(\003)3516 1698 y FA(\),)32 b(etc.)-47 1798 y(When)d(\003)f(is)g(in\014nite,)i(an)e(elemen)n(t)g(in)h(the)g (direct)f(tensor)f(pro)r(duct)i(has)f(only)g(\014nitely)h(man)n(y)e (comp)r(onen)n(ts)h(di\013eren)n(t)-47 1897 y(from)g(one.)38 b(Since)29 b Fz(A)616 1909 y Fy(\003)694 1897 y FA(is)f(compact,)g(a)g (function)h(in)g Fz(E)5 b FA(\()p Fz(A)1788 1909 y Fy(\003)1838 1897 y FA(\))29 b(b)r(eing)f(con)n(tin)n(uous)f(ma)n(y)h(b)r(e)h(appro) n(ximated)d(arbitrarily)-47 1997 y(w)n(ell)h(b)n(y)g(functions)h(dep)r (ending)g(on)g(\014nitely)g(man)n(y)f(v)-5 b(ariables,)26 b(hence)i(b)n(y)f(elemen)n(ts)g(in)h(the)g(tensor)f(pro)r(duct)p 3495 1943 V 3495 2011 4 68 v 3524 2011 V 3495 2014 33 4 v -47 2279 a Fr(Corollary)k(B.7)41 b Fs(L)l(et)32 b Fz(p;)14 b(q)29 b Ft(2)e FA(\003)32 b Fs(\(an)g(index)g(set\))f(and)h (let)g Fz(Q)1915 2291 y Fw(p)1985 2279 y Fs(and)g Fz(Q)2214 2291 y Fw(q)2283 2279 y Fs(b)l(e)f(b)l(ounde)l(d)i(line)l(ar)f(op)l(er) l(ators)h(on)f Fz(E)5 b FA(\()p Fz(A)p FA([)p Fz(\032)p FA(]\))p Fs(.)-47 2390 y(Then)36 b Fz(Q)241 2402 y Fw(p)315 2390 y Fs(and)g Fz(Q)548 2402 y Fw(q)620 2390 y Fs(extends)g(isometric) l(al)t(ly)i(to)e(op)l(er)l(ators)1908 2369 y FA(^)1889 2390 y Fz(Q)1955 2402 y Fw(p)2029 2390 y Fs(and)2216 2369 y FA(^)2197 2390 y Fz(Q)2263 2402 y Fw(q)2335 2390 y Fs(on)g Fz(E)5 b FA(\()p Fz(A)2620 2402 y Fy(\003)2670 2390 y FA(\))p Fs(.)57 b(When)36 b Fz(p)e Ft(6)p FA(=)g Fz(q)k Fs(the)e(extende)l(d)-47 2509 y(op)l(er)l(ators)25 b(c)l(ommute)g(while)h(if)f Fz(p)e FA(=)g Fz(q)28 b Fs(the)c(extension) h(of)g(the)g(pr)l(o)l(duct,)2198 2490 y Fu(\\)2192 2509 y Fz(Q)2258 2521 y Fw(p)2296 2509 y Fz(Q)2362 2521 y 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y(v)n(ector.)59 b(By)35 b(h)n(yp)r(othesis)g Fz(\024)h FA(=)g(sup)1123 1498 y Fw(p)p Fv(2)p Fy(\012)1267 1478 y Ft(j)p Fz(g)1330 1490 y Fw(p)1369 1478 y Ft(j)g Fz(<)g FA(\()p Fz(\025)24 b Ft(\000)f FA(1\))p Fz(\032)36 b FA(and)f(the)h(ab)r(o)n(v)n(e)e(Lemma)h(sho)n(ws)g(that)g(for)g(eac)n (h)g Fz(p)h Ft(2)h FA(\003)-47 1599 y Fz(\036)2 1568 y Fw(n;w)2 1619 y(p)117 1599 y FA(\()p Fz(x)p FA(\))24 b(:=)380 1577 y(^)363 1599 y Fz( )417 1611 y Fw(p)455 1599 y FA(\()p Fz(w)546 1611 y Fw(p)599 1599 y FA(+)14 b Fz(n)728 1611 y Fw(p)780 1599 y Ft(\000)f Fz(g)898 1611 y Fw(p)936 1599 y FA(\()p Fz(x)p FA(\)\))27 b(is)e(real-analytic)f (\(when)i Fz(w)1969 1611 y Fy(\003)2044 1599 y FA(is)f(real\))g(and)g (maps)2711 1578 y Fx(b)2691 1599 y Fz(A)q FA([)p Fz(\032)p FA(])2843 1611 y Fy(\003)2918 1599 y FA(in)n(to)3103 1578 y Fx(b)3084 1599 y Fz(A)p FA([\()p Fz(\032)14 b FA(+)g Fz(\024)p FA(\))p Fz(=\025)p FA(].)36 b(The)-47 1720 y(natural)25 b(inclusion)g(of)698 1699 y Fx(b)678 1720 y Fz(A)q FA([\()p 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b(unique)f(\014xed)g(p)r(oin)n(t)h(is)f(clearly)f (a)g(preimage)g(of)h Fz(F)3517 2031 y Fy(\003)3599 2019 y FA(and)-47 2118 y(dep)r(ends)e(real-analytically)d(on)j Fz(w)1061 2130 y Fy(\003)1111 2118 y FA(.)40 b(A)29 b(homotop)n(y)f (argumen)n(t)g(\(letting)h(the)g(coupling)f(go)g(to)h(zero\))f(sho)n (ws)g(that)h(b)n(y)-47 2218 y(v)-5 b(arying)26 b Fz(n)298 2230 y Fy(\003)375 2218 y FA(w)n(e)h(obtain)h(exactly)1042 2156 y Fx(Q)1120 2243 y Fw(p)p Fv(2)p Fy(\003)1263 2218 y Fz(d)1306 2188 y Fw(o)1343 2218 y FA(\()p Fz(f)1416 2230 y Fw(p)1454 2218 y FA(\))g(preimages.)-47 2492 y FB(References)-6 2674 y FA([1])41 b(V.)27 b(Baladi,)f(M.)g(Degli)g(Esp) r(osti,)g(S.)g(Isola,)g(E.)g(J\177)-42 b(arv)n(enp\177)g(a\177)g(a,)24 b(A.)i(Kupiainen:)36 b(The)26 b(sp)r(ectrum)g(of)h(w)n(eakly)e(coupled) 123 2774 y(map)j(lattices,)f Fs(J.)j(Math.)h(Pur)l(es)f(Appl.)f Fr(77)p FA(,)e(539-584,)d(\(1998\).)-6 2940 y([2])41 b(G.)28 b(Birkho\013:)37 b(Lattice)27 b(Theory)-7 b(,)27 b(3rd)g(edn.,)g(Amer.)h(Math.)g(So)r(c.,)f(\(1967\).)-6 3106 y([3])41 b(J.)28 b(Bricmon)n(t)f(and)g(A.)h(Kupiainen:)37 b(Coupled)27 b(Analytic)h(Maps,)f Fs(Nonline)l(arity)36 b Fr(8)p FA(,)27 b(379-396,)e(1995.)-6 3272 y([4])41 b(J.)d(Bricmon)n(t)f(and)g(A.)h(Kupiainen:)57 b(High)38 b(T)-7 b(emp)r(erature)37 b(Expansions)f(and)i(Dynamical)f(Systems,)j Fs(Comm.)123 3372 y(Math.)31 b(Phys.)f Fr(178)p FA(,)d(703-732,)d (1996.)-6 3538 y([5])41 b(L.A.)34 b(Bunimo)n(vic)n(h,)f(Y.G.)h(Sinai:) 47 b(Space-time)33 b(c)n(haos)e(in)i(coupled)g(map)g(lattices,)h Fs(Nonline)l(arity)41 b Fr(1)p FA(,)34 b(491-516,)123 3637 y(1988.)-6 3803 y([6])41 b(L.)28 b(Carleson)e(and)h(T.)h(Gamelin:) 37 b(Complex)27 b(Dynamics,)h(Springer,)e(1993.)-6 3969 y([7])41 b(A.)28 b(Douady)g(\(priv)-5 b(ate)27 b(comm)n(unication,)g (1998\).)-6 4135 y([8])41 b(T.)28 b(Fisc)n(her)f(and)g(H.H.)i(Rugh:)37 b(T)-7 b(ransfer)26 b(Op)r(erators)g(for)h(Coupled)g(Analytic)h(Maps,) 123 4268 y Fs(Er)l(g.)j(Th.)g(&)e(Dyn.)h(Syst.)e Fr(20)p FA(,)f(109-144)d(\(2000\).)-6 4434 y([9])41 b(P)-7 b(.)35 b(F)-7 b(errero)33 b(and)i(B.)f(Sc)n(hmitt:)52 b(Pro)r(duits)35 b(al)n(\023)-39 b(eatoires)31 b(d'op)n(\023)-39 b(erateurs)33 b(matrices)h(de)h(transfert,)h Fs(Pr)l(ob.)h(Th.)h(and)123 4534 y(R)l(el.)30 b(Fields.)g Fr(79)p FA(,)d(227-248)d(\(1988\).)-47 4700 y([10])40 b(R.B.)29 b(Israel:)37 b(High-temp)r(erature)28 b(analyticit)n(y)g(in)h(classical)e(lattice)h(systems,)g Fs(Comm.)k(Math.)g(Phys.)e Fr(50)p FA(,)e(245-)123 4800 y(257)f(\(1976\).)-47 4966 y([11])40 b(M.)28 b(Jiang:)36 b(Equilibrium)27 b(states)h(for)f(lattice)g(mo)r(dels)h(of)f(h)n(yp)r (erb)r(olic)g(t)n(yp)r(e,)h Fs(Nonline)l(arity)36 b Fr(8)p FA(,)28 b(631-659)c(\(1995\).)-47 5132 y([12])40 b(M.)28 b(Jiang)e(and)h(Y)-7 b(a.B.)27 b(P)n(esin:)36 b(Equilibrium)27 b(Measures)f(for)g(Coupled)h(Map)g(Lattices:)37 b(Existence,)26 b(Uniqueness)123 5231 y(and)i(Finite-Dimensional)g(Appro)n(ximations,)e Fs(Comm.)31 b(Math.)g(Phys.)e Fr(193)p FA(,)e(675-711)d(\(1998\).)-47 5397 y([13])40 b(K.)28 b(Kanek)n(o)e(\(ed.\):)37 b(Theory)27 b(and)g(Applications)h(of)f(Coupled)h(map)f(Lattices,)g Fs(J.)j(Wiley)p FA(,)f(\(1993\).)1801 5777 y(38)p eop %%Page: 39 39 39 38 bop -47 71 a FA([14])40 b(G.)33 b(Keller)f(and)g(M.)h(K)r(\177) -44 b(unzle:)46 b(T)-7 b(ransfer)32 b(Op)r(erators)e(for)i(Coupled)h (Map)f(Lattices,)h Fs(Er)l(g.Th.)k(&)d(Dyn.Sys.)f Fr(12)p FA(,)123 170 y(297-318,)25 b(\(1992\).)-47 337 y([15])40 b(C.)28 b(Liv)n(erani,)e(Deca)n(y)h(of)h(Correlations,)d Fs(A)n(nnals)30 b(Math.)f Fr(142)p FA(,)e(239-301)d(\(1995\).)-47 503 y([16])40 b(C.)31 b(Maes)f(and)g(A.)g(V)-7 b(an)31 b(Mo\013aert:)42 b(Sto)r(c)n(hastic)30 b(Stabilit)n(y)g(of)g(W)-7 b(eakly)30 b(Coupled)g(Lattice)h(Maps,)f Fs(Nonline)l(arity)123 602 y Fr(10)p FA(,)e(715-730,)c(\(1997\).)-47 768 y([17])40 b(D.L.)32 b(V)-7 b(olevic)n(h:)44 b(The)31 b(Sinai-Bo)n(w)n(en-Ruelle)e (Measure)h(for)h(a)g(Multidimensional)g(Lattice)g(of)g(in)n(teracting)g (Hy-)123 868 y(p)r(erb)r(olic)d(Mappings,)f Fs(R)n(uss.)i(A)n(kad.)i (Dokl.)f(Math.)g Fr(47)p FA(,)d(117-121)d(\(1993\).)1801 5777 y(39)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0101040825184--