Content-Type: multipart/mixed; boundary="-------------9909190432436" This is a multi-part message in MIME format. ---------------9909190432436 Content-Type: text/plain; name="99-342.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-342.keywords" anomalous transport, multifractal analysis ---------------9909190432436 Content-Type: application/postscript; name="lowshort2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="lowshort2.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.85 Copyright 1999 Radical Eye Software %%Title: lowshort2.dvi %%Pages: 7 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -Z lowshort2 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 1999.09.19:1113 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul 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Ff(X)1509 2766 y Fi(n)p Fh(\025)p Fr(0)1676 2666 y Fl(n)1734 2625 y Fi(q)1805 2549 y Ff(Z)1888 2575 y Fi(T)1851 2738 y Fr(0)1970 2599 y Fl(dt)p 1970 2643 86 4 v 1978 2734 a(T)2082 2666 y Fj(jh)p Fl(n)p Fj(j)p Fl(e)2280 2625 y Fh(\000)p Fi({tH)2450 2666 y Fj(j)p Fn(0)p Fj(ij)2594 2625 y Fr(2)2665 2666 y Fn(.)1081 b(\(3\))-118 2921 y(W)-8 b(e)31 b(recall)e(that)h(all) e(di\013usion)i(exp)s(onen)m(ts)i(lie)d(in)g(the)i(in)m(terv)-5 b(al)29 b([0)p Fl(;)17 b Fn(1])30 b(if)f(the)i(o\013-diagonal)c(elemen) m(ts)k(of)f(the)-118 3041 y(Hamiltonian)e Fl(H)39 b Fn(fall)29 b(o\013)i(su\016cien)m(tly)h(rapidly)-8 b(.)42 b(Finally)-8 b(,)29 b(the)j(di\013usion)e(exp)s(onen)m(ts)k(are)d(non-decreasing,) -118 3161 y(con)m(tin)m(uous)i(functions)g(of)f Fl(q)t Fn(,)g(except)i(p)s(ossibly)e(at)h Fl(q)e Fn(=)d(0.)-118 3353 y Fk(Theorem)34 b Fe(Supp)-5 b(ose)35 b(that)i(the)f(gener)-5 b(alize)g(d)35 b(dimensions)g(exist)h(for)g(al)5 b(l)36 b Fl(q)e Fj(2)d Fk(R)36 b Fe(and)f(ar)-5 b(e)36 b(\014nite)g(for)g 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y(eigenfunctions)32 b Fl(P)582 4635 y Fi(n)662 4620 y Fn(b)s(e)g(de\014ned)i(via)e (functional)f(calculus)h(b)m(y)714 4861 y Fj(h)p Fl(n)p Fj(j)p Fl(f)11 b Fn(\()p Fl(H)d Fn(\))p Fj(j)p Fn(0)p Fj(i)54 b Fn(=)1364 4744 y Ff(Z)1464 4861 y Fl(d\026)p Fn(\()p Fl(E)6 b Fn(\))17 b Fl(P)1808 4876 y Fi(n)1853 4861 y Fn(\()p Fl(E)6 b Fn(\))17 b Fl(f)11 b Fn(\()p Fl(E)6 b Fn(\))32 b(,)228 b Fl(f)38 b Fj(2)28 b Fl(C)2774 4876 y Fr(0)2813 4861 y Fn(\()p Fk(R)p Fn(\))k(,)-118 5058 y(and)h(supp)s(ose)g(that)g(there)g(exists)g(a)g Fl(\015)f Fj(2)d Fk(R)j Fn(suc)m(h)i(that)e(for)g(all)f Fl(E)38 b Fn(in)32 b(the)h(sp)s(ectrum)g(of)f Fl(H)40 b Fn(the)33 b(b)s(ound)1167 5216 y Ff(X)1146 5400 y Fi(n)p Fh(\024)p Fi(N)1324 5299 y Fl(P)1387 5314 y Fi(n)1433 5299 y Fn(\()p Fl(E)6 b Fn(\))1587 5258 y Fr(2)1682 5299 y Fj(\024)56 b Fl(c)17 b(N)1962 5258 y Fi(\015)2039 5299 y Fn(,)228 b Fl(c)27 b Fj(2)i Fk(R)j Fn(,)1172 b(\(4\))1865 5635 y(2)p eop %%Page: 3 3 3 2 bop -118 29 a Fn(holds)27 b(for)g(all)f Fl(N)38 b Fj(2)28 b Fk(N)p Fn(.)42 b(Then)29 b(an)e(impro)m(v)m(emen)m(t)h(of)f (the)h(pro)s(of)f(b)s(elo)m(w)h(sho)m(ws)h(that)e Fl(\014)3068 -7 y Fh(\000)3062 54 y Fi(q)3155 29 y Fj(\025)h Fl(D)3341 44 y Fi(\026)3388 29 y Fn(\(1)12 b Fj(\000)g Fl(q)t(=\015)5 b Fn(\))p Fl(=\015)g Fn(.)-118 150 y(If)24 b(no)m(w)h Fl(\015)k Fn(is)24 b(small)e(and)i(the)h(thermo)s(dynamics)e(of)h Fl(\026)g Fn(is)f(imp)s(ortan)m(t)g(so)h(that)g Fl(D)2776 165 y Fi(\026)2823 150 y Fn(\()p Fl(q)t Fn(\))g(increases)h (considerably)-118 270 y(for)34 b(negativ)m(e)i Fl(q)t Fn(,)f(this)g(b)s(ound)g(can)g(b)s(e)g(b)s(etter)h(than)f(the)g(ab)s(o) m(v)m(e.)52 b(Actually)34 b(suc)m(h)i(a)f Fl(\015)40 b Fn(exists)c(and)f(can)g(b)s(e)-118 390 y(calculated)c(for)h(1)p Fl(D)i Fn(quasicrystals)f([8].)43 b(Ho)m(w)m(ev)m(er,)35 b(it)c(turns)i(out)f(to)f(b)s(e)i(rather)f(large.)42 b(This)32 b(results)h(from)-118 511 y(the)i(h)m(yp)s(othesis)i(that)e (\(4\))f(holds)h(uniformly)e(in)h(energy)-8 b(.)52 b(More)36 b(detailed)e(analysis)g(of)h(the)h(link)e(b)s(et)m(w)m(een)-118 631 y(a)e(lo)s(cal)f(exp)s(onen)m(t)j Fl(\015)5 b Fn(\()p Fl(E)h Fn(\))32 b(and)h(the)g(sp)s(ectral)g(prop)s(erties)g(near)f Fl(E)39 b Fn(w)m(as)34 b(made)e(b)m(y)i(Jitormisk)-5 b(a)m(y)m(a)31 b(and)h(Last)-118 751 y([9].)-118 1126 y Fo(3)161 b(Pro)t(of)54 b(of)g(the)f(result)-118 1345 y Fn(First)23 b(w)m(e)i(remark)e(that)h(algebraic)e(gro)m(wth)i(exp)s (onen)m(ts)i(of)d(time)f(a)m(v)m(erages)k(are)d(equal)h(to)g(gro)m(wth) g(exp)s(onen)m(ts)-118 1465 y(of)32 b(gaussian)g(a)m(v)m(erages:)-118 1633 y Fk(Lemma)39 b Fe(L)-5 b(et)41 b Fl(f)51 b Fe(b)-5 b(e)41 b(a)f(non-ne)-5 b(gative)39 b(me)-5 b(asur)g(able)40 b(function)h(satisfying)f Fl(f)11 b Fn(\()p Fl(t)p Fn(\))38 b Fj(\024)h Fl(C)7 b(t)3114 1597 y Fi(n)3202 1633 y Fe(for)41 b(some)f Fl(C)45 b(>)39 b Fn(0)-118 1753 y Fe(and)34 b Fl(n)28 b Fj(\025)g Fn(0)p Fe(.)45 b(Then)624 2068 y Fn(lim)17 b(inf)663 2127 y Fi(T)10 b Fh(!1)921 1981 y Fn(log)1063 1885 y Ff(\020)1113 1910 y(R)1168 1937 y Fi(T)1152 2007 y Fr(0)1240 1981 y Fl(dt)17 b(f)11 b Fn(\()p Fl(t)p Fn(\))1513 1885 y Ff(\021)p 921 2045 642 4 v 1105 2136 a Fn(log\()p Fl(T)j Fn(\))1627 2068 y(=)55 b(lim)17 b(inf)1797 2127 y Fi(T)10 b Fh(!1)2055 1981 y Fn(log)2198 1885 y Ff(\020)2247 1910 y(R)2302 1937 y Fh(1)2286 2007 y Fr(0)2394 1981 y Fl(dt)17 b(e)2542 1945 y Fh(\000)p Fi(t)2622 1922 y Fd(2)2656 1945 y Fi(=)p Fr(4)p Fi(T)2777 1922 y Fd(2)2817 1981 y Fl(f)11 b Fn(\()p Fl(t)p Fn(\))2987 1885 y Ff(\021)p 2055 2045 982 4 v 2409 2136 a Fn(log\()p Fl(T)j Fn(\))3081 2068 y Fe(.)-118 2276 y(Similar)34 b(e)-5 b(qualities)34 b(hold)g(for)h Fn(lim)17 b(sup)33 b Fe(and)h(other)h(gr)-5 b(owth)35 b(exp)-5 b(onents.)-118 2443 y Fk(Pro)s(of:)54 b Fn(Let)39 b Fl(\013)g Fn(and)f Fl(\014)44 b Fn(denote)39 b(the)f(exp)s(onen)m(ts) j(on)d(the)g(left)g(and)g(righ)m(t)f(hand)i(side)f(resp)s(ectiv)m(ely) -8 b(.)62 b(The)-118 2564 y(inequalit)m(y)1148 2712 y Ff(Z)1231 2738 y Fi(T)1194 2900 y Fr(0)1303 2829 y Fl(dt)17 b(f)11 b Fn(\()p Fl(t)p Fn(\))54 b Fj(\024)i Fl(e)1808 2788 y Fr(4)1864 2712 y Ff(Z)1947 2738 y Fh(1)1911 2900 y Fr(0)2039 2829 y Fl(dt)17 b(e)2187 2788 y Fh(\000)p Fi(t)2267 2764 y Fd(2)2301 2788 y Fi(=)p Fr(4)p Fi(T)2422 2764 y Fd(2)2462 2829 y Fl(f)11 b Fn(\()p Fl(t)p Fn(\))-118 3015 y(implies)30 b(that)i Fl(\013)d Fj(\024)f Fl(\014)6 b Fn(.)43 b(On)33 b(the)g(other)f(hand,)h(w)m(e)h(ha)m(v)m(e)637 3234 y Ff(Z)720 3260 y Fh(1)683 3423 y Fr(0)811 3351 y Fl(dt)17 b(e)959 3310 y Fh(\000)p Fi(t)1039 3286 y Fd(2)1074 3310 y Fi(=)p Fr(4)p Fi(T)1195 3286 y Fd(2)1234 3351 y Fl(f)11 b Fn(\()p Fl(t)p Fn(\))83 b Fj(\024)1648 3234 y Ff(Z)1731 3260 y Fi(T)1782 3237 y Fd(1+)p Fc(\017)1694 3423 y Fr(0)1910 3351 y Fl(dtf)11 b Fn(\()p Fl(t)p Fn(\))38 b(+)h Fl(C)2412 3234 y Ff(Z)2495 3260 y Fh(1)2458 3423 y Fi(T)2509 3404 y Fd(1+)p Fc(\017)2637 3351 y Fl(dt)17 b(e)2785 3310 y Fh(\000)p Fi(t)2865 3286 y Fd(2)2900 3310 y Fi(=)p Fr(4)p Fi(T)3021 3286 y Fd(2)3060 3351 y Fl(t)3095 3310 y Fi(n)1487 3602 y Fj(\024)1648 3485 y Ff(Z)1731 3511 y Fi(T)1782 3488 y Fd(1+)p Fc(\017)1694 3674 y Fr(0)1910 3602 y Fl(dtf)11 b Fn(\()p Fl(t)p Fn(\))38 b(+)h Fl(C)2396 3561 y Fh(0)2419 3602 y Fl(e)2464 3561 y Fh(\000)p Fi(T)2570 3538 y Fc(\017)2600 3561 y Fi(=)p Fr(4)2675 3602 y Fl(T)2746 3561 y Fi(n)2825 3602 y Fn(.)-118 3817 y(This)33 b(implies)d(that)i Fl(\014)h Fj(\024)c Fn(\(1)21 b(+)h Fl(\017)p Fn(\))p Fl(\013)34 b Fn(for)e(an)m(y)h Fl(\017)28 b(>)g Fn(0.)2023 b Fb(2)28 3985 y Fn(Hence)42 b(w)m(e)g(shall)d(pro)m(v)m(e)j(a)f(lo)m(w)m(er)g(b)s(ound)g(on)f(the)i (gaussian)e(mean.)68 b(Moreo)m(v)m(er)42 b(w)m(e)g(note)f(that)f Fl(t)i Fj(7!)-118 4105 y(h)p Fl(n)p Fj(j)p Fl(e)52 4069 y Fh(\000)p Fi({tH)223 4105 y Fj(j)p Fn(0)p Fj(i)26 b Fn(is)h(a)g(symmetric)f(function,)i(so)g(that)f(w)m(e)h(can)g (symmetrize)e(time)g(and)i(gaussian)f(means.)41 b(Then)-118 4226 y(w)m(e)j(translate)e(the)i(problem)e(of)g(pro)m(ving)h(a)g(lo)m (w)m(er)g(b)s(ound)g(to)g(that)f(of)h(pro)m(ving)g(an)f(upp)s(er)i(b)s (ounds)g(on)-118 4346 y(presence)35 b(probabilit)m(y)-8 b(,)30 b(notably)-8 b(,)32 b(for)g(an)m(y)h Fl(N)10 b Fn(\()p Fl(T)k Fn(\))28 b Fj(\025)g Fn(0,)33 b(w)m(e)g(ha)m(v)m(e:)465 4584 y Ff(X)459 4767 y Fi(n)p Fh(\025)p Fr(0)636 4667 y Fl(n)694 4626 y Fi(q)876 4600 y Fn(1)p 770 4644 262 4 v 770 4736 a(2)p Fl(T)890 4664 y Fj(p)p 972 4664 59 4 v 972 4736 a Fl(\031)1058 4550 y Ff(Z)1141 4577 y Fh(1)1104 4739 y(\0001)1250 4667 y Fl(dt)28 b(e)1409 4626 y Fh(\000)p Fi(t)1489 4603 y Fd(2)1524 4626 y Fi(=)p Fr(4)p Fi(T)1645 4603 y Fd(2)1701 4667 y Fj(jh)p Fl(n)p Fj(j)p Fl(e)1899 4626 y Fh(\000)p Fi({tH)2069 4667 y Fj(j)p Fn(0)p Fj(ij)2213 4626 y Fr(2)633 4951 y Fj(\025)83 b Fl(N)10 b Fn(\()p Fl(T)k Fn(\))1028 4910 y Fi(q)1173 4868 y Ff(X)1100 5057 y Fi(n>N)7 b Fr(\()p Fi(T)j Fr(\))1527 4884 y Fn(1)p 1420 4928 262 4 v 1420 5020 a(2)p Fl(T)1540 4948 y Fj(p)p 1623 4948 59 4 v 72 x Fl(\031)1708 4834 y Ff(Z)1791 4861 y Fh(1)1755 5023 y(\0001)1901 4951 y Fl(dt)27 b(e)2059 4910 y Fh(\000)p Fi(t)2139 4887 y Fd(2)2174 4910 y Fi(=)p Fr(4)p Fi(T)2295 4887 y Fd(2)2351 4951 y Fj(jh)p Fl(n)p Fj(j)p Fl(e)2549 4910 y Fh(\000)p Fi({tH)2720 4951 y Fj(j)p Fn(0)p Fj(ij)2864 4910 y Fr(2)634 5289 y Fn(=)83 b Fl(N)10 b Fn(\()p Fl(T)k Fn(\))1028 5248 y Fi(q)1083 5118 y Ff(0)1083 5267 y(@)1156 5289 y Fn(1)22 b Fj(\000)1399 5206 y Ff(X)1326 5394 y Fi(n)p Fh(\024)p Fi(N)7 b Fr(\()p Fi(T)j Fr(\))1753 5222 y Fn(1)p 1647 5266 262 4 v 1647 5357 a(2)p Fl(T)1767 5286 y Fj(p)p 1849 5286 59 4 v 1849 5357 a Fl(\031)1935 5172 y Ff(Z)2018 5198 y Fh(1)1981 5361 y(\0001)2127 5289 y Fl(dt)28 b(e)2286 5248 y Fh(\000)p Fi(t)2366 5224 y Fd(2)2401 5248 y Fi(=)p Fr(4)p Fi(T)2522 5224 y Fd(2)2578 5289 y Fj(jh)p Fl(n)p Fj(j)p Fl(e)2776 5248 y Fh(\000)p Fi({tH)2946 5289 y Fj(j)p Fn(0)p Fj(ij)3090 5248 y Fr(2)3129 5118 y Ff(1)3129 5267 y(A)3250 5289 y Fn(.)1865 5635 y(3)p eop %%Page: 4 4 4 3 bop -118 22 a Fn(In)41 b(order)h(to)f(pro)m(v)m(e)h(an)f(upp)s(er)h (b)s(ound)f(on)h(the)f(second)i(expression)f(in)f(the)g(paren)m (thesis,)k(w)m(e)d(split)e(the)-118 142 y(w)m(a)m(v)m(e)c(pac)m(k)m(et) g(as)e(in)g([12])g(in)g(a)g(small,)e(fast)i(di\013using)f(part,)i(and)g (in)e(a)h(rest,)i(but)e(w)m(e)h(mak)m(e)g(this)f(splitting)-118 263 y(time)e(dep)s(enden)m(t)j(as)f(in)f([7)o(].)47 b(F)-8 b(or)32 b(a)h(giv)m(en)h(real)e(Borel)h(set)h Fl(I)8 b Fn(,)34 b(let)e Fl(\037)2419 278 y Fi(I)2493 263 y Fn(denote)i(its)f(indicator)f(function)g(and)-118 383 y(set)h Fj(j)p Fl( )125 398 y Fi(I)165 383 y Fj(i)28 b Fn(=)f Fl(\037)396 398 y Fi(I)436 383 y Fn(\()p Fl(H)8 b Fn(\))p Fj(j)p Fn(0)p Fj(i)p Fn(.)42 b(W)-8 b(e)33 b(c)m(ho)s(ose,)h(for)e(a)g(giv)m(en)h Fl(\013)28 b Fj(2)g Fk(R)p Fn(,)796 636 y Fl(I)8 b Fn(\()p Fl(T)14 b Fn(\))55 b(=)1180 515 y Ff(\032)1243 636 y Fl(E)33 b Fj(2)28 b Fk(R)1570 512 y Ff(\014)1570 562 y(\014)1570 611 y(\014)1570 661 y(\014)1626 519 y(Z)1725 636 y Fl(d\026)p Fn(\()p Fl(E)1951 595 y Fh(0)1974 636 y Fn(\))17 b Fl(e)2074 595 y Fh(\000)p Fr(\()p Fi(E)t Fh(\000)p Fi(E)2323 572 y Fa(0)2344 595 y Fr(\))2371 572 y Fd(2)2406 595 y Fi(T)2457 572 y Fd(2)2524 636 y Fl(<)27 b(T)2698 595 y Fh(\000)p Fi(\013)2802 515 y Ff(\033)2924 636 y Fn(,)-118 859 y(so)32 b(that)g(the)h(large)e(deviation)g(b)s(ound)h(\(11\))g(giv)m(es)g Fj(k)p Fl( )1900 875 y Fi(I)5 b Fr(\()p Fi(T)10 b Fr(\))2046 859 y Fj(k)2096 823 y Fr(2)2163 859 y Fn(=)28 b Fl(\026)p Fn(\()p Fl(I)8 b Fn(\()p Fl(T)14 b Fn(\)\))27 b Fj(\031)h Fl(T)2803 823 y Fi(f)2837 831 y Fc(\026)2878 823 y Fr(\()p Fi(\013)p Fr(\))p Fh(\000)p Fi(\013)3082 859 y Fn(.)44 b(Using)32 b(the)g(decom-)-118 980 y(p)s(osition)f Fj(j)p Fn(0)p Fj(i)c Fn(=)g Fj(j)p Fl( )591 995 y Fi(I)5 b Fr(\()p Fi(T)10 b Fr(\))737 980 y Fj(i)22 b Fn(+)g Fj(j)p Fl( )987 995 y Fi(I)5 b Fr(\()p Fi(T)10 b Fr(\))1128 976 y Fc(c)1165 980 y Fj(i)p Fn(,)32 b(w)m(e)i(then)f(obtain:)357 1173 y(1)p 251 1217 262 4 v 251 1308 a(2)p Fl(T)371 1237 y Fj(p)p 453 1237 59 4 v 453 1308 a Fl(\031)612 1157 y Ff(X)539 1345 y Fi(n)p Fh(\024)p Fi(N)7 b Fr(\()p Fi(T)j Fr(\))849 1123 y Ff(Z)932 1149 y Fh(1)896 1312 y(\0001)1042 1240 y Fl(dt)27 b(e)1200 1199 y Fh(\000)p Fi(t)1280 1175 y Fd(2)1316 1199 y Fi(=)p Fr(4)p Fi(T)1437 1175 y Fd(2)1476 1240 y Fj(jh)p Fl(n)p Fj(j)p Fl(e)1674 1199 y Fh(\000)p Fi({tH)1844 1240 y Fj(j)p Fn(0)p Fj(ij)1988 1199 y Fr(2)2109 1240 y Fj(\024)56 b Fn(1)22 b Fj(\000)h(k)p Fl( )2526 1256 y Fi(I)5 b Fr(\()p Fi(T)10 b Fr(\))2672 1240 y Fj(k)2722 1199 y Fr(2)2783 1240 y Fn(+)22 b Fl(A)p Fn(\()p Fl(T)14 b Fn(\))22 b(+)g Fl(B)5 b Fn(\()p Fl(T)14 b Fn(\))32 b(,)-118 1502 y(where)667 1641 y Fl(A)p Fn(\()p Fl(T)14 b Fn(\))56 b(=)1190 1574 y(1)p 1084 1618 262 4 v 1084 1710 a(2)p Fl(T)1204 1638 y Fj(p)p 1286 1638 59 4 v 1286 1710 a Fl(\031)1445 1558 y Ff(X)1372 1747 y Fi(n)p Fh(\024)p Fi(N)7 b Fr(\()p Fi(T)j Fr(\))1683 1524 y Ff(Z)1766 1551 y Fh(1)1729 1713 y(\0001)1875 1641 y Fl(dt)28 b(e)2034 1600 y Fh(\000)p Fi(t)2114 1577 y Fd(2)2149 1600 y Fi(=)p Fr(4)p Fi(T)2270 1577 y Fd(2)2309 1641 y Fj(jh)p Fl(n)p Fj(j)p Fl(e)2507 1600 y Fh(\000)p Fi({tH)2677 1641 y Fj(j)p Fl( )2768 1657 y Fi(I)5 b Fr(\()p Fi(T)10 b Fr(\))2914 1641 y Fj(ij)2981 1600 y Fr(2)3052 1641 y Fn(,)-118 1903 y(and)353 2043 y Fl(B)5 b Fn(\()p Fl(T)14 b Fn(\))56 b(=)858 1975 y(1)p 776 2020 213 4 v 776 2111 a Fl(T)847 2040 y Fj(p)p 929 2040 59 4 v 929 2111 a Fl(\031)1015 2043 y Fj(<)p Fl(e)1222 1960 y Ff(X)1149 2148 y Fi(n)p Fh(\024)p Fi(N)7 b Fr(\()p Fi(T)j Fr(\))1460 1926 y Ff(Z)1543 1952 y Fh(1)1506 2114 y(\0001)1652 2043 y Fl(dt)27 b(e)1810 2002 y Fh(\000)p Fi(t)1890 1978 y Fd(2)1926 2002 y Fi(=)p Fr(4)p Fi(T)2047 1978 y Fd(2)2086 2043 y Fj(h)p Fl(n)p Fj(j)p Fl(e)2256 2002 y Fh(\000)p Fi({tH)2427 2043 y Fj(j)p Fl( )2518 2058 y Fi(I)5 b Fr(\()p Fi(T)10 b Fr(\))2663 2043 y Fj(i)28 b(h)p Fl( )2832 2058 y Fi(I)5 b Fr(\()p Fi(T)10 b Fr(\))2973 2039 y Fc(c)3010 2043 y Fj(j)p Fl(e)3083 2002 y Fi({tH)3198 2043 y Fj(j)p Fl(n)p Fj(i)33 b Fn(.)-118 2310 y(Let)g(us)g(no)m(w)g(\014rst)g(calculate)f(and)g(b)s(ound)h Fl(A)p Fn(\()p Fl(T)14 b Fn(\):)480 2650 y Fl(A)p Fn(\()p Fl(T)g 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b(that)g(purp)s(ose,)i(w)m(e)g(b)s(ound)e Fl(A)p Fn(\()p Fl(T)14 b Fn(\),)29 b Fl(B)3711 525 y Fr(1)3751 510 y Fn(\()p Fl(T)14 b Fn(\))-118 631 y(and)37 b Fl(B)150 646 y Fr(2)189 631 y Fn(\()p Fl(T)14 b Fn(\))37 b(b)m(y)h(replacing)d (the)j(domains)d(of)i(in)m(tergration)e Fl(I)8 b Fn(\()p Fl(T)14 b Fn(\),)37 b Fl(I)8 b Fn(\()p Fl(T)14 b Fn(\))37 b(and)g Fl(I)8 b Fn(\()p Fl(T)14 b Fn(\))3039 595 y Fi(c)3110 631 y Fn(in)36 b(\(5\),)h(\(6\))g(and)g(\(7\))-118 751 y(resp)s(ectiv)m(ely)d(b)m(y)f Fk(R)p Fn(.)43 b(As)33 b(the)g(generalized)f(eigenfunctions)h(are)f(normalized,)f(one)i(th)m (us)g(obtains)1011 992 y Fl(A)p Fn(\()p Fl(T)14 b Fn(\))22 b(+)g Fl(B)1425 1007 y Fr(1)1465 992 y Fn(\()p Fl(T)14 b Fn(\))21 b(+)i Fl(B)1806 1007 y Fr(2)1845 992 y Fn(\()p Fl(T)14 b Fn(\))55 b Fj(\024)h Fl(c)17 b(T)2310 951 y Fh(\000)p Fi(\013)2430 992 y Fl(N)10 b Fn(\()p Fl(T)k Fn(\))33 b(.)-118 1166 y(Th)m(us)h(w)m(e)g(get:)216 1342 y Ff(X)209 1525 y Fi(n)p Fh(\025)p Fr(0)387 1425 y Fl(n)445 1384 y Fi(q)616 1358 y Fn(1)p 509 1402 262 4 v 509 1494 a(2)p Fl(T)629 1422 y Fj(p)p 712 1422 59 4 v 72 x Fl(\031)797 1308 y Ff(Z)880 1334 y Fh(1)843 1497 y(\0001)990 1425 y Fl(dt)27 b(e)1148 1384 y Fh(\000)p Fi(t)1228 1361 y Fd(2)1263 1384 y Fi(=)p Fr(4)p Fi(T)1384 1361 y Fd(2)1440 1425 y Fj(jh)p Fl(n)p Fj(j)p Fl(e)1638 1384 y Fh(\000)p Fi({tH)1809 1425 y Fj(j)p Fn(0)p Fj(ij)1953 1384 y Fr(2)2046 1425 y Fj(\025)56 b Fl(N)10 b Fn(\()p Fl(T)k Fn(\))2414 1384 y Fi(q)2452 1425 y Fn(\()p Fl(\026)p Fn(\()p Fl(I)8 b Fn(\()p Fl(T)14 b Fn(\)\))21 b Fj(\000)i Fl(c)17 b(T)3074 1384 y Fh(\000)p Fi(\013)3194 1425 y Fl(N)10 b Fn(\()p Fl(T)k Fn(\)\))33 b(.)-118 1727 y(No)m(w)g(the)g(large)e(deviation)g(b) s(ound)i(\(11\))f(implies)e(that)i(for)g(an)m(y)h Fl(\016)f(>)27 b Fn(0)32 b(there)i(exists)f(a)f(p)s(ositiv)m(e)g(constan)m(t)-118 1847 y Fl(C)7 b Fn(\()p Fl(\016)t Fn(\))35 b Fl(<)h Fj(1)h Fn(suc)m(h)h(that)f Fl(\026)p Fn(\()p Fl(I)8 b Fn(\()p Fl(T)14 b Fn(\)\))35 b Fj(\025)h Fl(C)7 b Fn(\()p Fl(\016)t Fn(\))p Fl(T)1558 1811 y Fi(f)1592 1819 y Fc(\026)1633 1811 y Fr(\()p Fi(\013)p Fr(\))p Fh(\000)p Fi(\013)p Fh(\000)p Fi(\016)1926 1847 y Fn(.)58 b(Finally)34 b(w)m(e)39 b(c)m(ho)s(ose)f Fl(N)10 b Fn(\()p Fl(T)k Fn(\))36 b(=)f Fl(T)3263 1811 y Fi(f)3297 1819 y Fc(\026)3338 1811 y Fr(\()p Fi(\013)p Fr(\))p Fh(\000)p Fi(\016)3531 1847 y Fl(C)7 b Fn(\()p Fl(\016)t Fn(\))p Fl(=)p Fn(2)p Fl(c)p Fn(.)-118 1968 y(This)29 b(c)m(hoice)g(assures)h(the)g(p)s(ositivit)m (y)d(of)h(the)h(righ)m(t)f(hand)i(side)e(in)g(the)i(latter)d(equation.) 42 b(Th)m(us)31 b(w)m(e)f(obtain:)343 2156 y Ff(X)337 2338 y Fi(n)p Fh(\025)p Fr(0)514 2239 y Fl(n)572 2198 y Fi(q)743 2171 y Fn(1)p 637 2215 262 4 v 637 2307 a(2)p Fl(T)757 2236 y Fj(p)p 839 2236 59 4 v 839 2307 a Fl(\031)925 2121 y Ff(Z)1008 2148 y Fh(1)971 2310 y(\0001)1117 2239 y Fl(dt)e(e)1276 2198 y Fh(\000)p Fi(t)1356 2174 y Fd(2)1391 2198 y Fi(=)p Fr(4)p Fi(T)1512 2174 y Fd(2)1568 2239 y Fj(jh)p Fl(n)p Fj(j)p Fl(e)1766 2198 y Fh(\000)p Fi({tH)1936 2239 y Fj(j)p Fn(0)p Fj(ij)2080 2198 y Fr(2)2174 2239 y Fj(\025)2316 2171 y Fl(C)7 b Fn(\()p Fl(\016)t Fn(\))p 2316 2215 200 4 v 2392 2307 a(2)2543 2239 y Fl(T)2614 2198 y Fr(\()p Fi(q)r Fr(+1\))p Fi(f)2826 2206 y Fc(\026)2868 2198 y Fr(\()p Fi(\013)p Fr(\))p Fh(\000)p Fi(\013)p Fh(\000)p Fr(\()p Fi(q)r Fr(+1\))p Fi(\016)3372 2239 y Fn(.)-118 2489 y(Up)37 b(to)f(no)m(w)h Fl(\013)f Fj(2)f Fk(R)h Fn(w)m(as)h(arbitrary)-8 b(.)55 b(Maximising)34 b(the)j(righ)m(t)f(hand)h(side)g(o)m(v)m(er)g Fl(\013)e Fj(2)g Fk(R)i Fn(and)f(taking)g(the)-118 2610 y(limit)29 b Fl(\016)j Fj(!)27 b Fn(0,)32 b(one)h(gets)g(the)g(theorem.)-118 2990 y Fo(App)t(endix:)71 b(Large)54 b(deviation)g(theory)f(for)h (singular)g(measures)-118 3209 y Fn(Here)35 b(presen)m(t)h(the)f(form)e (of)h(m)m(ultifractal)d(analysis)i(used)j(in)d(the)i(ab)s(o)m(v)m(e)g (pro)s(of.)48 b(It)35 b(is)f(directly)f(based)j(on)-118 3329 y(the)41 b(G\177)-49 b(artner-Ellis)38 b(large)i(deviation)g (theorem)h([3])g(whic)m(h)g(w)m(e)h(repro)s(duce)g(for)e(the)i(con)m(v) m(enience)h(of)d(the)-118 3450 y(reader.)65 b(Recall)38 b(that)h(b)m(y)i(de\014nition)e(a)g(p)s(oin)m(t)g Fl(x)h Fj(2)g Fk(R)f Fn(is)g(an)h(exp)s(osed)h(p)s(oin)m(t)e(of)g(a)g(real)g (function)g(\003)3762 3413 y Fh(\003)3841 3450 y Fn(if)-118 3570 y(there)33 b(exists)h(a)e Fl(\025)27 b Fj(2)h Fk(R)p Fn(,)33 b(called)e(the)i(exp)s(osing)g(h)m(yp)s(erplane)g(of)f Fl(x)p Fn(,)h(suc)m(h)h(that)959 3811 y Fl(\025y)25 b Fj(\000)e Fn(\003)1257 3770 y Fh(\003)1296 3811 y Fn(\()p Fl(y)t Fn(\))54 b Fl(>)h(\025x)23 b Fj(\000)g Fn(\003)1912 3770 y Fh(\003)1951 3811 y Fn(\()p Fl(x)p Fn(\))32 b(,)228 b Fj(8)56 b Fl(y)31 b Fj(6)p Fn(=)c Fl(x)33 b Fn(.)-118 3985 y Fk(Theorem)i Fn([3])g Fe(L)-5 b(et)37 b Fn(\()p Fl(A)774 4000 y Fi(t)804 3985 y Fn(\))842 4000 y Fi(t)p Fh(2)p Fg(R)975 4009 y Fd(+)1067 3985 y Fe(b)-5 b(e)37 b(a)g(family)g(of)g(r)-5 b(e)g(al)37 b(r)-5 b(andom)36 b(variables)g(on)h(a)g(pr)-5 b(ob)g(ability)37 b(sp)-5 b(ac)g(e)36 b Fn(\(\012)p Fl(;)17 b Fk(P)p Fn(\))p Fe(.)-118 4105 y(Their)34 b(char)-5 b(acteristic)34 b(functions)h(ar)-5 b(e)34 b(de\014ne)-5 b(d)34 b(as)877 4359 y Fn(\003)945 4374 y Fi(t)975 4359 y Fn(\()p Fl(\025)p Fn(\))55 b(=)g(log)1437 4237 y Ff(\022)1498 4241 y(Z)1597 4359 y Fl(d)p Fk(P)p Fn(\()p Fl(!)t Fn(\))17 b Fl(e)1928 4318 y Fi(\025A)2022 4326 y Fc(t)2048 4318 y Fr(\()p Fi(!)r Fr(\))2154 4237 y Ff(\023)2266 4359 y Fe(,)234 b Fl(\025)28 b Fj(2)g Fk(R)35 b Fe(.)-118 4565 y(We)g(supp)-5 b(ose)34 b(that)h(for)g(al)5 b(l)34 b Fl(\025)28 b Fj(2)g Fk(R)35 b Fe(the)g(fol)5 b(lowing)33 b(limit)i(exists:)1388 4825 y Fn(\003\()p Fl(\025)p Fn(\))55 b(=)70 b(lim)1775 4880 y Fi(t)p Fh(!1)1968 4758 y Fn(1)p 1968 4802 49 4 v 1975 4893 a Fl(t)2027 4825 y Fn(\003)2095 4840 y Fi(t)2124 4825 y Fn(\()p Fl(t\025)p Fn(\))35 b Fe(,)1416 b Fn(\(8\))-118 5025 y Fe(and)37 b(further)h(that)f Fn(\003)h Fe(is)f(\014nite)g(in)g(a)g(neighb)-5 b(orho)g(o)g(d)36 b(of)h Fn(0)p Fe(.)52 b(Then)37 b Fn(\003)g Fe(is)g(a)h(c)-5 b(onvex)36 b(function)h(and)g(so)g(is)h(its)-118 5145 y(L)-5 b(e)g(gendr)g(e)34 b(tr)-5 b(ansform)34 b Fn(\003)804 5109 y Fh(\003)843 5145 y Fe(.)-118 5289 y(\(i\))g(F)-7 b(or)34 b(any)h(close)-5 b(d)34 b(set)h Fl(F)41 b Fj(\032)28 b Fk(R)p Fe(,)35 b(one)f(has)1865 5635 y Fn(5)p eop %%Page: 6 6 6 5 bop 918 93 a Fn(lim)17 b(sup)984 167 y Fi(t)p Fh(!1)1243 25 y Fn(1)p 1243 70 49 4 v 1250 161 a Fl(t)1319 93 y Fn(log\()p Fk(P)p Fn(\()p Fj(f)p Fl(A)1721 108 y Fi(t)1777 93 y Fj(2)28 b Fl(F)14 b Fj(g)p Fn(\)\))55 b Fj(\024)h(\000)29 b Fn(inf)2356 152 y Fi(x)p Fh(2)p Fi(F)2514 93 y Fn(\003)2582 52 y Fh(\003)2621 93 y Fn(\()p Fl(x)p Fn(\))35 b Fe(.)956 b Fn(\(9\))-118 336 y Fe(\(ii\))34 b(F)-7 b(or)34 b(any)h(op)-5 b(en)34 b(set)h Fl(G)28 b Fj(\032)g Fk(R)p Fe(,)34 b(we)h(have)885 596 y Fn(lim)17 b(inf)936 651 y Fi(t)p Fh(!1)1181 529 y Fn(1)p 1181 573 V 1188 664 a Fl(t)1257 596 y Fn(log\()p Fk(P)p Fn(\()p Fj(f)p Fl(A)1659 611 y Fi(t)1715 596 y Fj(2)28 b Fl(G)p Fj(g)p Fn(\)\))55 b Fj(\025)h(\000)81 b Fn(inf)2294 655 y Fi(x)p Fh(2)p Fi(G)p Fh(\\F)2557 596 y Fn(\003)2625 555 y Fh(\003)2664 596 y Fn(\()p Fl(x)p Fn(\))35 b Fe(,)864 b Fn(\(10\))28 799 y Fe(wher)-5 b(e)34 b Fj(F)45 b Fe(is)34 b(the)h(set)g(of)g(exp)-5 b(ose)g(d)34 b(p)-5 b(oints)34 b(of)h Fn(\003)1771 763 y Fh(\003)1845 799 y Fe(having)e(an)i(exp)-5 b(osing)33 b(hyp)-5 b(erplane)34 b(in)h(the)g(interior)f(of)28 920 y(the)h(set)g Fj(f)p Fl(\025)27 b Fj(2)i Fk(R)17 b Fj(j)g Fn(\003\()p Fl(\025)p Fn(\))25 b Fl(<)j Fj(1g)p Fe(.)28 1087 y Fn(W)-8 b(e)42 b(w)m(an)m(t)h(to)e(analyse)h(the)h(fractal)d(prop)s(erties)i(of)f (probabilit)m(y)f(measure)i Fl(\026)f Fn(on)h Fk(R)f Fn(with)h(the)g(large)-118 1208 y(deviation)31 b(theory)j(in)d(\()p Fk(R)p Fl(;)17 b(\026)p Fn(\).)43 b(Herefore)33 b(w)m(e)g(study)h(the)f (random)f(v)-5 b(ariables)1058 1495 y Fl(A)1131 1510 y Fi(t)1160 1495 y Fn(\()p Fl(E)6 b Fn(\))56 b(=)1511 1428 y(log\()1675 1357 y Ff(R)1747 1428 y Fl(d\026)p Fn(\()p Fl(E)1973 1392 y Fh(0)1995 1428 y Fn(\))17 b Fl(e)2095 1392 y Fh(\000)p Fr(\()p Fi(E)t Fh(\000)p Fi(E)2344 1368 y Fa(0)2366 1392 y Fr(\))2393 1368 y Fd(2)2428 1392 y Fi(e)2461 1368 y Fa(\000)p Fd(2)p Fc(t)2571 1428 y Fn(\))p 1511 1472 1098 4 v 2042 1564 a Fl(t)2651 1495 y Fn(.)-118 1695 y(Under)36 b(the)g(h)m(yp)s(othesis)h(that)f(the)g (app)s(earing)e(limits)f(exist,)k(the)f(asso)s(ciated)f(rate)h (function)f(is)g(the)h(Leg-)-118 1816 y(endre)d(transform)f(of)618 2193 y(\003\()p Fl(\025)p Fn(\))82 b(=)99 b(lim)1060 2248 y Fi(t)p Fh(!1)1253 2126 y Fn(1)p 1253 2170 49 4 v 1260 2262 a Fl(t)1329 2193 y Fn(log)1471 2047 y Ff( )1537 2076 y(Z)1637 2193 y Fl(d\026)p Fn(\()p Fl(E)6 b Fn(\))1933 2072 y Ff(\022)1994 2076 y(Z)2094 2193 y Fl(d\026)p Fn(\()p Fl(E)2320 2152 y Fh(0)2343 2193 y Fn(\))17 b Fl(e)2443 2152 y Fh(\000)p Fr(\()p Fi(E)t Fh(\000)p Fi(E)2692 2129 y Fa(0)2713 2152 y Fr(\))2740 2129 y Fd(2)2775 2152 y Fi(e)2808 2129 y Fa(\000)p Fd(2)p Fc(t)2918 2072 y Ff(\023)2979 2095 y Fh(\000)p Fi(\025)3079 2047 y Ff(!)901 2562 y Fn(=)111 b(lim)1060 2621 y Fi(T)10 b Fh(!1)1279 2451 y Fn(log)1421 2329 y Ff(\022)1483 2380 y(R)1554 2451 y Fl(d\026)p Fn(\()p Fl(E)c Fn(\))1851 2354 y Ff(\020)1901 2380 y(R)1973 2451 y Fl(d\026)p Fn(\()p Fl(E)2199 2414 y Fh(0)2221 2451 y Fn(\))17 b Fl(e)2321 2414 y Fh(\000)p Fr(\()p Fi(E)t Fh(\000)p Fi(E)2570 2391 y Fa(0)2591 2414 y Fr(\))2618 2391 y Fd(2)2653 2414 y Fi(T)2704 2391 y Fd(2)2743 2354 y Ff(\021)2793 2377 y Fh(\000)p Fi(\025)2893 2329 y Ff(\023)p 1279 2539 1676 4 v 1980 2630 a Fn(log\()p Fl(T)d Fn(\))2996 2562 y(.)-118 2838 y(Actually)31 b(the)i(function)g (\003)f(is)g(link)m(ed)g(to)g Fl(\034)1474 2853 y Fi(\026)1521 2838 y Fn(\()p Fl(q)t Fn(\))h(and)f Fl(D)1947 2853 y Fi(\026)1994 2838 y Fn(\()p Fl(q)t Fn(\))g(as)h(de\014ned)h(in)e(\(1\)) g(b)m(y:)520 3079 y(\003\()p Fl(\025)p Fn(\))55 b(=)g Fl(\025D)1045 3094 y Fi(\026)1092 3079 y Fn(\(1)22 b Fj(\000)g Fl(\025)p Fn(\))56 b(=)f Fj(\000)p Fl(\034)1701 3094 y Fi(\026)1748 3079 y Fn(\(1)22 b Fj(\000)h Fl(\025)p Fn(\))32 b(,)228 b(\003)2407 3037 y Fh(\003)2446 3079 y Fn(\()p Fl(x)p Fn(\))56 b(=)f Fl(x)22 b Fj(\000)h Fl(f)2989 3094 y Fi(\026)3036 3079 y Fn(\()p Fl(x)p Fn(\))33 b(,)-118 3253 y(where)j Fl(f)214 3268 y Fi(\026)295 3253 y Fn(is)f(the)g (scaling)f(function)g(of)g(the)i(measure)f Fl(\026)f Fn(de\014ned)j(to)d(b)s(e)h(the)g(Legendre)h(transform)e(of)h Fl(\034)3824 3268 y Fi(\026)3871 3253 y Fn(.)-118 3373 y(T)-8 b(o)37 b(see)i(this,)f(let)e(us)i(sho)m(w)g Fl(\034)979 3388 y Fi(\026)1026 3373 y Fn(\()p Fl(q)t Fn(\))e(=)f Fj(\000)p Fn(\003\(1)25 b Fj(\000)h Fl(q)t Fn(\))37 b(for)g(all)e Fl(q)k Fj(2)d Fk(R)p Fj(nf)p Fn(0)p Fj(g)p Fn(.)57 b(As)37 b(the)h(c)m(haracteristic)f(function)-118 3494 y(on)32 b(a)f(ball)f(of)h(size)h(1)p Fl(=T)45 b Fn(used)33 b(in)f(the)g (de\014nition)f(\(1\))g(is)g(less)h(or)g(equal)g(to)f(the)h(function)g Fl(ee)3281 3458 y Fh(\000)p Fr(\()p Fi(E)t Fh(\000)p Fi(E)3530 3434 y Fa(0)3551 3458 y Fr(\))3578 3434 y Fd(2)3614 3458 y Fi(T)3665 3434 y Fd(2)3703 3494 y Fn(,)g(the)-118 3614 y(inequalities)g Fl(\034)438 3629 y Fi(\026)485 3614 y Fn(\()p Fl(q)t Fn(\))e Fj(\024)h(\000)p Fn(\003\(1)24 b Fj(\000)f Fl(q)t Fn(\))34 b(for)g Fl(q)g(<)d Fn(1)j(and)g Fl(\034)1872 3629 y Fi(\026)1919 3614 y Fn(\()p Fl(q)t Fn(\))c Fj(\025)h(\000)p Fn(\003\(1)23 b Fj(\000)h Fl(q)t Fn(\))34 b(for)g Fl(q)g(>)c Fn(1)k(follo)m(w)f(immediately)-8 b(.)-118 3734 y(F)g(or)32 b(the)h(con)m(v)m(erse,)i(w)m(e)e(note)g (that)g(an)m(y)g Fl(\017)28 b(>)f Fn(0:)493 3858 y Ff(Z)593 3975 y Fl(d\026)p Fn(\()p Fl(E)819 3934 y Fh(0)841 3975 y Fn(\))17 b Fl(e)941 3934 y Fh(\000)p Fr(\()p Fi(E)t Fh(\000)p Fi(E)1190 3911 y Fa(0)1212 3934 y Fr(\))1239 3911 y Fd(2)1274 3934 y Fi(T)1325 3911 y Fd(2)1419 3975 y Fj(\024)1552 3858 y Ff(Z)1651 3975 y Fl(d\026)p Fn(\()p Fl(E)1877 3934 y Fh(0)1900 3975 y Fn(\))g Fl(\037)p Fn(\()p Fj(j)p Fl(E)27 b Fj(\000)c Fl(E)2359 3934 y Fh(0)2382 3975 y Fj(j)28 b(\024)g Fl(T)2614 3934 y Fh(\000)p Fr(1+)p Fi(\017)2791 3975 y Fn(\))22 b(+)g(2)p Fl(e)3043 3934 y Fh(\000)p Fi(T)3149 3911 y Fc(\017)3216 3975 y Fn(.)-118 4180 y(This)33 b(implies)d(that)i(\(1)22 b Fj(\000)h Fl(\017)p Fn(\))p Fl(\034)975 4195 y Fi(\026)1022 4180 y Fn(\()p Fl(q)t Fn(\))k Fj(\025)h(\000)p Fn(\003\(1)22 b Fj(\000)h Fl(q)t Fn(\))32 b(for)g Fl(q)g(<)27 b Fn(1)33 b(and)f(\(1)22 b Fj(\000)h Fl(\017)p Fn(\))p Fl(\034)2674 4195 y Fi(\026)2721 4180 y Fn(\()p Fl(q)t Fn(\))28 b Fj(\024)g(\000)p Fn(\003\(1)22 b Fj(\000)g Fl(q)t Fn(\))33 b(for)f Fl(q)f(>)d Fn(1.)28 4347 y(Under)e(the)g(h)m(yp)s(othesis)h (that)e Fl(D)1236 4362 y Fi(\026)1282 4347 y Fn(\()p Fl(q)t Fn(\))j Fl(<)f Fj(1)e Fn(for)g(some)g Fl(q)32 b(<)27 b Fn(0,)g(w)m(e)f(can)g(no)m(w)g(disp)s(ose)g(of)f(the)g(ab)s(o) m(v)m(e)i(large)-118 4468 y(deviation)38 b(b)s(ounds.)64 b(In)39 b(particular,)g(for)f(the)i(sets)g Fj(f)p Fl(A)1974 4483 y Fi(t)2043 4468 y Fl(>)e(\013)q Fj(g)p Fn(,)i Fl(\013)g Fj(\025)f Fl(f)2603 4483 y Fi(\026)2650 4468 y Fn(\()p Fl(\013)q Fn(\),)h(the)g(lo)m(w)m(er)f(b)s(ound)g(implies)-118 4588 y(that:)367 4903 y(lim)17 b(inf)406 4962 y Fi(T)10 b Fh(!1)664 4816 y Fn(log)806 4720 y Ff(\020)856 4816 y Fl(\026)932 4720 y Ff(\020)o(n)1036 4816 y Fl(E)34 b Fj(2)28 b Fk(R)1353 4717 y Ff(\014)1353 4766 y(\014)1353 4816 y(\014)1397 4745 y(R)1469 4816 y Fl(d\026)p Fn(\()p Fl(E)1695 4780 y Fh(0)1718 4816 y Fn(\))p Fl(e)1801 4780 y Fh(\000)p Fr(\()p Fi(E)t Fh(\000)p Fi(E)2050 4757 y Fa(0)2071 4780 y Fr(\))2098 4757 y Fd(2)2133 4780 y Fi(T)2184 4757 y Fd(2)2251 4816 y Fl(<)f(T)2425 4780 y Fh(\000)p Fi(\013)2539 4720 y Ff(o)q(\021)o(\021)p 664 4879 2030 4 v 1542 4971 a Fn(log\()p Fl(T)14 b Fn(\))2759 4903 y Fj(\025)56 b Fl(f)2940 4918 y Fi(\026)2987 4903 y Fn(\()p Fl(\013)q Fn(\))21 b Fj(\000)i Fl(\013)33 b Fn(.)355 b(\(11\))1865 5635 y(6)p eop %%Page: 7 7 7 6 bop -118 52 a Fo(References)28 272 y Fn([1])49 b(J.-M.)e (Barbaroux,)k(H.)c(Sc)m(h)m(ulz-Baldes,)k Fm(Anomalous)45 b(transp)s(ort)i(in)f(presence)j(of)e(self-simalar)180 392 y(sp)s(ectra)p Fn(,)33 b(to)g(app)s(ear)f(in)g(Annales)h(Inst.)g (H.)g(P)m(oincar)m(\023)-46 b(e,)32 b(No)m(v.)i(1999.)28 593 y([2])49 b(J.-M.)58 b(Com)m(b)s(es,)65 b Fm(Connection)58 b(b)s(et)m(w)m(een)i(quan)m(tum)f(dynamics)e(and)h(sp)s(ectral)g(prop)s (erties)g(of)180 714 y(time)35 b(ev)m(olution)h(op)s(erators)p Fn(,)h(in)f Fm(Di\013eren)m(tial)e(Equations)j(with)f(Applications)f (to)h(Mathematical)180 834 y(Ph)m(ysics)p Fn(,)e(Ames)f(W.F.,)g(Harell) e(E.M.,)i(Hero)s(d)g(J.V.)g(Eds,)g(Academic)g(Press,)h(Boston)f (\(1993\).)28 1036 y([3])49 b(A.)32 b(Dem)m(b)s(o,)f(O.)g(Zeitouni,)f Fm(Large)h(Deviation)f(T)-8 b(ec)m(hniques)34 b(and)d(Applications)p Fn(,)f(\(Jones)j(and)e(Bar-)180 1156 y(lett,)h(Boston,)h(London,)f (1993\).)28 1358 y([4])49 b(I.)d(Guarneri,)i Fm(Sp)s(ectral)d(prop)s (erties)h(of)g(quan)m(tum)g(di\013usion)e(on)i(discrete)g(lattices)p Fn(,)j(Europh)m(ys.)180 1478 y(Lett.,)32 b Fk(10)p Fn(,)g(95-100,)f (\(1989\);)g Fm(On)g(an)h(estimate)f(concerning)h(quan)m(tum)g (di\013usion)e(in)h(the)h(presence)180 1599 y(of)g(a)g(fractal)g(sp)s (ectrum)p Fn(,)h(Europh)m(ys.)h(Lett.,)f Fk(21)p Fn(,)g(729-733,)e (\(1993\).)28 1800 y([5])49 b(I.)24 b(Guarneri,)g(G.)g(Man)m(tica,)h Fm(Multifractal)c(Energy)j(Sp)s(ectra)h(and)e(Their)h(Dynamical)d (Implications)p Fn(,)180 1920 y(Ph)m(ys.)34 b(Rev.)g(Lett.)e Fk(73)p Fn(,)h(3379-3383)d(\(1994\).)28 2122 y([6])49 b(I.)29 b(Guarneri,)g(H.)g(Sc)m(h)m(ulz-Baldes,)h Fm(Upp)s(er)f(b)s (ounds)g(for)g(quan)m(tum)g(dynamics)f(go)m(v)m(erned)j(b)m(y)e(Jacobi) 180 2242 y(matrices)j(with)g(self-similar)c(sp)s(ectra)p Fn(,)34 b(to)e(app)s(ear)g(in)g(Rev.)i(Math.)e(Ph)m(ys.,)j(No)m(v.)e (1999.)28 2444 y([7])49 b(I.)41 b(Guarneri,)h(H.)f(Sc)m(h)m (ulz-Baldes,)j Fm(Lo)m(w)m(er)e(b)s(ounds)f(on)g(w)m(a)m(v)m(e)i(pac)m (k)m(et)f(propagation)e(b)m(y)h(pac)m(king)180 2564 y(dimensions)32 b(of)g(sp)s(ectral)g(measures)p Fn(,)i(Elect.)e(J.)h(Math.)g(Ph)m(ys.)h Fk(5)p Fn(,)f(\(1999\).)28 2766 y([8])49 b(B.)28 b(Io)s(c)m(h)m(um,)i (L.)e(Ra)m(ymond,)h(D.)f(T)-8 b(estard,)30 b Fm(Resistance)f(of)f (one-dimensional)e(quasicrystals)p Fn(,)j(Ph)m(ys-)180 2886 y(ica)j Fk(A)37 b(187)p Fn(,)32 b(353-368)f(\(1992\).)28 3088 y([9])49 b(S.)25 b(Jitorminsk)-5 b(a)m(y)m(a,)26 b(Y.)f(Last,)i Fm(Dimensional)22 b(Hausdor\013)k(prop)s(erties)f(of)f (singular)g(con)m(tin)m(uous)i(sp)s(ec-)180 3208 y(tra)p Fn(,)32 b(Ph)m(ys.)j(Rev.)e(Lett.)g Fk(76)p Fn(,)f(1765-1769)f (\(1996\).)-21 3410 y([10])49 b(R.)43 b(Ketzmeric)m(k,)k(K.)d(Kruse,)j (S.)d(Kraut,)i(T.)e(Geisel,)i Fm(What)d(Determines)g(the)h(Spreading)g (of)f(a)180 3530 y(W)-8 b(a)m(v)m(e)34 b(P)m(ac)m(k)m(et?)p Fn(,)h(Ph)m(ys.)f(Rev.)f(Lett.)g Fk(79)p Fn(,)g(1959-1962)d(\(1997\).) -21 3732 y([11])49 b(A.)41 b(Kiselev,)h(Y.)f(Last,)i Fm(Solutions,)f(sp)s(ectrum,)h(and)e(dynamics)g(for)f(Sc)m(hr\177)-49 b(odinger)41 b(op)s(erators)g(on)180 3852 y(in\014nite)32 b(domains)p Fn(,)f(preprin)m(t,)i(mp-arc)e(99-206.)-21 4054 y([12])49 b(Y.)25 b(Last,)i Fm(Quan)m(tum)e(dynamics)g(and)h (decomp)s(osition)d(of)i(singular)e(con)m(tin)m(uous)j(sp)s(ectra)p Fn(,)i(J.)d(F)-8 b(unct.)180 4174 y(Anal.)32 b Fk(142)p Fn(,)h(402-445)d(\(1996\).)-21 4376 y([13])49 b(G.)27 b(Man)m(tica,)i Fm(Quan)m(tum)e(in)m(termittency)g(in)g(almost)f(p)s (erio)s(dic)h(systems)i(deriv)m(ed)f(from)f(their)g(sp)s(ec-)180 4496 y(tral)32 b(prop)s(erties)p Fn(,)h(Ph)m(ysica)g Fk(D)38 b(103)p Fn(,)33 b(576-589,)e(\(1997\);)i Fm(W)-8 b(a)m(v)m(e)34 b(Propagation)d(in)h(Almost-P)m(erio)s(dic)180 4617 y(Structures)p Fn(,)i(Ph)m(ysica)g Fk(D)j(109)p Fn(,)c(113-127)d(\(1997\).)-21 4818 y([14])49 b(F.)33 b(Pi)m(\023)-46 b(ec)m(hon,)34 b Fm(Anomalous)e(Di\013usion)f(Prop)s (erties)i(of)g(W)-8 b(a)m(v)m(e)35 b(P)m(ac)m(k)m(ets)g(on)e(Quasip)s (erio)s(dic)e(Chains)p Fn(,)180 4938 y(Ph)m(ys.)j(Rev.)g(Lett.)e Fk(76)p Fn(,)h(4372-4375)d(\(1996\).)-21 5140 y([15])49 b(H.)31 b(Sc)m(h)m(ulz-Baldes,)h(J.)g(Bellissard,)e Fm(Anomalous)g (transp)s(ort:)42 b(a)31 b(mathematical)d(framew)m(ork)p Fn(,)k(Rev.)180 5260 y(Math.)h(Ph)m(ys.)h Fk(10)p Fn(,)f(1-46)e (\(1998\).)1865 5635 y(7)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9909190432436--