%!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: mocor.dvi %%Pages: 15 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -Z mocor %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 1999.01.06:1400 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 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b(als)34 b(all)f(con)m(tained)j(in)e(a)h(closed)g(in)m(terv)-5 b(al)34 b Fn(I)2394 810 y Fs(0)2433 846 y Fo(.)51 b(Let)36 b Fn(S)k Fo(b)s(e)c(a)f(smo)s(oth)f(real)g(function)-118 967 y(suc)m(h)28 b(that,)g(for)f(all)e Fn(l)30 b Fo(=)d(1)p Fn(;)17 b(:)g(:)g(:)f(;)h(L)p Fo(,)28 b(the)g(restriction)e Fn(S)1839 982 y Fm(l)1892 967 y Fo(of)g Fn(S)33 b Fo(to)27 b Fn(I)2255 931 y Fs(1)2247 991 y Fm(l)2321 967 y Fo(is)g(bijectiv)m(e) g(from)f Fn(I)3072 931 y Fs(1)3064 991 y Fm(l)3138 967 y Fo(to)h Fn(I)3303 931 y Fs(0)3369 967 y Fo(with)f(smo)s(oth)-118 1087 y(in)m(v)m(erse)38 b Fn(S)275 1046 y Fg(\000)p Fs(1)269 1112 y Fm(l)406 1087 y Fo(\(in)e(particular,)g(w)m(e)h(assume)h(the)f (deriv)-5 b(ativ)m(e)36 b(of)g Fn(S)43 b Fo(to)36 b(b)s(e)h(b)s(ounded) g(a)m(w)m(a)m(y)h(from)e(0)g(and)h Fk(1)-118 1208 y Fo(on)31 b(the)h(in)m(terv)-5 b(als)31 b Fn(I)624 1171 y Fs(1)616 1232 y Fm(l)663 1208 y Fo(\).)43 b(F)-8 b(urthermore,)31 b(the)h(image)e(of)h Fn(I)1971 1171 y Fs(0)2010 1208 y Fk(n)2077 1141 y Fl(S)2146 1167 y Fm(L)2146 1232 y(l)q Fs(=1)2279 1208 y Fn(I)2330 1171 y Fs(1)2322 1232 y Fm(l)2401 1208 y Fo(under)h Fn(S)37 b Fo(is)31 b(supp)s(osed)i(to)e(ha)m(v)m(e)i (empt)m(y)-118 1328 y(in)m(tersection)f(with)h Fn(I)679 1292 y Fs(0)718 1328 y Fo(.)28 1448 y(W)-8 b(e)32 b(call)e Ff(co)s(des)i Fo(the)g(one-sided)f(sequences)k(of)30 b(sym)m(b)s(ols)i(tak)m(en)g(from)e Fk(f)p Fo(1)p Fn(;)17 b(:)g(:)g(:)e(;)i(L)p Fk(g)32 b Fo(and)f(denote)h(the)g(set)-118 1569 y(of)41 b(co)s(des)h(of)f(length)g Fn(N)52 b Fo(b)m(y)43 b(\006)1046 1533 y Fm(N)1046 1593 y(L)1155 1569 y Fo(and)e(the)h(set)g (of)f(co)s(des)i(of)e(in\014nite)f(length)h(b)m(y)h(\006)3064 1584 y Fm(L)3117 1569 y Fo(.)70 b(F)-8 b(or)41 b(all)e Fn(N)54 b Fk(2)43 b Fe(N)p Fo(,)-118 1689 y Fn(S)-52 1653 y Fg(\000)p Fm(N)70 1689 y Fo(\()p Fn(I)159 1653 y Fs(0)198 1689 y Fo(\))28 b(consists)g(of)f Fn(L)790 1653 y Fm(N)885 1689 y Fo(closed,)i(disjoin)m(t)d(in)m(terv)-5 b(als)27 b Fn(I)1975 1653 y Fm(N)1967 1714 y(\033)2070 1689 y Fo(=)h Fn(S)2240 1653 y Fg(\000)p Fs(1)2234 1714 y Fm(\033)2274 1723 y Fh(1)2346 1689 y Fk(\016)12 b Fn(:)17 b(:)g(:)11 b Fk(\016)h Fn(S)2662 1653 y Fg(\000)p Fs(1)2656 1714 y Fm(\033)2696 1725 y Fj(N)2758 1689 y Fo(\()p Fn(I)2847 1653 y Fs(0)2886 1689 y Fo(\),)29 b Fn(\033)i Fo(=)d(\()p Fn(\033)3263 1704 y Fs(1)3319 1689 y Fn(:)17 b(:)g(:)f(\033)3505 1704 y Fm(N)3573 1689 y Fo(\))28 b Fk(2)g Fo(\006)3803 1653 y Fm(N)3803 1714 y(L)3871 1689 y Fo(,)-118 1809 y(whic)m(h)33 b(w)m(e)g(call)d(the)j(in)m(terv)-5 b(als)32 b(of)f(the)i Fn(N)10 b Fo(th)33 b(generation.)43 b(W)-8 b(e)32 b(further)h(assume)g(that)f(there)h(exist)f(p)s(ositiv)m(e)-118 1930 y(constan)m(ts)45 b Fn(a)h(<)g Fo(1)d(and)g Fn(c)g Fo(so)h(that,)i(for)c(an)m(y)i Fn(N)57 b Fk(2)46 b Fe(N)p Fo(,)g(all)41 b(in)m(terv)-5 b(als)42 b(of)h(the)h Fn(N)10 b Fo(th)44 b(generation)f(satisfy)-118 2050 y Fk(j)p Fn(I)-39 2014 y Fm(N)-47 2075 y(\033)28 2050 y Fk(j)28 b(\024)g Fn(c)17 b(a)299 2014 y Fm(N)399 2050 y Fo(where)35 b Fk(j)p Fn(I)8 b Fk(j)32 b Fo(denotes)i(the)f(length)g(of)f(the)h(in)m (terv)-5 b(al)32 b Fn(I)8 b Fo(.)45 b(No)m(w)33 b Fn(J)k Fo(=)2814 1984 y Fl(T)2883 2071 y Fm(N)7 b Fg(\025)p Fs(0)3057 2050 y Fn(S)3123 2014 y Fg(\000)p Fm(N)3245 2050 y Fo(\()p Fn(I)3334 2014 y Fs(0)3373 2050 y Fo(\))33 b(is)f(a)h(fractal)-118 2171 y(set)k(whic)m(h)g(is)f(in)m(v)-5 b(arian)m(t)36 b(under)h Fn(S)6 b Fo(,)37 b Ff(i.e.)56 b Fn(S)6 b Fo(\()p Fn(J)j Fo(\))34 b(=)g Fn(J)9 b Fo(.)56 b(If)36 b Fn(S)43 b Fo(is)36 b(a)g(p)s(olynomial)d(map)j(satisfying)g (the)h(ab)s(o)m(v)m(e)-118 2291 y(h)m(yp)s(othesis,)29 b(then)d Fn(J)36 b Fo(is)25 b(called)g(a)h(real)f(h)m(yp)s(erb)s(olic)h (Julia)e(set.)42 b(In)27 b(the)f(case)h(where)h(the)e(maps)g Fn(S)3424 2306 y Fm(l)3476 2291 y Fo(are)g(linear,)-118 2411 y(w)m(e)34 b(deal)d(with)i(a)f(linear)f(iterated)h(function)g (system)i(and)e(call)f(the)i(rep)s(eller)f(a)g(linear)f(Can)m(tor)i (set.)28 2532 y(The)g(dynamical)e(system)i(\()p Fn(J)n(;)17 b(S)6 b Fo(\))31 b(is)h(conjugated)g(to)g(the)g(shift)g(on)g(\006)2593 2547 y Fm(L)2678 2532 y Fo(b)m(y)g(the)h(co)s(ding)e(map)g Fn(E)j Fk(2)28 b Fn(J)37 b Fk(7!)-118 2652 y Fn(\033)t Fo(\()p Fn(E)6 b Fo(\))28 b Fk(2)g Fo(\006)287 2667 y Fm(L)339 2652 y Fo(.)44 b(Giv)m(en)32 b(a)g(shift-in)m(v)-5 b(arian)m(t,)31 b(ergo)s(dic)g(measure)i(on)g(\006)2354 2667 y Fm(L)2406 2652 y Fo(,)g(the)g(p)s(oin)m(t)m(wise)f(dimensions) 1129 2923 y Fn(d)1180 2938 y Fm(\026)1227 2923 y Fo(\()p Fn(E)6 b Fo(\))55 b(=)g(lim)1568 2980 y Fm(\017)p Fg(!)p Fs(0)1729 2856 y Fo(log\()p Fn(\026)p Fo(\([)p Fn(E)28 b Fk(\000)23 b Fn(\017;)17 b(E)28 b Fo(+)22 b Fn(\017)p Fo(]\)\))p 1729 2900 912 4 v 2064 2991 a(log\()p Fn(\017)p Fo(\))3773 2923 y(\(1\))-118 3148 y(of)44 b(its)h(pullbac)m(k)f (measure)i Fn(\026)e Fo(on)h Fn(J)54 b Fo(exist)46 b Fn(\026)p Fo(-almost)c(surely)k(and)f(are)g Fn(\026)p Fo(-almost)d(surely)k(equal)f(to)f(the)-118 3268 y(information)31 b(or)k(Hausdor\013)g(dimension)e(dim)1607 3283 y Fs(H)1664 3268 y Fo(\()p Fn(\026)p Fo(\))h(of)h Fn(\026)f Fo([10].)50 b(The)36 b(latter)d(is)i(furthermore)f(equal)h(to)f(the)-118 3388 y(quotien)m(t)42 b(of)f(the)i(dynamical)d(en)m(trop)m(y)j Fk(E)9 b Fo(\()p Fn(\026)p Fo(\))41 b(and)h(the)g(Ly)m(apuno)m(v)h(exp) s(onen)m(t)h(\003\()p Fn(\026)p Fo(\))d(of)g(the)h(dynamical)-118 3509 y(system)j(\()p Fn(J)n(;)17 b(S;)g(\026)p Fo(\))43 b([10].)77 b(A)44 b(m)m(ultifractal)c(prop)s(ert)m(y)45 b(of)e Fn(\026)p Fo(,)k(that)d(is)f(\014ner)h(c)m(haracteristic)g(than) g(just)g(its)-118 3629 y(Hausdor\013)33 b(dimension,)e(is)h(giv)m(en)h (b)m(y)g(its)f(singularit)m(y)f(sp)s(ectrum)1017 3870 y Fn(f)1065 3885 y Fm(\026)1111 3870 y Fo(\()p Fn(\013)q Fo(\))55 b(=)h(dim)1599 3885 y Fm(H)1683 3870 y Fo(\()p Fk(f)p Fn(E)33 b Fk(2)c Fn(J)42 b Fk(j)16 b Fn(d)2162 3885 y Fm(\026)2208 3870 y Fo(\()p Fn(E)6 b Fo(\))28 b(=)g Fn(\013)10 b Fk(g)p Fo(\))49 b(,)-118 4044 y(where)36 b Fn(d)217 4059 y Fm(\026)263 4044 y Fo(\()p Fn(E)6 b Fo(\))31 b(=)g Fn(\013)k Fo(means)g(that)f(the)h(limit)d(in)h(\(1\))i (exists)g(and)g(is)f(equal)g(to)h Fn(\013)g Fo(and,)g(b)m(y)h(con)m(v)m (en)m(tion,)g(the)-118 4165 y(Hausdor\013)c(dimension)f(of)h(an)g(empt) m(y)h(set)g(is)e(equal)h(to)g Fk(\0001)p Fo(.)44 b(F)-8 b(or)31 b(a)h(Gibbs)g(measure)g Fn(\026)g Fo(on)g Fn(J)41 b Fo(in)m(tro)s(duced)-118 4285 y(in)c(Section)g(3.3,)i(it)e(can)h(b)s (e)f(sho)m(wn)i(that)f Fn(f)1528 4300 y Fm(\026)1612 4285 y Fo(is)f(a)h(conca)m(v)m(e)h(analytic)e(function)g(\(see,)j(for)d (example,)i([4]\).)-118 4405 y(Its)33 b(Legendre)h(transform)d Fn(\034)934 4420 y Fm(\026)1014 4405 y Fo(allo)m(ws)g(to)h(de\014ne)i (the)f(generalized)f(dimensions)g Fn(D)2958 4420 y Fm(\026)3005 4405 y Fo(\()p Fn(q)t Fo(\))g(b)m(y)711 4676 y Fn(D)792 4691 y Fm(\026)839 4676 y Fo(\()p Fn(q)t Fo(\))55 b(=)68 b(lim)1148 4737 y Fm(q)1182 4718 y Fi(0)1205 4737 y Fg(!)p Fm(q)1338 4609 y Fn(\034)1380 4624 y Fm(\026)1427 4609 y Fo(\()p Fn(q)1512 4573 y Fg(0)1535 4609 y Fo(\))p 1336 4653 241 4 v 1336 4745 a Fn(q)1383 4716 y Fg(0)1428 4745 y Fk(\000)22 b Fo(1)1619 4676 y(,)227 b Fn(\034)1915 4691 y Fm(\026)1962 4676 y Fo(\()p Fn(q)t Fo(\))55 b(=)73 b(inf)2272 4736 y Fm(\013)p Fg(2)p Fc(R)2425 4676 y Fo(\()p Fn(\013)q(q)25 b Fk(\000)e Fn(f)2742 4691 y Fm(\026)2789 4676 y Fo(\()p Fn(\013)q Fo(\)\))32 b(.)-118 5096 y Fd(2.2)135 b(Jacobi)46 b(matrices)-118 5281 y Fo(Once)41 b(the)f(measure)g Fn(\026)g Fo(on)g Fn(J)49 b Fo(is)39 b(\014xed,)k(w)m(e)e(construct)g (the)g(Hamiltonian)36 b(as)k(the)h(Jacobi)e(matrix)f(of)i Fn(\026)p Fo(.)-118 5401 y(Let)46 b Fn(P)133 5416 y Fm(n)180 5401 y Fo(,)j Fn(n)i Fk(\025)g Fo(0,)e(denote)d(the)h(orthogonal)d(and) i(normalized)e(p)s(olynomials)f(asso)s(ciated)i(to)h Fn(\026)p Fo(.)83 b(They)-118 5521 y(form)42 b(a)i(Hilb)s(ert)e(basis)i Fk(B)50 b Fo(=)d(\()p Fn(P)1153 5536 y Fm(n)1200 5521 y Fo(\))1238 5536 y Fm(n)p Fg(2)p Fc(N)1439 5521 y Fo(in)c Fn(L)1630 5485 y Fs(2)1670 5521 y Fo(\()p Fe(R)p Fn(;)17 b(\026)p Fo(\))43 b(and)g(satisfy)h(a)g(three)g(term)g(recurrence)h (relation)1865 5800 y(3)p eop %%Page: 4 4 4 3 bop -118 28 a Fn(E)6 b(P)23 43 y Fm(n)70 28 y Fo(\()p Fn(E)g Fo(\))27 b(=)h Fn(t)390 43 y Fm(n)p Fs(+1)527 28 y Fn(P)590 43 y Fm(n)p Fs(+1)727 28 y Fo(\()p Fn(E)6 b Fo(\))22 b(+)f Fn(v)1047 43 y Fm(n)1094 28 y Fn(P)1157 43 y Fm(n)1204 28 y Fo(\()p Fn(E)6 b Fo(\))21 b(+)h Fn(t)1512 43 y Fm(n)1559 28 y Fn(P)1622 43 y Fm(n)p Fg(\000)p Fs(1)1759 28 y Fo(\()p Fn(E)6 b Fo(\),)32 b Fn(n)c Fk(\025)g Fo(0,)k(where)i Fn(v)2600 43 y Fm(n)2674 28 y Fk(2)29 b Fe(R)i Fo(and)i Fn(t)3109 43 y Fm(n)3183 28 y Fk(\025)c Fo(0)j(are)g(b)s(ounded)-118 148 y(sequences,)44 b(and)39 b Fn(P)619 163 y Fg(\000)p Fs(1)752 148 y Fo(=)f(0.)63 b(Therefore)40 b(the)g(isomorphism)c(of)j Fn(L)2388 112 y Fs(2)2427 148 y Fo(\()p Fe(R)p Fn(;)17 b(\026)p Fo(\))38 b(on)m(to)h Fn(`)2995 112 y Fs(2)3035 148 y Fo(\()p Fe(N)p Fo(\))f(asso)s(ciated)h(with)-118 268 y(the)e(basis)f Fk(B)j Fo(carries)e(the)f(op)s(erator)g(of)g(m)m (ultiplication)31 b(b)m(y)38 b Fn(E)k Fo(in)35 b Fn(L)2464 232 y Fs(2)2504 268 y Fo(\()p Fe(R)p Fn(;)17 b(\026)p Fo(\))35 b(in)m(to)h(the)g(self-adjoin)m(t)f(\014nite)-118 389 y(di\013erence)e(op)s(erator)f Fn(H)40 b Fo(de\014ned)34 b(on)f Fn(`)1345 353 y Fs(2)1384 389 y Fo(\()p Fe(N)p Fo(\))f(b)m(y:)729 629 y Fn(H)8 b Fk(j)p Fn(n)p Fk(i)55 b Fo(=)g Fn(t)1164 644 y Fm(n)p Fs(+1)1301 629 y Fk(j)p Fn(n)22 b Fo(+)g(1)p Fk(i)g Fo(+)g Fn(v)1762 644 y Fm(n)1809 629 y Fk(j)p Fn(n)p Fk(i)g Fo(+)g Fn(t)2089 644 y Fm(n)2136 629 y Fk(j)p Fn(n)g Fk(\000)h Fo(1)p Fk(i)32 b Fo(,)227 b Fn(n)28 b Fk(\025)h Fo(1)j(,)755 b(\(2\))-118 803 y(and)32 b Fn(H)8 b Fk(j)p Fo(0)p Fk(i)26 b Fo(=)i Fn(t)441 818 y Fs(1)480 803 y Fk(j)p Fo(1)p Fk(i)20 b Fo(+)g Fn(v)759 818 y Fs(0)799 803 y Fk(j)p Fo(0)p Fk(i)p Fo(.)42 b(Then)33 b Fn(\026)e Fo(is)h(the)g(sp)s(ectral)f(measure)h(of)g Fn(H)39 b Fo(asso)s(ciated)31 b(to)h Fk(j)p Fo(0)p Fk(i)p Fo(,)f(also)g(called)f(its)-118 923 y(LDOS.)-118 1259 y Fd(2.3)135 b(Di\013usion)46 b(exp)t(onen)l(ts)-118 1444 y Fo(The)f(propagation)e(of)h(w)m(a)m(v)m(e)j(pac)m(k)m(ets)f (initially)41 b(lo)s(calized)h(on)j(the)g(state)g Fk(j)p Fo(0)p Fk(i)e Fo(is)h(c)m(haracterized)i(b)m(y)f(the)-118 1564 y(gro)m(wth)33 b(exp)s(onen)m(ts)564 1835 y Fn(\014)625 1794 y Fs(+)619 1860 y Fm(\013)739 1835 y Fo(=)55 b(lim)17 b(sup)924 1914 y Fm(T)10 b Fg(!1)1212 1768 y Fo(log\()p Fn(M)1470 1783 y Fm(\013)1520 1768 y Fo(\()p Fn(T)k Fo(\)\))p 1212 1812 493 4 v 1297 1904 a(log\()p Fn(T)1532 1875 y Fm(\013)1581 1904 y Fo(\))1747 1835 y(,)228 b Fn(\014)2063 1794 y Fg(\000)2057 1860 y Fm(\013)2177 1835 y Fo(=)55 b(lim)17 b(inf)2347 1894 y Fm(T)10 b Fg(!1)2621 1768 y Fo(log\()p Fn(M)2879 1783 y Fm(\013)2929 1768 y Fo(\()p Fn(T)k Fo(\)\))p 2621 1812 V 2706 1904 a(log\()p Fn(T)2941 1875 y Fm(\013)2990 1904 y Fo(\))3156 1835 y(,)590 b(\(3\))-118 2059 y(of)32 b(the)h(time-a)m(v)m(eraged)f(momen)m(ts)g(of)g(the)h(p)s (osition)e(op)s(erator)h(on)g Fk(B)767 2324 y Fn(M)861 2339 y Fm(\013)911 2324 y Fo(\()p Fn(T)14 b Fo(\))55 b(=)1251 2241 y Fl(X)1244 2424 y Fm(n)p Fg(\025)p Fs(0)1411 2324 y Fn(n)1469 2283 y Fm(\013)1552 2207 y Fl(Z)1635 2234 y Fm(T)1598 2396 y Fs(0)1716 2257 y Fn(dt)p 1716 2301 86 4 v 1724 2393 a(T)1829 2324 y Fk(jh)p Fn(n)p Fk(j)p Fn(e)2027 2283 y Fg(\000)p Fm({tH)2197 2324 y Fk(j)p Fo(0)p Fk(ij)2341 2283 y Fs(2)2412 2324 y Fo(,)228 b Fn(\013)28 b Fk(6)p Fo(=)f(0)33 b(.)-118 2775 y Fd(2.4)135 b(Results)46 b(and)f(commen)l(ts)-118 2959 y Fo(The)37 b(presen)m(tation)g(of)f(our)g(most)g(general)f(result)i(is)e(p)s(ostp) s(oned)i(to)f(Section)g(3.4)g(in)g(order)g(to)g(a)m(v)m(oid)g(the)-118 3080 y(use)j(of)f(hea)m(vy)i(notations)e(at)g(this)g(p)s(oin)m(t.)61 b(W)-8 b(e)39 b(rather)f(discuss)i(the)f(result)f(for)g(t)m(w)m(o)h(in) m(teresting)g(sp)s(ecial)-118 3200 y(cases.)-118 3426 y Fe(Theorem)e(1)49 b Fb(L)-5 b(et)28 b Fn(H)36 b Fb(b)-5 b(e)28 b(a)g(Julia)h(matrix,)g(namely)f Fn(S)34 b Fb(is)28 b(a)g(p)-5 b(olynomial)27 b(map)h(and)f Fn(\026)h Fo(=)f Fn(\026)3228 3441 y Fs(0)3296 3426 y Fb(is)h(the)g(b)-5 b(alanc)g(e)g(d)-118 3547 y(me)g(asur)g(e)34 b(of)h(maximal)e(entr)-5 b(opy.)45 b(Set)35 b Fn(\013)1386 3562 y Fm(c)1448 3547 y Fo(=)28 b(1)22 b Fk(\000)h Fo(\()p Fn(\034)1803 3562 y Fm(\026)1845 3571 y Fh(0)1884 3547 y Fo(\))1922 3511 y Fg(\000)p Fs(1)2016 3547 y Fo(\()p Fk(\000)p Fo(2\))p Fb(.)45 b(Then)1040 3881 y Fn(\014)1101 3840 y Fs(+)1095 3906 y Fm(\013)1215 3881 y Fk(\024)1347 3682 y Fl(8)1347 3756 y(>)1347 3781 y(<)1347 3931 y(>)1347 3956 y(:)1463 3760 y Fn(D)1544 3775 y Fm(\026)1586 3784 y Fh(0)1625 3760 y Fo(\(1)22 b Fk(\000)g Fn(\013)q Fo(\))35 b Fb(,)118 b Fo(0)27 b Fn(<)h(\013)g Fk(\024)g Fn(\013)2554 3775 y Fm(c)2624 3760 y Fb(,)1519 4000 y Fo(1)22 b Fk(\000)1700 3961 y Fm(\013)1745 3969 y Fj(c)1777 3961 y Fg(\000)p Fs(2)p 1700 3977 168 4 v 1761 4035 a Fm(\013)1912 4000 y Fb(,)260 b Fn(\013)28 b Fk(\025)g Fn(\013)2459 4015 y Fm(c)2529 4000 y Fb(.)3773 3881 y Fo(\(4\))-118 4227 y Fe(Remark)37 b(1)49 b Fo(Theorem)34 b(1)g(com)m(bined)g(with)f (Guarneri's)h(inequalit)m(y)f(\(notably)g Fn(\014)2958 4191 y Fg(\000)2952 4252 y Fm(\013)3047 4227 y Fk(\025)e Fo(dim)3317 4242 y Fs(H)3374 4227 y Fo(\()p Fn(\026)p Fo(\))f(=)g Fn(D)3726 4242 y Fm(\026)3773 4227 y Fo(\(1\))-118 4347 y(for)45 b Fn(\013)51 b(>)g Fo(0\))45 b(implies)f(that)h(transp)s (ort)h(is)f(anomalous)g(in)g(Julia)f(matrices.)83 b(In)46 b(fact,)j Fn(D)3339 4362 y Fm(\026)3386 4347 y Fo(\()p Fn(q)t Fo(\))h Fk(\024)g Fo(1)c(for)-118 4468 y(all)39 b Fn(q)46 b Fk(2)d Fe(R)d Fo(and)i(dim)709 4483 y Fs(H)766 4468 y Fo(\()p Fn(\026)863 4483 y Fs(0)902 4468 y Fo(\))g Fk(\025)h Fo(log\()p Fn(L)p Fo(\))p Fn(=)p Fo(\003)1487 4483 y Fs(max)1663 4468 y Fo(where)g(\003)2022 4483 y Fs(max)2199 4468 y Fo(=)f(log\(max)2663 4483 y Fm(E)t Fg(2)p Fm(J)2831 4468 y Fk(j)p Fn(S)2925 4432 y Fg(0)2948 4468 y Fo(\()p Fn(E)6 b Fo(\))p Fk(j)p Fo(\).)68 b(Numerical)39 b(and)-118 4588 y(theoretical)30 b(analysis)g(using)h(GM-mac)m(hinery)g ([13])g(indicates)g(that)g(the)g(equalit)m(y)g Fn(\014)3041 4603 y Fm(\013)3118 4588 y Fo(=)d Fn(D)3303 4603 y Fm(\026)3345 4612 y Fh(0)3384 4588 y Fo(\(1)19 b Fk(\000)h Fn(\013)q Fo(\))31 b(ma)m(y)-118 4709 y(hold)k(for)g(all)f Fn(\013)h(>)e Fo(0.)53 b(Whether)37 b(the)g(equalit)m(y)e(actually)g(holds)h(for)f (all)f Fn(\013)j Fo(or)e Fn(\014)2882 4724 y Fm(\013)2965 4709 y Fk(!)e Fo(1)j(as)g Fn(\013)e Fk(!)f(1)j Fo(\(as)g(is)-118 4829 y(the)d(case)h(in)d(our)i(upp)s(er)g(b)s(ound\))g(is)f(an)h(in)m (teresting)f(question.)44 b(Tw)m(o)34 b(facts,)f(b)s(oth)f(v)m (eri\014ed)i(for)e(real)f(Julia)-118 4949 y(sets)37 b(generated)f(b)m (y)h Fn(S)6 b Fo(\()p Fn(E)g Fo(\))32 b(=)h Fn(E)1101 4913 y Fs(2)1165 4949 y Fk(\000)24 b Fn(\025)36 b Fo(with)f Fn(\025)e(>)g Fo(2,)j(indicate)e(that)i(our)f(upp)s(er)i(b)s(ound)e(is) h(probably)f(not)-118 5070 y(tigh)m(t)f(for)g Fn(\013)e(>)f(\013)534 5085 y Fm(c)569 5070 y Fo(.)50 b(First)34 b(of)g(all,)g Fn(\013)1227 5085 y Fm(c)1296 5070 y Fo(con)m(v)m(erges)j(to)e(2)f (from)g(ab)s(o)m(v)m(e)h(as)g Fn(\025)c Fk(!)h Fo(2)i(\(while)g(n)m (umerical)f(results)-118 5190 y(giv)m(e)h(the)h(equalit)m(y)f Fn(\014)682 5205 y Fm(\013)762 5190 y Fo(=)c Fn(D)949 5205 y Fm(\026)991 5214 y Fh(0)1030 5190 y Fo(\(1)23 b Fk(\000)h Fn(\013)q Fo(\))34 b(for)g(m)m(uc)m(h)g(larger)f(v)-5 b(alue)34 b(of)g Fn(\013)h Fo(if)e Fn(\025)h Fo(is)g(close)g(to)g(2)g ([13)o(]\);)i(second)f(of)-118 5311 y(all,)30 b(the)i(curv)m(e)i (de\014ned)f(b)m(y)g(the)f(upp)s(er)g(b)s(ound)h(\(4\))e(has)h(a)g (discon)m(tin)m(uous)g(deriv)-5 b(ativ)m(e)32 b(at)f Fn(\013)3331 5326 y Fm(c)3366 5311 y Fo(.)43 b(W)-8 b(e)32 b(discuss)-118 5431 y(the)f(problems)f(arising)f(for)h(large)g Fn(\013)h Fo(with)g(more)f(tec)m(hnical)g(details)g(in)g(Remark)g(6)h (of)f(Section)h(3.1.)42 b(Let)31 b(us)-118 5551 y(note)g(that,)g(for)g (quadratic)g(Julia)e(sets,)k Fn(\013)1434 5566 y Fm(c)1496 5551 y Fk(\030)28 b Fo(log\()p Fn(\025)p Fo(\))p Fn(=)p Fo(2)j(for)f(large)g Fn(\025)p Fo(.)43 b(Finally)-8 b(,)29 b(w)m(e)j(remark)e(that)h(w)m(e)h(kno)m(w)1865 5800 y(4)p eop %%Page: 5 5 5 4 bop -118 28 a Fo(of)28 b(no)h(theoretical)f(w)m(ork)i(ab)s(out)e (negativ)m(e)i(momen)m(ts)e(\()p Fn(\013)g(<)g Fo(0\).)42 b(Ho)m(w)m(ev)m(er,)32 b(one)d(can)g(sho)m(w)i Fn(\014)3352 43 y Fm(\013)3429 28 y Fk(\030)d Fn(D)3615 43 y Fm(\026)3662 28 y Fo(\(2\))p Fn(=\013)-118 148 y Fo(for)k(large)f(negativ)m(e)i Fn(\013)g Fo(using)g(results)g(from)e([12,)h(2,)h(17)o(].)28 358 y(Julia)k(matrices)g(exhibit)h(an)g(exact)h(renormalization)c(prop) s(ert)m(y)k(in)e(ph)m(ysical)h(space)i([3)o(],)g(so)f(that)f(all)-118 479 y(in)m(termittency)28 b(is)f(due)i(to)f(the)h(thermo)s(dynamics)e (of)h(the)h(supp)s(ort)f(of)g(the)h(sp)s(ectral)f(measure.)42 b(As)29 b(already)-118 599 y(p)s(oin)m(ted)47 b(out)g(in)g(the)h(In)m (tro)s(duction,)j(a)c(second)i(reason)f(for)f(in)m(termittency)g(is)g (due)h(to)f(the)h(in)m(terpla)m(y)-118 719 y(b)s(et)m(w)m(een)40 b(Hamiltonian)34 b(and)j(p)s(osition)f(op)s(erator.)58 b(An)38 b(extreme)h(example)e(of)g(this)g(is)h(giv)m(en)f(b)m(y)i(sp)s (ectra)-118 840 y(supp)s(orted)d(on)f(uniform)e(Can)m(tor)i(sets)h(for) e(whic)m(h)i(the)f(thermo)s(dynamics)f(is)h(\015at)f(and)h(cannot)g(b)s (e)h(at)e(the)-118 960 y(origin)40 b(of)i(in)m(termittency)-8 b(,)44 b(but)e(for)g(whic)m(h)g(the)h(quan)m(tum)f(motion)f(is)g(nev)m (ertheless)k(in)m(termitten)m(t)c([13)o(].)-118 1080 y(Our)f(pro)s(of)f(do)s(es)i(not)f(allo)m(w)e(to)i(exhibit)g(and)g (analyse)g(these)h(\014ne)g(prop)s(erties,)h(ho)m(w)m(ev)m(er,)j(w)m(e) c(obtain)e(a)-118 1201 y(sub-ballistic)30 b(b)s(ound)j(on)f(dynamics.) -118 1411 y Fe(Theorem)37 b(2)49 b Fb(L)-5 b(et)34 b Fn(J)42 b Fb(b)-5 b(e)34 b(a)f(uniform)g(Cantor)g(set,)h(that)g(is,)f Fn(S)40 b Fb(has)33 b(two)g(line)-5 b(ar)33 b(br)-5 b(anches)33 b Fn(S)3327 1426 y Fs(1)3400 1411 y Fb(and)g Fn(S)3648 1426 y Fs(2)3721 1411 y Fb(with)-118 1531 y(slop)-5 b(e)34 b(e)-5 b(qual)35 b(to)g Fk(\006)p Fn(e)607 1495 y Fs(\003)695 1531 y Fb(and)g(let)g Fn(\026)f Fb(b)-5 b(e)35 b(an)f(arbitr)-5 b(ary)35 b(er)-5 b(go)g(dic)34 b(me)-5 b(asur)g(e)35 b(on)f Fn(J)9 b Fb(.)45 b(We)35 b(set)295 1826 y Fo(\007)55 b(=)558 1680 y Fl( )623 1826 y Fo(arcsinh)944 1680 y Fl( )1020 1759 y Fn(e)1065 1723 y Fs(\003)1140 1759 y Fk(\000)23 b Fo(2)p 1020 1803 269 4 v 1130 1894 a(4)1298 1680 y Fl(!!)1430 1703 y Fg(\000)p Fs(1)1541 1705 y Fl(\022)1602 1709 y(Z)1702 1826 y Fn(d)p Fk(N)15 b Fo(\()p Fn(E)6 b Fo(\))32 b(log\()p Fn(e)2245 1785 y Fs(\003)2320 1826 y Fk(\000)23 b Fn(E)6 b Fo(\))22 b Fk(\000)h Fo(log\(cap\()p Fn(J)9 b Fo(\)\))3145 1705 y Fl(\023)3228 1826 y Fo(+)22 b(1)35 b Fb(.)333 b Fo(\(5\))-118 2050 y Fb(Then)896 2406 y Fn(\014)957 2364 y Fs(+)951 2430 y Fm(\013)1071 2406 y Fk(\024)1204 2181 y Fl(8)1204 2256 y(>)1204 2281 y(>)1204 2306 y(<)1204 2455 y(>)1204 2480 y(>)1204 2505 y(:)1536 2229 y Fs(log)q(\(\007\))p 1536 2252 201 4 v 1612 2310 a(\003)1782 2275 y Fb(,)324 b Fo(0)27 b Fn(<)h(\013)g Fk(\024)2580 2236 y Fs(2\003)p 2521 2252 V 2521 2310 a(log)q(\(\007\))2767 2275 y Fb(,)1319 2531 y Fo(1)22 b(+)1503 2492 y Fs(2)p 1498 2508 46 4 v 1498 2566 a Fm(\013)1576 2531 y Fk(\000)1772 2492 y Fs(2\003)p 1685 2508 259 4 v 1685 2566 a Fm(\013)12 b Fs(log)q(\(\007\))1988 2531 y Fb(,)203 b Fn(\013)28 b Fk(\025)2485 2492 y Fs(2\003)p 2426 2508 201 4 v 2426 2566 a(log)r(\(\007\))2672 2531 y Fb(.)3773 2406 y Fo(\(6\))-118 2755 y Fe(Remark)37 b(2)49 b Fo(It)34 b(w)m(as)h(sho)m(wn)g(in)f(ref.)48 b([10])34 b(that)g(\007)g(as)g(giv)m(en)h(in)e(\(5\))h(div)m(erges)h (linearly)d(in)h(\003)h(in)g(the)g(limit)-118 2876 y(\003)43 b Fk(!)g(1)p Fo(.)70 b(Therefore)43 b(the)f(b)s(ound)g(\(6\))g(is)f (strictly)g(b)s(etter)h(than)g(the)g(ballistic)d(b)s(ound)j(for)f (su\016cien)m(tly)-118 2996 y(large)34 b(\003.)52 b(As)36 b(long)e(as)h(the)h(Hausdor\013)g(dimension)e(of)h Fn(\026)g Fo(is)g(bigger)f(than)h(0)h(\(for)e(example,)i(for)f(Bernoulli)-118 3116 y(measures\),)49 b(Guarneri's)c(inequalit)m(y)g([7])g(com)m(bined) g(with)h(Theorem)f(2)h(sho)m(ws)h(that)e(the)h(transp)s(ort)f(is)-118 3237 y(anomalous)33 b(in)i(these)i(mo)s(dels.)50 b(Note)36 b(that)f(the)g(b)s(ound)h(\(6\))f(do)s(es)g(not)h(dep)s(end)g(on)f(the) h(measure)g Fn(\026)p Fo(,)f(but)-118 3357 y(only)d(on)g(the)h(DOS.)g (This)f(is,)h(ho)m(w)m(ev)m(er,)i(an)d(artefact)g(of)h(our)f(pro)s(of.) -118 3825 y Fp(3)161 b(Pro)t(of)54 b(of)g(upp)t(er)f(b)t(ounds)-118 4073 y Fd(3.1)135 b(Resolving)47 b(the)e(sp)t(ectrum)f(at)i(di\013eren) l(t)g(scales)-118 4257 y Fo(The)29 b(main)d(result)h(of)h(this)f (section,)i(notably)e(that)h(Prop)s(osition)e(1)i(holds)f(giv)m(en)h (Hyp)s(othesis)h(I)e(and)h(I)s(I,)g(can)-118 4378 y(b)s(e)39 b(directly)e(transp)s(osed)j(to)e(other)h(Hamiltonians)c(and)k(other)f (exp)s(onen)m(tially)g(lo)s(calized)e(initial)f(states.)-118 4498 y(F)-8 b(or)40 b(sak)m(e)j(of)e(simplicit)m(y)-8 b(,)40 b(w)m(e)j(form)m(ulate)c(nev)m(ertheless)44 b(ev)m(erything)e (only)f(for)g(the)h(mo)s(del)d(describ)s(ed)j(in)-118 4619 y(Sections)33 b(2.1)f(and)h(2.2.)28 4786 y(The)45 b(follo)m(wing)c(h)m(yp)s(othesis)k(allo)m(w)d(to)i(separate)g(t)m(w)m (o)h(reasons)g(for)e(in)m(termittency:)65 b(one)44 b(link)m(ed)g(to) -118 4907 y(the)39 b(thermo)s(dynamics)g(of)f(the)i(LDOS,)e(the)i (other)f(to)g(the)h(asymptotic)e(prop)s(erties)h(of)g(the)g (generalized)-118 5027 y(eigenfunctions.)-118 5195 y Fe(Hyp)s(othesis)e(I)32 b Ff(Let)h Fo(\003)781 5158 y Fg(\003)820 5195 y Fo(\()p Fn(:;)17 b(\026)p Fo(\))27 b(:)h Fe(R)f Fk(!)h Fo([0)p Fn(;)17 b Fk(1)p Fo(])31 b Ff(b)s(e)i(a)f(con)m(v)m(ex)j(function)d(suc)m(h)i(that,)f(for)f(all) e Fn(\025)e Fk(\024)g Fo(\003\()p Fn(\026)p Fo(\))p Ff(,)801 5485 y Fn(\026)877 5339 y Fl( )o(\()1009 5485 y Fn(E)34 b Fk(2)28 b Fn(J)1305 5335 y Fl(\014)1305 5385 y(\014)1305 5435 y(\014)1305 5485 y(\014)1305 5535 y(\014)1349 5485 y Fn(\025)g Fk(\025)g(\000)1626 5405 y Fo(log)q(\()p Fk(j)p Fn(I)1870 5369 y Fm(N)1862 5432 y(\033)r Fs(\()p Fm(E)t Fs(\))2018 5405 y Fk(j)p Fo(\))p 1626 5462 458 4 v 1811 5553 a Fn(N)2104 5339 y Fl(\)!)2292 5485 y Fk(\024)56 b Fn(ae)2521 5444 y Fg(\000)p Fm(N)7 b Fs(\003)2688 5420 y Fi(\003)2724 5444 y Fs(\()p Fm(\025;\026)p Fs(\))2919 5485 y Ff(,)827 b Fo(\(7\))1865 5800 y(5)p eop %%Page: 6 6 6 5 bop -118 28 a Ff(for)47 b(some)h(constan)m(t)g Fn(a)54 b Fk(\025)g Fo(0)p Ff(.)89 b(W)-8 b(e)48 b(further)g(supp)s(ose)h(that) f(there)g(is)g Fo(0)53 b Fn(<)h Fo(\003)2928 43 y Fs(min)3103 28 y Fk(\024)g Fo(\003\()p Fn(\026)p Fo(\))47 b Ff(suc)m(h)i(that)-118 148 y Fo(\003)-50 112 y Fg(\003)-11 148 y Fo(\()p Fn(:;)17 b(\026)p Fo(\))27 b(:)h([\003)372 163 y Fs(min)493 148 y Fn(;)17 b Fo(\003\()p Fn(\026)p Fo(\)])27 b Fk(!)g Fo([0)p Fn(;)17 b Fk(1)p Fo(\))32 b Ff(is)f(a)h Fn(C)1466 112 y Fs(1)1537 148 y Ff(and)g(strictly)f(con)m(v)m(ex)j(bijection)d (and)h(that)g Fo(\003)3244 112 y Fg(\003)3283 148 y Fo(\()p Fn(\025;)17 b(\026)p Fo(\))27 b(=)h Fk(1)j Ff(for)-118 268 y Fn(\025)c(<)h Fo(\003)138 283 y Fs(min)259 268 y Ff(.)-118 436 y Fe(Hyp)s(othesis)37 b(I)s(I)32 b Ff(F)-8 b(or)32 b Fn(\021)f(>)d Fo(0)p Ff(,)k(let)g Fo(\007\()p Fn(\021)t Fo(\))27 b Fn(>)h Fo(0)k Ff(and)h Fn(c)p Fo(\()p Fn(\021)t Fo(\))27 b Fn(<)h Fk(1)k Ff(b)s(e)g(suc)m(h)i(that)f(for)f (all)e Fn(n)j Ff(and)g Fn(N)43 b Ff(v)m(erifying)1585 677 y Fn(n)56 b Fk(\025)g Fo(\007\()p Fn(\021)t Fo(\))2036 636 y Fm(N)2135 677 y Ff(,)-118 847 y(the)33 b(follo)m(wing)d(b)s(ound) j(holds:)1297 1004 y Fl(X)1257 1200 y Fm(\033)r Fg(2)p Fs(\006)1397 1177 y Fj(N)1397 1224 y(L)1500 1087 y Fk(jh)p Fn(\037)1628 1106 y Fm(I)1664 1087 y Fj(N)1659 1123 y(\033)1725 1087 y Fk(j)p Fn(n)p Fk(ij)1878 1046 y Fs(2)1972 1087 y Fk(\024)56 b Fn(c)p Fo(\()p Fn(\021)t Fo(\))p Fn(n)2333 1046 y Fg(\000)p Fm(\021)2462 1087 y Ff(,)1284 b Fo(\(8\))-118 1365 y Ff(where)34 b Fk(j)p Fn(\037)253 1383 y Fm(I)289 1364 y Fj(N)284 1400 y(\033)350 1365 y Fk(i)28 b Fo(=)f Fn(\037)581 1383 y Fm(I)617 1364 y Fj(N)612 1400 y(\033)679 1365 y Fo(\()p Fn(H)8 b Fo(\))p Fk(j)p Fo(0)p Fk(i)p Ff(,)31 b Fn(\037)1079 1380 y Fm(I)1152 1365 y Ff(denoting)h(the)h(c)m (haracteristic)f(function)g(of)g(the)h(in)m(terv)-5 b(al)31 b Fn(I)8 b Ff(.)-118 1628 y Fe(Remark)37 b(3)49 b Fo(In)40 b(the)g(next)h(section,)h(the)e(function)f(\003)1937 1592 y Fg(\003)1976 1628 y Fo(\()p Fn(:;)17 b(\026)p Fo(\))40 b(will)d(b)s(e)j(determined)g(to)f(b)s(e)h(the)h(Legendre)-118 1749 y(transform)d(of)g(the)h(generalized)f(Ly)m(apuno)m(v)i(exp)s (onen)m(ts.)64 b(In)39 b(Section)f(3.3,)i(w)m(e)g(furthermore)e(sho)m (w)i(that)-118 1869 y(\003)-50 1833 y Fg(\003)-11 1869 y Fo(\()p Fn(:;)17 b(\026)p Fo(\))29 b(is)g(giv)m(en)h(in)f(terms)h(of) f(the)h(scaling)e(function)i(of)f(the)h(maximal)c(en)m(trop)m(y)31 b(measure)f(whenev)m(er)j Fn(\026)c Fo(is)-118 1989 y(a)35 b(Gibbs)g(measure.)53 b(A)m(t)36 b(least)g(in)f(this)g(situation,)g (all)e(the)j(ab)s(o)m(v)m(e)h(h)m(yp)s(othesis)g(on)e(\003)3082 1953 y Fg(\003)3121 1989 y Fo(\()p Fn(:;)17 b(\026)p Fo(\))35 b(are)h(satis\014ed.)-118 2110 y(A)30 b(closer)g(insp)s (ection)f(of)h(the)g(pro)s(of)f(b)s(elo)m(w)h(sho)m(ws)i(that)e(w)m (eak)m(er)h(results)g(can)f(b)s(e)g(obtained)g(under)h(w)m(eak)m(er) -118 2230 y(h)m(yp)s(othesis)h(\(no)g(di\013eren)m(tiabilit)m(y)-8 b(,)29 b(for)i(example\).)42 b(Let)32 b(us)g(note)g(righ)m(t)e(a)m(w)m (a)m(y)j(that)e(\003)3109 2194 y Fg(\003)3149 2230 y Fo(\(\003\()p Fn(\026)p Fo(\))p Fn(;)17 b(\026)p Fo(\))26 b(=)h(0)k(and)-118 2351 y(that)42 b(b)s(oth)f(\003)410 2314 y Fg(\003)449 2351 y Fo(\()p Fn(:;)17 b(\026)p Fo(\))42 b(and)g(\(\003)1002 2314 y Fg(\003)1041 2351 y Fo(\))1079 2314 y Fg(0)1102 2351 y Fo(\()p Fn(:;)17 b(\026)p Fo(\))41 b(are)h(bijections)f(from)g(\(\003)2313 2366 y Fs(min)2434 2351 y Fn(;)17 b Fo(\003\()p Fn(\026)p Fo(\)])41 b(to)h(their)f(resp)s (ectiv)m(e)j(images;)-118 2471 y(further)33 b(\003)278 2435 y Fg(\003)317 2471 y Fo(\()p Fn(:;)17 b(\026)p Fo(\))32 b(is)g(discon)m(tin)m(uous)h(at)f(\003)1448 2486 y Fs(min)1570 2471 y Fo(.)-118 2687 y Fe(Remark)37 b(4)49 b Fo(Hyp)s(othesis)31 b(I)s(I)g(is)g(in)f(a)h(more)f(general)g(form)g(than)h(w)m(e)h(can)f (actually)f(pro)m(v)m(e)i(it.)42 b(The)32 b(results)-118 2807 y(from)g(Section)g(3.2)h(of)f(ref.)45 b([10)o(])33 b(sho)m(w)h(that,)f(for)g(the)g(c)m(hoice)g(\007)g(giv)m(en)g(as)g(in)f (Section)h(3.4,)g(the)g(b)s(ound)g(\(8\))-118 2928 y(holds)40 b(for)f(all)f Fn(\021)44 b(>)d Fo(0.)65 b(Th)m(us)42 b(\007)e(is)f(indep)s(enden)m(t)j(of)d Fn(\021)44 b Fo(in)39 b(this)h(situation.)64 b(T)-8 b(o)41 b(obtain)e(a)g(smaller,)h(but)-118 3048 y Fn(\021)t Fo(-dep)s(enden)m(t)33 b(\007)g(is)e(a)h(tough)h(task) g(for)e(whic)m(h)i(only)f(n)m(umerical)f(results)i(exist)f([13].)44 b(As)32 b(w)m(e)i(cannot)e(pro)m(v)m(e)-118 3169 y(suc)m(h)37 b(a)f(h)m(yp)s(othesis)h(for)e(the)h(momen)m(t,)g(w)m(e)h(restrict)f (ourselv)m(es)h(to)e(the)i(case)f(of)g(an)f Fn(\021)t Fo(-indep)s(enden)m(t)i(\007)e(in)-118 3289 y(Prop)s(osition)f(1)i(b)s (elo)m(w,)g(b)s(ecause)h(it)e(simpli\014es)f(considerably)i(the)g(pro)s (of.)53 b(Ho)m(w)m(ev)m(er,)39 b(w)m(e)e(cannot)f(obtain)-118 3409 y(an)m(y)29 b(result)f(on)g(in)m(termittency)g(due)g(to)g(Hyp)s (othesis)h(I)s(I)g(in)e(this)h(w)m(a)m(y)h(\()p Ff(cf.)42 b Fo(Theorem)29 b(2)f(on)g(uniform)e(Can)m(tor)-118 3530 y(sets\).)45 b(F)-8 b(or)31 b(Julia)g(matrices,)h(one)h(can)g(c)m(ho)s (ose)g(\007\()p Fn(\021)t Fo(\))27 b(=)h Fn(L)33 b Fo(for)f(all)e Fn(\021)i(>)27 b Fo(0)32 b([10].)28 3746 y(The)f(follo)m(wing)26 b(prop)s(osition)i(is)h(obtained)g(b)m(y)h(com)m(bining)e(the)h(tec)m (hnique)i(\\resolving)e(the)g(sp)s(ectrum")-118 3866 y([10,)h(Prop)s(osition)f(1])h(and)h(the)g(argumen)m(t)f(in)f(Section)i (3.4)f(of)g(the)g(PhD)h(thesis)g(of)f(one)g(of)g(the)h(authors)g([1)o (].)-118 4083 y Fe(Prop)s(osition)k(1)49 b Fb(We)34 b(supp)-5 b(ose)33 b Fo(Hyp)s(othesis)g(I)h Fb(and)f Fo(I)s(I)h Fb(to)g(b)-5 b(e)33 b(veri\014e)-5 b(d)33 b(for)h(a)f(given)g Fo(\007)h Fb(indep)-5 b(endent)33 b(of)g Fn(\021)t Fb(.)-118 4203 y(L)-5 b(et)29 b Fo(\003)112 4218 y Fm(R)197 4203 y Fk(2)f Fo(\(\003)397 4218 y Fs(min)519 4203 y Fn(;)17 b Fo(\003\()p Fn(\026)p Fo(\)])28 b Fb(b)-5 b(e)29 b(the)g(solution)f (of)h Fo(\(\003)1668 4167 y Fg(\003)1707 4203 y Fo(\))1745 4167 y Fg(0)1768 4203 y Fo(\(\003)1874 4218 y Fm(R)1932 4203 y Fn(;)17 b(\026)p Fo(\))27 b(=)g Fk(\000)p Fo(2)j Fb(if)e(it)i(exists)e(and)h Fo(\003)3051 4218 y Fm(R)3136 4203 y Fo(=)e(\003\()p Fn(\026)p Fo(\))h Fb(otherwise.)-118 4323 y(L)-5 b(et)35 b(us)g(further)g(set)1336 4594 y Fn(\013)1398 4609 y Fm(c)1488 4594 y Fo(=)1629 4527 y(\003)1697 4491 y Fg(\003)1736 4527 y Fo(\(\003)1842 4542 y Fm(R)1900 4527 y Fn(;)17 b(\026)p Fo(\))k(+)h(2\003)2277 4542 y Fm(R)p 1629 4571 706 4 v 1834 4662 a Fo(log)17 b(\(\007\))2379 4594 y Fb(,)1364 b Fo(\(9\))-118 4820 y Fb(and)34 b(let)h Fn(z)253 4835 y Fm(\013)331 4820 y Fk(2)28 b Fo([\003)520 4835 y Fs(min)641 4820 y Fn(;)17 b Fo(\003\()p Fn(\026)p Fo(\)])34 b Fb(b)-5 b(e)35 b(the)g(solution)f(of)1022 5061 y Fn(\013)17 b Fo(log)g(\(\007\))55 b(=)g(\003)1650 5019 y Fg(\003)1689 5061 y Fo(\()p Fn(z)1772 5076 y Fm(\013)1822 5061 y Fn(;)17 b(\026)p Fo(\))k Fk(\000)i Fn(z)2129 5076 y Fm(\013)2179 5061 y Fo(\(\003)2285 5019 y Fg(\003)2324 5061 y Fo(\))2362 5019 y Fg(0)2385 5061 y Fo(\()p Fn(z)2468 5076 y Fm(\013)2518 5061 y Fn(;)17 b(\026)p Fo(\))34 b Fb(,)1001 b Fo(\(10\))-118 5230 y Fb(if)35 b(it)g(exists,)f(and)g Fn(z)604 5245 y Fm(\013)682 5230 y Fo(=)27 b(\003\()p Fn(\026)p Fo(\))35 b Fb(otherwise.)43 b(Then,)34 b(for)h Fo(0)28 b Fn(<)f(\013)h Fk(\024)h Fn(\013)2372 5245 y Fm(c)2406 5230 y Fb(,)1091 5501 y Fn(\014)1152 5460 y Fs(+)1146 5526 y Fm(\013)1266 5501 y Fk(\024)56 b Fn(\024)1455 5516 y Fm(\013)1560 5501 y Fk(\021)1703 5434 y Fn(\013)17 b Fo(log)g(\(\007\))22 b Fk(\000)g Fo(\003)2266 5398 y Fg(\003)2306 5434 y Fo(\()p Fn(z)2389 5449 y Fm(\013)2438 5434 y Fn(;)17 b(\026)p Fo(\))p 1703 5478 876 4 v 2062 5570 a Fn(\013)q(z)2170 5585 y Fm(\013)2623 5501 y Fb(,)1071 b Fo(\(11\))1865 5800 y(6)p eop %%Page: 7 7 7 6 bop -118 22 a Fb(while,)34 b(for)h Fn(\013)28 b Fk(\025)g Fn(\013)574 37 y Fm(c)609 22 y Fb(,)1305 282 y Fn(\014)1366 241 y Fs(+)1360 307 y Fm(\013)1480 282 y Fk(\024)56 b Fn(\024)1669 297 y Fm(\013)1774 282 y Fk(\021)g Fo(1)22 b Fk(\000)2087 215 y Fn(\013)2149 230 y Fm(c)2206 215 y Fk(\000)h Fo(2)p 2087 259 268 4 v 2190 351 a Fn(\013)2400 282 y Fb(.)1294 b Fo(\(12\))-118 541 y Fe(Pro)s(of.)42 b Fo(Let)28 b(0)g Fk(\024)g Fn(\024)g(<)f Fo(1,)i Fn(\032)f(>)g Fo(0)g(and)g Fn(\017)h(>)e Fo(0)h(b)s(e)h(suc)m(h)h(that)e(\(1)14 b Fk(\000)g Fn(\024)p Fo(\))p Fn(=\017)27 b Fk(2)h Fe(N)p Fo(.)42 b(W)-8 b(e)29 b(in)m(tro)s(duce)f(the)h(monoton)-118 661 y(sequence)p 290 608 59 4 v 39 w Fn(n)348 676 y Fm(k)391 661 y Fo(\()p Fn(T)14 b Fo(\))34 b(=)g Fn(\032T)803 625 y Fm(\024)p Fs(+)p Fm(k)r(\017)970 661 y Fo(,)k Fn(k)f Fk(\025)e Fo(0,)j(as)e(w)m(ell)g(as)h(the)g(presence)i(probabilities)33 b(in)j(the)h(rings)f(limited)e(b)m(y)-118 782 y(the)f(radii)p 278 729 V 31 w Fn(n)336 797 y Fm(k)r Fg(\000)p Fs(1)469 782 y Fo(\()p Fn(T)14 b Fo(\))32 b(and)p 837 729 V 32 w Fn(n)896 797 y Fm(k)938 782 y Fo(\()p Fn(T)14 b Fo(\):)889 1047 y Fn(B)963 1062 y Fm(k)1006 1047 y Fo(\()p Fn(T)g Fo(\))55 b(=)1581 964 y Fl(X)p 1339 1114 43 4 v 1339 1152 a Fm(n)1382 1164 y Fj(k)q Fi(\000)p Fh(1)1498 1152 y Fs(\()p Fm(T)10 b Fs(\))p Fm()32 b Fo(0)j(whenev)m(er)i Fn(\032)f Fo(is)f(su\016cien)m (tly)h(large.)50 b(More)36 b(precisely)-8 b(,)36 b(let)e Fn(X)3518 1973 y Fm(\015)3563 1958 y Fo(,)i Fn(\015)h(>)32 b Fo(0,)-118 2078 y(b)s(e)j(the)g(Banac)m(h)g(space)g(of)f Fn(`)953 2042 y Fs(2)993 2078 y Fo(\()p Fe(N)p Fo(\)-v)m(ectors)h Fn( )j Fo(suc)m(h)e(that)e Fk(k)p Fn( )t Fk(k)2225 2093 y Fm(\015)2300 2078 y Fo(=)d(sup)2554 2102 y Fm(n)p Fg(\025)p Fs(0)2708 2078 y Fk(jh)p Fn( )t Fk(j)p Fn(n)p Fk(ij)17 b Fo(exp)o(\()p Fn(\015)5 b(n)p Fo(\))31 b Fn(<)g Fk(1)p Fo(.)49 b(As)35 b Fn(H)-118 2199 y Fo(is)d(a)g(b)s(ounded)i(op)s (erator)e(on)g Fn(X)1069 2214 y Fm(\015)1113 2199 y Fo(,)h(w)m(e)h(ha)m (v)m(e)41 2455 y Fl(X)9 2639 y Fm(n)p Fg(\025)p Fm(\032T)210 2538 y Fn(n)268 2497 y Fm(\013)334 2421 y Fl(Z)417 2448 y Fm(T)380 2610 y Fs(0)499 2471 y Fn(dt)p 499 2515 V 507 2607 a(T)611 2538 y Fk(jh)p Fn(n)p Fk(j)p Fn(e)809 2497 y Fg(\000)p Fm({H)5 b(t)980 2538 y Fk(j)p Fo(0)p Fk(ij)1124 2497 y Fs(2)1217 2538 y Fk(\024)1383 2455 y Fl(X)1350 2639 y Fm(n)p Fg(\025)p Fm(\032T)1551 2538 y Fn(n)1609 2497 y Fm(\013)1676 2421 y Fl(Z)1759 2448 y Fm(T)1722 2610 y Fs(0)1840 2471 y Fn(dt)p 1840 2515 V 1848 2607 a(T)1953 2538 y(e)1998 2497 y Fg(\000)p Fs(2)p Fm(n\015)t Fs(+2)p Fm(t)p Fg(k)p Fm(H)g Fg(k)2419 2505 y Fj(\015)2520 2538 y Fk(\024)2696 2471 y Fo(\()p Fn(\032\015)g(T)14 b Fo(\))2949 2435 y Fs([)p Fm(\013)p Fs(])p 2663 2515 409 4 v 2663 2607 a Fo(4)p Fn(\015)5 b(T)14 b Fk(k)p Fn(H)8 b Fk(k)3028 2622 y Fm(\015)3097 2538 y Fn(e)3142 2497 y Fg(\000)p Fs(2)p Fm(\032\015)t(T)i Fs(+2)p Fm(T)g Fg(k)p Fm(H)5 b Fg(k)3633 2505 y Fj(\015)3711 2538 y Fo(,)-118 2801 y(where)30 b([)p Fn(\013)q Fo(])f(is)g(the)g(smallest)f (in)m(teger)h(larger)f(than)h Fn(\013)q Fo(.)42 b(No)m(w)29 b(the)h(latter)e(expression)i(is)f(uniformly)e(b)s(ounded)-118 2922 y(in)32 b Fn(T)46 b Fo(for)32 b(an)m(y)h Fn(\032)28 b(>)g Fk(k)p Fn(H)8 b Fk(k)803 2937 y Fm(\015)846 2922 y Fn(=\015)d Fo(.)28 3042 y(In)41 b(order)f(to)g(b)s(ound)h(the)f (second)i(summand)d(in)h(\(13\),)h(w)m(e)h(pro)s(ceed)f(as)f(in)g([10)o (,)j(Prop)s(osition)38 b(1])i(for)-118 3162 y(eac)m(h)28 b Fn(B)170 3177 y Fm(k)213 3162 y Fo(\()p Fn(T)14 b Fo(\))27 b(separately)-8 b(.)43 b(So)27 b(for)g(eac)m(h)i Fn(k)i Fk(\025)d Fo(1,)g(let)f(us)i(supp)s(ose)g Fn(N)38 b Fo(and)27 b Fn(T)42 b Fo(to)27 b(b)s(e)h(link)m(ed)f(b)m(y)i(some)e(relation)-118 3283 y(c)m(hosen)34 b(later)e(on)g(\(see)i(equation)e(\(17\))g(b)s(elo) m(w\))g(and)h(let)f(us)h(set,)h(for)e(giv)m(en)g Fn(\025)2767 3298 y Fm(c)2802 3283 y Fo(\()p Fn(k)s Fo(\))c Fk(\024)g Fo(\003\()p Fn(\026)p Fo(\),)1099 3523 y Fk(j)p Fn( )1190 3538 y Fm(N)s(;k)1311 3523 y Fo(\()p Fn(t)p Fo(\))p Fk(i)55 b Fo(=)1796 3440 y Fl(X)1648 3629 y Fm(\033)r Fg(2J)1785 3640 y Fj(N)1843 3629 y Fs(\()p Fm(\025)1911 3637 y Fj(c)1943 3629 y Fs(\()p Fm(k)r Fs(\)\))2098 3523 y Fn(e)2143 3482 y Fg(\000)p Fm({E)2277 3459 y Fj(N)2273 3499 y(\033)2334 3482 y Fm(t)2364 3523 y Fk(j)p Fn(\037)2453 3542 y Fm(I)2489 3523 y Fj(N)2484 3559 y(\033)2550 3523 y Fk(i)32 b Fo(,)-118 3779 y(where)j Fn(E)243 3743 y Fm(N)237 3804 y(\033)345 3779 y Fo(is)e(some)h(p)s(oin)m(t)f(in)h Fn(I)1113 3743 y Fm(N)1105 3804 y(\033)1214 3779 y Fo(and)g Fk(J)1473 3794 y Fm(N)1540 3779 y Fo(\()p Fn(\025)p Fo(\))g(is)f(the)i(set)g(of)e Fn(\033)i Fk(2)30 b Fo(\006)2498 3743 y Fm(N)2498 3804 y(L)2600 3779 y Fo(satisfying)j Fk(\000)17 b Fo(log\(\()p Fk(j)p Fn(I)3411 3743 y Fm(N)3403 3804 y(\033)3478 3779 y Fk(j)p Fo(\))p Fn(=)-5 b(N)40 b Fk(\025)31 b Fn(\025)p Fo(.)-118 3900 y(Let)d(us)h(further)f(divide)f(the)i(in)m(terv)-5 b(al)27 b([)p Fn(\025)1377 3915 y Fm(c)1411 3900 y Fo(\()p Fn(k)s Fo(\))p Fn(;)17 b Fo(\003\()p Fn(\026)p Fo(\)])27 b(in)m(to)h Fn(Q)f Fo(=)h(\(\003\()p Fn(\026)p Fo(\))13 b Fk(\000)g Fn(\025)2645 3915 y Fm(c)2679 3900 y Fo(\()p Fn(k)s Fo(\)\))p Fn(=\016)31 b Fk(2)d Fe(N)g Fo(in)m(terv)-5 b(als)27 b(of)h(equal)-118 4020 y(length)41 b Fn(\016)t Fo(.)69 b(Then)42 b(the)g(v)m(ector)g Fk(j)p Fn( )1164 4035 y Fm(N)s(;k)1286 4020 y Fo(\()p Fn(t)p Fo(\))p Fk(i)f Fo(appro)m(ximates)f(the)i(time)e(ev)m(olution)g(of)h Fk(j)p Fo(0)p Fk(i)f Fo(in)h(Hilb)s(ert)e(space)-118 4140 y(norm:)147 4439 y Fk(kj)p Fn( )288 4454 y Fm(N)s(;k)410 4439 y Fo(\()p Fn(t)p Fo(\))p Fk(i)22 b(\000)g Fn(e)726 4398 y Fg(\000)p Fm({H)5 b(t)897 4439 y Fk(j)p Fo(0)p Fk(ik)1063 4398 y Fs(2)1185 4439 y Fo(=)1465 4356 y Fl(X)1316 4544 y Fm(\033)r Fg(2J)1453 4555 y Fj(N)1511 4544 y Fs(\()p Fm(\025)1579 4552 y Fj(c)1612 4544 y Fs(\()p Fm(k)r Fs(\)\))1750 4322 y Fl(Z)1796 4510 y Fm(I)1832 4492 y Fj(N)1827 4527 y(\033)1910 4439 y Fn(d\026)p Fo(\()p Fn(E)h Fo(\))17 b Fk(j)p Fn(e)2264 4398 y Fm({E)t(t)2393 4439 y Fk(\000)23 b Fn(e)2538 4398 y Fm({E)2617 4374 y Fj(N)2613 4414 y(\033)2674 4398 y Fm(t)2704 4439 y Fk(j)2732 4398 y Fs(2)2793 4439 y Fo(+)3056 4356 y Fl(X)2891 4544 y Fm(\033)r Fg(2J)3028 4555 y Fj(N)3086 4544 y Fs(\()p Fm(\025)3154 4552 y Fj(c)3187 4544 y Fs(\()p Fm(k)r Fs(\)\))3307 4525 y Fj(c)3357 4439 y Fn(\026)p Fo(\()p Fn(I)3505 4398 y Fm(N)3497 4464 y(\033)3571 4439 y Fo(\))265 4810 y Fk(\024)443 4639 y Fl(0)443 4788 y(@)516 4696 y Fm(Q)p Fg(\000)p Fs(1)529 4727 y Fl(X)527 4909 y Fm(j)t Fs(=1)1261 4727 y Fl(X)706 4915 y Fm(\033)r Fg(2J)843 4926 y Fj(N)901 4915 y Fs(\()p Fm(\025)969 4923 y Fj(c)1002 4915 y Fs(\()p Fm(k)r Fs(\)+)p Fm(j)t(\016)r Fs(\))1243 4896 y Fj(c)1275 4915 y Fg(nJ)1358 4926 y Fj(N)1416 4915 y Fs(\()p Fm(\025)1484 4923 y Fj(c)1517 4915 y Fs(\()p Fm(k)r Fs(\)+\()p Fm(j)t Fg(\000)p Fs(1\))p Fm(\016)r Fs(\))1902 4896 y Fj(c)1952 4810 y Fo(+)2227 4727 y Fl(X)2045 4915 y Fm(\033)r Fg(2J)2182 4926 y Fj(N)2240 4915 y Fs(\(\003\()p Fm(\026)p Fs(\))p Fg(\000)p Fm(\016)r Fs(\))2530 4639 y Fl(1)2530 4788 y(A)2619 4810 y Fn(\026)p Fo(\()p Fn(I)2767 4769 y Fm(N)2759 4834 y(\033)2834 4810 y Fo(\))17 b Fn(t)2924 4769 y Fs(2)2987 4810 y Fo(max)2980 4877 y Fm(E)t Fg(2)p Fm(I)3119 4858 y Fj(N)3114 4893 y(\033)3192 4810 y Fk(j)p Fn(E)28 b Fk(\000)23 b Fn(E)3498 4769 y Fm(N)3492 4834 y(\033)3565 4810 y Fk(j)3593 4769 y Fs(2)498 5118 y Fo(+)28 b Fn(\026)p Fo(\()p Fk(J)767 5133 y Fm(N)833 5118 y Fo(\()p Fn(\025)928 5133 y Fm(c)963 5118 y Fo(\()p Fn(k)s Fo(\)\))1131 5077 y Fm(c)1165 5118 y Fo(\))265 5464 y Fk(\024)443 5351 y Fm(Q)p Fg(\000)p Fs(1)456 5381 y Fl(X)455 5563 y Fm(j)t Fs(=1)605 5464 y Fn(ae)701 5423 y Fg(\000)p Fm(N)7 b Fs(\003)868 5399 y Fi(\003)905 5423 y Fs(\()p Fm(\025)973 5431 y Fj(c)1005 5423 y Fs(\()p Fm(k)r Fs(\)+)p Fm(j)t(\016)o(;\026)p Fs(\))1309 5464 y Fn(e)1354 5423 y Fg(\000)p Fs(2)p Fm(N)g Fs(\()p Fm(\025)1575 5431 y Fj(c)1608 5423 y Fs(\()p Fm(k)r Fs(\)+\()p Fm(j)t Fg(\000)p Fs(1\))p Fm(\016)r Fs(\))1999 5464 y Fn(t)2034 5423 y Fs(2)2096 5464 y Fo(+)22 b Fn(e)2239 5423 y Fg(\000)p Fm(N)7 b Fs(\(\003\()p Fm(\026)p Fs(\))p Fg(\000)p Fm(\016)r Fs(\))2651 5464 y Fn(t)2686 5423 y Fs(2)2748 5464 y Fo(+)22 b Fn(ae)2942 5423 y Fg(\000)p Fm(N)7 b Fs(\003)3109 5399 y Fi(\003)3145 5423 y Fs(\()p Fm(\025)3213 5431 y Fj(c)3246 5423 y Fs(\()p Fm(k)r Fs(\))p Fm(;\026)p Fs(\))3466 5464 y Fo(,)231 b(\(14\))1865 5800 y(7)p eop %%Page: 8 8 8 7 bop -118 28 a Fo(where)34 b(w)m(e)f(ha)m(v)m(e)h(used)g(the)f(b)s (ound)g(\(7\).)43 b(F)-8 b(or)32 b(a)g(giv)m(en)h Fn(r)d Fk(\025)e Fn(T)11 b(=)p Fo(\(2)p Fn(\031)t Fo(\),)32 b(let)g(us)h(\014rst)g(b)s(ound)g Fn(B)3311 43 y Fm(k)3354 28 y Fo(\()p Fn(T)14 b Fo(\))32 b(b)m(y)-101 384 y(2)-52 267 y Fl(Z)31 293 y Fm(T)-6 455 y Fs(0)96 316 y Fn(dt)p 96 360 86 4 v 104 452 a(T)208 384 y Fk(kj)p Fn( )349 399 y Fm(N)s(;k)471 384 y Fo(\()p Fn(t)p Fo(\))p Fk(i)22 b(\000)g Fn(e)787 343 y Fg(\000)p Fm({H)5 b(t)958 384 y Fk(j)p Fo(0)p Fk(ik)1124 343 y Fs(2)1185 384 y Fo(+)22 b(2)1557 301 y Fl(X)p 1315 450 43 4 v 1315 489 a Fm(n)1358 501 y Fj(k)q Fi(\000)p Fh(1)1474 489 y Fs(\()p Fm(T)10 b Fs(\))p Fg(\024)p Fm(n)p Fg(\024)p 1732 450 V Fm(n)1776 501 y Fj(k)1814 489 y Fs(\()p Fm(T)g Fs(\))2125 301 y Fl(X)1936 489 y Fm(\033)n(;\033)2036 470 y Fi(0)2060 489 y Fg(2J)2155 500 y Fj(N)2212 489 y Fs(\()p Fm(\025)2280 497 y Fj(c)2313 489 y Fs(\()p Fm(k)r Fs(\)\))2417 384 y Fk(h)p Fn(n)p Fk(j)p Fn(\037)2603 402 y Fm(I)2639 383 y Fj(N)2634 419 y(\033)2700 384 y Fk(ih)p Fn(\037)2839 406 y Fm(I)2875 383 y Fj(N)2870 439 y(\033)2908 425 y Fi(0)2938 384 y Fk(j)p Fn(n)p Fk(i)3063 267 y Fl(Z)3145 293 y Fs(2)p Fm(\031)r(r)3109 455 y Fs(0)3272 316 y Fn(dt)p 3272 360 86 4 v 3280 452 a(T)3384 384 y(e)3429 341 y Fm({)p Fs(\()p Fm(E)3535 318 y Fj(N)3531 358 y(\033)3593 341 y Fg(\000)p Fm(E)3704 318 y Fj(N)3700 373 y(\033)3738 359 y Fi(0)3764 341 y Fs(\))p Fm(t)3853 384 y Fo(.)-118 697 y(No)m(w,)40 b(let)e(\003)353 712 y Fs(max)527 697 y Fo(b)s(e)g(the)h(minimal)c Fn(\025)j Fo(with)g(the)h(prop)s(ert)m(y)g Fk(j)p Fn(I)2205 661 y Fm(N)2197 722 y(\033)2272 697 y Fk(j)e(\025)h Fn(e)2497 661 y Fg(\000)p Fm(\025N)2699 697 y Fo(for)g(all)e Fn(\033)42 b Fk(2)c Fo(\006)3266 712 y Fm(L)3357 697 y Fo(and)h Fn(N)48 b Fk(2)38 b Fe(N)p Fo(.)-118 818 y(F)-8 b(or)43 b(\014xed)i Fn(N)10 b Fo(,)46 b(w)m(e)f(can)f(therefore)g(c)m(ho)s(ose)h(the)f Fn(E)1819 781 y Fm(N)1813 842 y(\033)1886 818 y Fo('s)g(all)e(to)h(b)s(e)h (elemen)m(ts)g(of)f(a)g(lattice)g(with)g(spacing)-118 938 y(1)p Fn(=r)30 b Fo(=)f Fn(e)204 902 y Fg(\000)p Fs(\003)308 910 y Fh(max)424 902 y Fm(N)524 938 y Fo(so)k(that)g(only)g (the)g(diagonal)e(terms)i Fn(\033)f Fo(=)c Fn(\033)2152 902 y Fg(0)2208 938 y Fo(remain)k(in)g(the)i(ab)s(o)m(v)m(e)f(sum)g (\(at)g(this)f(p)s(oin)m(t,)-118 1058 y(impro)m(v)m(emen)m(ts)f(are)g (p)s(ossible,)f(but)i(not)e(useful)h(if)f(\007)h(is)f(indep)s(enden)m (t)i(of)f Fn(\021)t Fo(\).)42 b(Using)31 b(\(14\))f(w)m(e)i(th)m(us)g (obtain)181 1445 y Fn(B)255 1460 y Fm(k)298 1445 y Fo(\()p Fn(T)14 b Fo(\))82 b Fk(\024)688 1332 y Fm(Q)p Fg(\000)p Fs(1)701 1362 y Fl(X)699 1544 y Fm(j)t Fs(=1)860 1378 y Fo(2)p Fn(aT)1031 1341 y Fs(2)p 860 1422 211 4 v 941 1513 a Fo(3)1080 1445 y Fn(e)1125 1404 y Fg(\000)p Fm(N)7 b Fs(\003)1292 1380 y Fi(\003)1328 1404 y Fs(\()p Fm(\025)1396 1412 y Fj(c)1429 1404 y Fs(\()p Fm(k)r Fs(\)+)p Fm(j)t(\016)o(;\026)p Fs(\))1733 1445 y Fn(e)1778 1404 y Fg(\000)p Fs(2)p Fm(N)g Fs(\()p Fm(\025)1999 1412 y Fj(c)2032 1404 y Fs(\()p Fm(k)r Fs(\)+\()p Fm(j)t Fg(\000)p Fs(1\))p Fm(\016)r Fs(\))2445 1445 y Fo(+)2553 1378 y(2)p Fn(T)2673 1341 y Fs(2)p 2553 1422 159 4 v 2608 1513 a Fo(3)2722 1445 y Fn(e)2767 1404 y Fg(\000)p Fs(2)p Fm(N)g Fs(\(\003\()p Fm(\026)p Fs(\))p Fg(\000)p Fm(\016)r Fs(\))3724 1673 y Fo(\(15\))715 1880 y(+)28 b(2)p Fn(ae)964 1839 y Fg(\000)p Fm(N)7 b Fs(\003)1131 1816 y Fi(\003)1167 1839 y Fs(\()p Fm(\025)1235 1847 y Fj(c)1268 1839 y Fs(\()p Fm(k)r Fs(\))p Fm(;\026)p Fs(\))1477 1880 y Fo(+)1612 1813 y(4)p Fn(\031)t(e)1765 1777 y Fs(\003)1814 1785 y Fh(max)1931 1777 y Fm(N)p 1612 1857 386 4 v 1770 1949 a Fn(T)2267 1797 y Fl(X)p 2025 1947 43 4 v 2025 1986 a Fm(n)2067 1998 y Fj(k)q Fi(\000)p Fh(1)2184 1986 y Fs(\()p Fm(T)j Fs(\))p Fg(\024)p Fm(n)p Fg(\024)p 2442 1947 V Fm(n)2485 1998 y Fj(k)2523 1986 y Fs(\()p Fm(T)g Fs(\))2794 1797 y Fl(X)2646 1986 y Fm(\033)r Fg(2J)2783 1997 y Fj(N)2841 1986 y Fs(\()p Fm(\025)2909 1994 y Fj(c)2941 1986 y Fs(\()p Fm(k)r Fs(\)\))3079 1880 y Fk(jh)p Fn(n)p Fk(j)p Fn(\037)3293 1899 y Fm(I)3329 1880 y Fj(N)3324 1916 y(\033)3390 1880 y Fk(ij)3457 1839 y Fs(2)3528 1880 y Fo(.)-118 2194 y(Putting)48 b(this)g(and)p 669 2141 59 4 v 49 w Fn(n)727 2209 y Fm(k)770 2194 y Fo(\()p Fn(T)14 b Fo(\))54 b(=)h Fn(\032T)1223 2158 y Fm(\024)p Fs(+)p Fm(k)r(\017)1438 2194 y Fo(in)m(to)48 b(\(13\),)k(w)m(e)e(obtain)d(for)h(some)h(constan)m(t)g Fn(c)g Fo(dep)s(ending)g(on)-118 2314 y Fn(a;)17 b(\032;)g(\013)q(;)g (\017;)g Fo(\003)329 2329 y Fs(max)463 2314 y Fn(;)g Fo(\007)33 b(and)f Fn(\015)5 b Fo(:)123 2709 y Fn(M)217 2724 y Fm(\013)267 2709 y Fo(\()p Fn(T)14 b Fo(\))83 b Fk(\024)g Fn(cT)770 2668 y Fm(\024\013)882 2709 y Fo(+)22 b Fn(cT)1093 2668 y Fm(\024\013)1200 2592 y Fs(\(1)p Fg(\000)p Fm(\024)p Fs(\))p Fm(=\017)1265 2626 y Fl(X)1260 2811 y Fm(k)r Fs(=1)1466 2709 y Fn(T)1537 2668 y Fm(k)r(\017\013)1670 2538 y Fl(0)1670 2688 y(@)1776 2596 y Fm(Q)1744 2626 y Fl(X)1742 2808 y Fm(j)t Fs(=1)1882 2709 y Fn(T)1953 2668 y Fs(2)1992 2709 y Fn(e)2037 2668 y Fg(\000)p Fm(N)7 b Fs(\003)2204 2645 y Fi(\003)2240 2668 y Fs(\()p Fm(\025)2308 2676 y Fj(c)2340 2668 y Fs(\()p Fm(k)r Fs(\)+)p Fm(j)t(\016)o(;\026)p Fs(\))2645 2709 y Fn(e)2690 2668 y Fg(\000)p Fs(2)p Fm(N)g Fs(\()p Fm(\025)2911 2676 y Fj(c)2944 2668 y Fs(\()p Fm(k)r Fs(\)+\()p Fm(j)t Fg(\000)p Fs(1\))p Fm(\016)r Fs(\))3724 2942 y Fo(\(16\))778 3174 y(+)28 b Fn(e)927 3133 y Fg(\000)p Fm(N)7 b Fs(\003)1094 3110 y Fi(\003)1130 3133 y Fs(\()p Fm(\025)1198 3141 y Fj(c)1230 3133 y Fs(\()p Fm(k)r Fs(\))p Fm(;\026)p Fs(\))1456 3174 y Fo(+)1581 3107 y(4)p Fn(\031)t(e)1734 3071 y Fs(\003)1783 3079 y Fh(max)1899 3071 y Fm(N)p 1581 3151 386 4 v 1738 3242 a Fn(T)2235 3091 y Fl(X)p 1993 3241 43 4 v 1993 3279 a Fm(n)2036 3291 y Fj(k)q Fi(\000)p Fh(1)2152 3279 y Fs(\()p Fm(T)j Fs(\))p Fg(\024)p Fm(n)p Fg(\024)p 2410 3241 V Fm(n)2453 3291 y Fj(k)2491 3279 y Fs(\()p Fm(T)g Fs(\))2762 3091 y Fl(X)2614 3279 y Fm(\033)r Fg(2J)2751 3290 y Fj(N)2809 3279 y Fs(\()p Fm(\025)2877 3287 y Fj(c)2910 3279 y Fs(\()p Fm(k)r Fs(\)\))3047 3174 y Fk(jh)p Fn(n)p Fk(j)p Fn(\037)3261 3193 y Fm(I)3297 3174 y Fj(N)3292 3209 y(\033)3358 3174 y Fk(ij)3425 3133 y Fs(2)3464 3003 y Fl(1)3464 3153 y(A)3586 3174 y Fo(.)-118 3475 y(No)m(w,)33 b(for)f(ev)m(ery)j(\014xed)e Fn(k)e Fk(\025)d Fo(1,)33 b(w)m(e)g(c)m(ho)s(ose)h Fn(N)43 b Fo(and)32 b Fn(T)47 b Fo(to)32 b(b)s(e)h(related)f(b)m(y)1518 3715 y(\007)1594 3674 y Fm(N)1717 3715 y Fo(=)55 b Fn(T)1919 3674 y Fm(\024)p Fs(+)p Fm(k)r(\017)p Fg(\000)p Fm(\017)2201 3715 y Fo(,)1496 b(\(17\))-118 3886 y(so)38 b(that)g(the)h(last)e(term)g(in)h(the)g (paren)m(thesis)h(in)f(\(16\))f(is)h(uniformly)e(b)s(ounded)j(in)e Fn(T)52 b Fo(b)m(y)39 b(Hyp)s(othesis)f(I)s(I.)-118 4006 y(W)-8 b(e)29 b(then)g(w)m(an)m(t)h(to)f(c)m(ho)s(ose)g Fn(\024)g Fo(in)f(suc)m(h)i(a)f(w)m(a)m(y)h(that)f(the)g(\014rst)g(and) g(second)h(term)e(in)g(\(16\))g(and)h(th)m(us)h Fn(M)3701 4021 y Fm(\013)3751 4006 y Fo(\()p Fn(T)14 b Fo(\))-118 4127 y(are)32 b(b)s(ounded)i(b)m(y)f Fn(cT)691 4091 y Fm(\013\024)781 4127 y Fo(.)44 b(This)32 b(imp)s(oses)g(the)h(t)m(w)m (o)g(conditions)265 4367 y(\()p Fn(k)s(\017\013)24 b Fo(+)e(2\))17 b(log)f(\(\007\))55 b Fk(\024)h Fo(\()p Fn(\024)22 b Fo(+)g Fn(k)s(\017)g Fk(\000)h Fn(\017)p Fo(\))17 b(\(\003)1795 4326 y Fg(\003)1834 4367 y Fo(\()p Fn(\025)1929 4382 y Fm(c)1964 4367 y Fo(\()p Fn(k)s Fo(\))22 b(+)g Fn(j)6 b(\016)n(;)17 b(\026)p Fo(\))22 b(+)g(2\()p Fn(\025)2706 4382 y Fm(c)2740 4367 y Fo(\()p Fn(k)s Fo(\))g(+)g(\()p Fn(j)28 b Fk(\000)23 b Fo(1\))p Fn(\016)t Fo(\)\))48 b(,)243 b(\(18\))1090 4658 y Fn(k)s(\017\013)17 b Fo(log)g(\(\007\))55 b Fk(\024)h Fo(\()p Fn(\024)22 b Fo(+)g Fn(k)s(\017)p Fo(\)\003)2158 4617 y Fg(\003)2198 4658 y Fo(\()p Fn(\025)2293 4673 y Fm(c)2327 4658 y Fo(\()p Fn(k)s Fo(\))p Fn(;)17 b(\026)p Fo(\))32 b(,)1067 b(\(19\))-118 4829 y(whic)m(h)38 b(ha)m(v)m(e)i(to)d(hold)g(for)h(all)e Fn(j)43 b Fo(=)36 b(1)p Fn(;)17 b(:)g(:)g(:)f(;)h(Q)37 b Fo(=)f(\(\003\()p Fn(\026)p Fo(\))25 b Fk(\000)i Fn(\025)2155 4844 y Fm(c)2189 4829 y Fo(\()p Fn(k)s Fo(\)\))p Fn(=\016)42 b Fo(and)c(all)e Fn(k)k Fo(=)c(1)p Fn(;)17 b(:)g(:)g(:)f(;)h Fo(\(1)25 b Fk(\000)i Fn(\024)p Fo(\))p Fn(=\017)p Fo(,)39 b(the)-118 4949 y(c)m(hoice)33 b(of)f(eac)m(h)h Fn(\025)559 4964 y Fm(c)594 4949 y Fo(\()p Fn(k)s Fo(\))28 b Fk(2)g Fo([0)p Fn(;)17 b Fo(\003\()p Fn(\026)p Fo(\)])31 b(still)f(b)s(eing)i(free.)28 5070 y(The)39 b(problem)e(is)h(no)m(w)g(to)g(determine)g(the)g(minimal) c Fn(\024)k Fo(suc)m(h)i(that)e(these)h(inequalities)d(hold)i(for)f (ap-)-118 5190 y(propriate)c(c)m(hoices)i(of)f Fn(\025)806 5205 y Fm(c)841 5190 y Fo(\()p Fn(k)s Fo(\).)48 b(F)-8 b(or)34 b(this)f(purp)s(ose,)j(w)m(e)f(study)h(their)e(con)m(tin)m(uum) g(limit)c Fn(\017;)17 b(\016)35 b Fk(!)30 b Fo(0.)48 b(Setting)-118 5311 y Fn(x)28 b Fo(=)g Fn(k)s(\017)p Fo(,)33 b Fn(\025)279 5326 y Fm(c)313 5311 y Fo(\()p Fn(x)p Fo(\))28 b(=)g Fn(\025)633 5326 y Fm(c)667 5311 y Fo(\()p Fn(k)s Fo(\))33 b(and)g Fn(\025)27 b Fo(=)h Fn(\025)1265 5326 y Fm(c)1299 5311 y Fo(\()p Fn(k)s Fo(\))22 b(+)g Fn(j)6 b(\016)t Fo(,)33 b(the)g(follo)m(wing)d(inequalities)937 5551 y(\()p Fn(x\013)23 b Fo(+)f(2\))17 b(log)f(\(\007\))55 b Fk(\024)h Fo(\()p Fn(\024)22 b Fo(+)g Fn(x)p Fo(\)\(\003)2212 5510 y Fg(\003)2252 5551 y Fo(\()p Fn(\025;)17 b(\026)p Fo(\))k(+)h(2)p Fn(\025)p Fo(\))32 b(,)914 b(\(20\))1865 5800 y(8)p eop %%Page: 9 9 9 8 bop 1127 73 a Fn(x\013)18 b Fo(log)e(\(\007\))56 b Fk(\024)f Fo(\()p Fn(\024)23 b Fo(+)f Fn(x)p Fo(\)\003)2120 32 y Fg(\003)2159 73 y Fo(\()p Fn(\025)2254 88 y Fm(c)2289 73 y Fo(\()p Fn(x)p Fo(\))p Fn(;)17 b(\026)p Fo(\))32 b(,)1104 b(\(21\))-118 248 y(ha)m(v)m(e)34 b(to)e(hold)g(for)g Fn(\025)27 b Fk(2)i Fo([)p Fn(\025)855 263 y Fm(c)889 248 y Fo(\()p Fn(x)p Fo(\))p Fn(;)17 b Fo(\003\()p Fn(\026)p Fo(\)])32 b(and)h Fn(x)28 b Fk(2)g Fo(\(0)p Fn(;)17 b Fo(1)k Fk(\000)i Fn(\024)p Fo(].)28 368 y(W)-8 b(e)33 b(\014rst)g(c)m(ho)s(ose)h Fn(\025)764 383 y Fm(c)798 368 y Fo(\()p Fn(x)p Fo(\))f(to)f(b)s(e)h(the)g(biggest)f Fn(\025)h Fo(suc)m(h)h(that)e(\(21\))g(is)g(satis\014ed:)774 621 y Fn(\025)831 636 y Fm(c)865 621 y Fo(\()p Fn(x)p Fo(\))56 b(=)f(sup)1347 500 y Fl(\032)1409 621 y Fn(\025)28 b Fk(\024)g Fo(\003\()p Fn(\026)p Fo(\))1834 497 y Fl(\014)1834 546 y(\014)1834 596 y(\014)1834 646 y(\014)1879 621 y Fo(\003)1947 580 y Fg(\003)1986 621 y Fo(\()p Fn(\025)p Fo(\))f Fk(\025)h Fo(log)17 b(\(\007\))2629 554 y Fn(x\013)p 2572 598 232 4 v 2572 690 a(\024)23 b Fo(+)f Fn(x)2814 500 y Fl(\033)2935 621 y Fo(.)762 b(\(22\))-118 834 y(Using)30 b(the)h(h)m(yp)s(othesis)g(on)f(\003)999 798 y Fg(\003)1039 834 y Fo(\()p Fn(:;)17 b(\026)p Fo(\),)30 b(it)f(can)i(b)s(e)f(v)m (eri\014ed)i(that)e Fn(\025)2310 849 y Fm(c)2344 834 y Fo(\()p Fn(x)p Fo(\))h(is)f(a)g(decreasing)h(function)f(in)f Fn(x)p Fo(.)44 b(On)-118 954 y(the)32 b(other)g(hand,)g(b)s(ecause)h (the)g(minimal)27 b(v)-5 b(alue)31 b(of)h(the)g(function)f Fn(\025)d Fk(7!)f Fo(\003)2669 918 y Fg(\003)2708 954 y Fo(\()p Fn(\025;)17 b(\026)p Fo(\))j(+)g(2)p Fn(\025)32 b Fo(in)f([\003)3406 969 y Fs(min)3527 954 y Fn(;)17 b Fo(\003\()p Fn(\026)p Fo(\)])31 b(is)-118 1075 y(tak)m(en)f(at)f (\003)328 1090 y Fm(R)415 1075 y Fo(and)g(this)g(function)g(is)g (increasing)f(on)i([\003)1942 1090 y Fm(R)1999 1075 y Fn(;)17 b Fo(\003\()p Fn(\026)p Fo(\)],)29 b(it)g(follo)m(ws)f(that)h (\(20\))f(is)h(alw)m(a)m(ys)h(satis\014ed)-118 1195 y(for)i Fn(\025)c Fk(2)g Fo([)p Fn(\025)294 1210 y Fm(c)328 1195 y Fo(\()p Fn(x)p Fo(\))p Fn(;)17 b Fo(\003\()p Fn(\026)p Fo(\)])32 b(if)916 1466 y Fn(\024)23 b Fo(+)f Fn(x)55 b Fk(\025)1711 1399 y Fo(\(2)22 b(+)g Fn(x\013)q Fo(\))17 b(log)f(\(\007\))p 1346 1443 1405 4 v 1346 1534 a(\(\003)1452 1506 y Fg(\003)1491 1534 y Fo(\()p Fn(:;)h(\026)p Fo(\))22 b(+)g(2)17 b(id)n(\)\(max)p Fk(f)p Fo(\003)2339 1549 y Fm(R)2396 1534 y Fn(;)g(\025)2497 1549 y Fm(c)2531 1534 y Fo(\()p Fn(x)p Fo(\))p Fk(g)p Fo(\))2793 1466 y(.)904 b(\(23\))-118 1696 y(In)33 b(order)g(to)f(treat)g(the)h(t)m(w)m (o)g(di\013eren)m(t)g(v)-5 b(alues)33 b(of)f(the)h(maxim)m(um)d(in)i (\(23\))g(separately)-8 b(,)33 b(w)m(e)g(in)m(tro)s(duce)637 1967 y Fn(x)692 1982 y Fm(\013)797 1967 y Fk(\021)56 b Fo(inf)6 b Fk(f)p Fn(x)28 b Fk(\025)g Fo(0)17 b Fk(j)g Fn(\025)1454 1982 y Fm(c)1487 1967 y Fo(\()p Fn(x)p Fo(\))28 b Fk(\024)h Fo(\003)1820 1982 y Fm(R)1877 1967 y Fk(g)55 b Fo(=)2343 1900 y Fn(\024)p Fo(\003)2467 1864 y Fg(\003)2506 1900 y Fo(\(\003)2612 1915 y Fm(R)2670 1900 y Fn(;)17 b(\026)p Fo(\))p 2123 1944 907 4 v 2123 2035 a Fn(\013)h Fo(log)e(\(\007\))22 b Fk(\000)h Fo(\003)2687 2007 y Fg(\003)2726 2035 y Fo(\(\003)2832 2050 y Fm(R)2889 2035 y Fn(;)17 b(\026)p Fo(\))3072 1967 y(.)-118 2197 y(F)-8 b(or)41 b Fn(x)k Fk(2)g Fo([)p Fn(x)359 2212 y Fm(\013)409 2197 y Fn(;)17 b Fo(1)28 b Fk(\000)h Fn(\024)p Fo(],)45 b(the)e(maxim)m(um)d(is)i(equal)g(to)g(\003)2006 2212 y Fm(R)2063 2197 y Fo(.)73 b(F)-8 b(or)41 b Fn(\013)k Fk(\024)g Fn(\013)2638 2212 y Fm(c)2673 2197 y Fo(,)f(the)f(inequalit)m (y)e(\(23\))h(is)g(then)-118 2318 y(most)37 b(di\016cult)f(to)h (satisfy)g(for)g(the)g(smallest)f(p)s(ossible)g Fn(x)p Fo(,)j(that)e(is)g Fn(x)f Fo(=)f Fn(x)2695 2333 y Fm(\013)2745 2318 y Fo(,)k(whereas,)h(for)c Fn(\013)h Fk(\025)f Fn(\013)3640 2333 y Fm(c)3675 2318 y Fo(,)i(this)-118 2438 y(is)32 b(the)i(case)g(for)e Fn(x)d Fo(=)f(1)22 b Fk(\000)h Fn(\024)p Fo(.)45 b(After)33 b(a)g(short)g(computation,)f(one)h(therefore)h (obtains)e(that)h(\(23\))f(holds)h(for)-118 2558 y Fn(x)28 b Fk(2)g Fo([)p Fn(x)141 2573 y Fm(\013)191 2558 y Fn(;)17 b Fo(1)22 b Fk(\000)g Fn(\024)p Fo(])33 b(only)f(if)637 2920 y Fn(\024)55 b Fk(\025)881 2695 y Fl(8)881 2770 y(>)881 2795 y(>)881 2820 y(<)881 2969 y(>)881 2994 y(>)881 3019 y(:)1181 2737 y Fm(\013)13 b Fs(log)q(\(\007\))q Fg(\000)p Fs(\003)1544 2714 y Fi(\003)1579 2737 y Fs(\(\003)1655 2748 y Fj(R)1706 2737 y Fm(;\026)p Fs(\))p 1181 2760 614 4 v 1416 2818 a Fm(\013)p Fs(\003)1510 2829 y Fj(R)1838 2783 y Fo(,)2214 2737 y Fs(\003)2263 2714 y Fi(\003)2299 2737 y Fs(\(\003)2375 2748 y Fj(R)2426 2737 y Fm(;\026)p Fs(\))p 2165 2760 399 4 v 2165 2818 a(\(1)p Fg(\000)p Fm(\024)p Fs(\))g(log)q(\()q(\007\))2602 2783 y Fk(\024)28 b Fn(\013)g Fk(\024)g Fn(\013)2964 2798 y Fm(c)3031 2783 y Fo(,)997 3051 y(1)22 b Fk(\000)1177 3005 y Fs(\003)1226 2981 y Fi(\003)1262 3005 y Fs(\(\003)1338 3016 y Fj(R)1389 3005 y Fm(;\026)p Fs(\)+2\003)1617 3016 y Fj(R)1668 3005 y Fg(\000)p Fs(2)11 b(log)r(\(\007\))p 1177 3028 794 4 v 1445 3085 a Fm(\013)h Fs(log)q(\(\007\))2013 3051 y Fo(,)391 b Fn(\013)28 b Fk(\025)h Fn(\013)2689 3066 y Fm(c)2756 3051 y Fo(,)3724 2920 y(\(24\))-118 3234 y(while)j(for)g(small)e Fn(\013)q Fo('s)j(no)f(condition)f(is)h(imp)s (osed)g(on)g Fn(\024)h Fo(b)s(ecause)h Fn(x)2386 3249 y Fm(\013)2464 3234 y Fn(>)27 b Fo(1)22 b Fk(\000)h Fn(\024)p Fo(.)28 3354 y(Next)35 b(w)m(e)g(study)g(\(23\))e(for)g Fn(x)e Fk(2)f Fo([0)p Fn(;)17 b(x)1397 3369 y Fm(\013)1446 3354 y Fo(\).)48 b(Using)33 b(the)h(de\014nition)f(of)h Fn(\025)2608 3369 y Fm(c)2642 3354 y Fo(\()p Fn(x)p Fo(\),)h(it)d (follo)m(ws)h(that)h(\(23\))f(holds)-118 3474 y(if)556 3745 y Fn(\024)22 b Fo(+)g Fn(x)56 b(>)1076 3678 y Fo(\(2)22 b(+)g Fn(x\013)q Fo(\))17 b(log)f(\(\007\))p 984 3722 859 4 v 984 3813 a(log)h(\()o(\007\))1330 3774 y Fm(x\013)p 1305 3790 136 4 v 1305 3848 a(\024)p Fs(+)p Fm(x)1472 3813 y Fo(+)22 b(2)p Fn(\025)1676 3828 y Fm(c)1711 3813 y Fo(\()p Fn(x)p Fo(\))2075 3745 y Fk(,)222 b Fn(\025)2454 3760 y Fm(c)2489 3745 y Fo(\()p Fn(x)p Fo(\))55 b Fn(>)2817 3678 y Fo(log)16 b(\(\007\))p 2817 3722 295 4 v 2848 3813 a Fn(\024)22 b Fo(+)g Fn(x)3153 3745 y Fo(.)-118 3992 y(Using)28 b(the)i(fact)e(that)h(\003)781 3956 y Fg(\003)820 3992 y Fo(\()p Fn(:;)17 b(\026)p Fo(\))28 b(is)g(decreasing)i(in)e([0)p Fn(;)17 b Fo(\003\()p Fn(\026)p Fo(\)])27 b(and)i(again)f(the)h(de\014nition)f(of)g Fn(\025)3305 4007 y Fm(c)3340 3992 y Fo(\()p Fn(x)p Fo(\),)h(it)f(is)h(th)m(us)-118 4112 y(su\016cien)m(t)34 b(that)1296 4390 y Fn(x\013)1424 4323 y Fo(log)16 b(\(\007\))p 1424 4367 V 1455 4459 a Fn(\024)22 b Fo(+)g Fn(x)1783 4390 y(<)56 b Fo(\003)1983 4349 y Fg(\003)2038 4244 y Fl( )2114 4323 y Fo(log)17 b(\()o(\007\))p 2114 4367 V 2145 4459 a Fn(\024)23 b Fo(+)f Fn(x)2418 4244 y Fl(!)3724 4390 y Fo(\(25\))-118 4627 y(holds)32 b(for)g Fn(x)c Fk(2)g Fo([0)p Fn(;)17 b(x)638 4642 y Fm(\013)688 4627 y Fo(\).)43 b(Let)33 b(us)g(set)g Fn(z)g Fo(=)27 b(log\(\007\))p Fn(=)p Fo(\()p Fn(\024)22 b Fo(+)g Fn(x)p Fo(\),)33 b(then)g(\(25\))f(is)g(equiv)-5 b(alen)m(t)33 b(to)623 4905 y Fn(\013)17 b Fo(log)g(\(\007\))22 b Fk(\000)h Fn(\024\013)q(z)60 b(<)55 b Fo(\003)1542 4864 y Fg(\003)1581 4905 y Fo(\()p Fn(z)t(;)17 b(\026)p Fo(\))32 b(,)228 b Fn(z)33 b Fk(2)2268 4759 y Fl(")2326 4838 y Fo(log)17 b(\(\007\))p 2326 4882 V 2333 4973 a Fn(\024)22 b Fo(+)g Fn(x)2564 4988 y Fm(\013)2630 4905 y Fn(;)2684 4838 y Fo(log)17 b(\(\007\))p 2684 4882 V 2803 4973 a Fn(\024)2988 4759 y Fl(#)3086 4905 y Fo(.)611 b(\(26\))-118 5136 y(The)39 b(righ)m(t)f(hand)h(side)g(is)f(con)m(v)m (ex)i(in)e Fn(z)t Fo(,)j(the)e(left)f(hand)h(side)f(decreasing)h(in)f Fn(\024)p Fo(.)61 b(The)40 b(minimal)34 b Fn(\024)k Fo(=)p 3792 5083 57 4 v 38 w Fn(\024)3848 5151 y Fm(\013)-118 5256 y Fo(suc)m(h)32 b(that)e(\(26\))g(holds)g(for)g Fn(z)i Fk(2)c Fo([0)p Fn(;)17 b Fo(\003\()p Fn(\026)p Fo(\)])30 b(or)g(\(25\))f(for)h(all)f Fn(x)f Fk(2)g Fe(R)i Fo(can)g(b)s(e)h(determined)f(b)m(y)i(equalizing)c(left)-118 5377 y(and)33 b(righ)m(t)e(hand)i(side)g(as)g(w)m(ell)e(as)i(their)f (deriv)-5 b(ativ)m(es)33 b(in)f Fn(x)p Fo(.)43 b(F)-8 b(or)32 b(a)h(giv)m(en)f Fn(\013)d(>)e Fo(0,)33 b(this)f(sho)m(ws)i (that)1865 5800 y(9)p eop %%Page: 10 10 10 9 bop 760 50 57 4 v 760 103 a Fn(\024)817 118 y Fm(\013)921 103 y Fo(=)1063 36 y Fn(\013)17 b Fo(log)f(\(\007\))22 b Fk(\000)h Fo(\003)1626 0 y Fg(\003)1665 36 y Fo(\()p Fn(z)1748 51 y Fm(\013)1798 36 y Fn(;)17 b(\026)p Fo(\))p 1063 80 876 4 v 1422 172 a Fn(\013)q(z)1530 187 y Fm(\013)1981 103 y Fo(,)p 2235 50 56 4 v 227 w Fn(x)2291 118 y Fm(\013)2396 103 y Fo(=)2537 36 y(\003)2605 0 y Fg(\003)2644 36 y Fo(\()p Fn(z)2727 51 y Fm(\013)2777 36 y Fn(;)g(\026)p Fo(\))p 2537 80 381 4 v 2648 172 a Fn(\013)q(z)2756 187 y Fm(\013)2959 103 y Fo(,)-118 318 y(where)38 b Fn(z)213 333 y Fm(\013)300 318 y Fo(is)e(determined)h(b)m(y)h(\(10\))e(and)p 1460 265 56 4 v 37 w Fn(x)1515 333 y Fm(\013)1602 318 y Fo(is)g(the)i(corresp)s(onding)e(v)-5 b(alue)37 b(of)f Fn(x)h Fo(at)g(whic)m(h)g(\(25\))g(is)f(most)-118 439 y(di\016cult)k(to)h(v)m(erify)-8 b(.)68 b(W)-8 b(e)42 b(note)f(that)g Fn(z)1368 454 y Fs(0)1449 439 y Fo(=)h(\003\()p Fn(\026)p Fo(\))e(and)h Fn(z)2053 454 y Fm(\013)2098 462 y Fj(c)2177 439 y Fo(=)h(\003)2363 454 y Fm(R)2420 439 y Fo(,)h(and)e(further)g(that)g Fn(\013)i Fk(7!)e Fn(z)3535 454 y Fm(\013)3626 439 y Fo(is)f(w)m(ell)-118 559 y(de\014ned)45 b(and)f(decreasing)g(due)h(to)f(the)g(con)m(v)m (exit)m(y)i(of)d(\003)2054 523 y Fg(\003)2093 559 y Fo(\()p Fn(:;)17 b(\026)p Fo(\).)77 b(If)p 2512 506 V 44 w Fn(x)2567 574 y Fm(\013)2663 559 y Fk(\024)48 b Fn(x)2843 574 y Fm(\013)2937 559 y Fo(whic)m(h)c(is)f(equiv)-5 b(alen)m(t)44 b(to)-118 679 y Fn(\013)28 b Fk(\024)g Fn(\013)139 694 y Fm(c)207 679 y Fo(and)k Fn(z)441 694 y Fm(\013)519 679 y Fk(\025)c Fo(\003)692 694 y Fm(R)749 679 y Fo(,)33 b(w)m(e)h(th)m(us)f(ha)m(v)m(e)h(up)s(on)f(taking)f(in)m(to)f(accoun)m (t)j(\(24\):)610 958 y Fn(\024)55 b(>)h Fo(max)1051 811 y Fl(\()1128 890 y Fn(\013)17 b Fo(log)g(\(\007\))k Fk(\000)i Fo(\003)1691 854 y Fg(\003)1730 890 y Fo(\()p 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Fm(\013)2372 1309 y Fn(\013)2435 1273 y Fs(2)2474 1309 y Fo(\))36 b(and)g Fn(g)2792 1273 y Fg(0)2788 1334 y Fm(r)2826 1309 y Fo(\()p Fn(\013)q Fo(\))d(=)h(\003)3176 1273 y Fg(\003)3215 1309 y Fo(\(\003)3321 1324 y Fm(R)3378 1309 y Fn(;)17 b(\026)p Fo(\))p Fn(=)p Fo(\(\003)3674 1324 y Fm(R)3730 1309 y Fn(\013)3793 1273 y Fs(2)3833 1309 y Fo(\),)-118 1430 y(so)33 b(that)f Fn(g)264 1394 y Fg(0)260 1454 y Fm(l)287 1430 y Fo(\()p Fn(\013)q Fo(\))27 b Fk(\024)h Fn(g)609 1394 y Fg(0)605 1454 y Fm(r)643 1430 y Fo(\()p Fn(\013)q Fo(\);)k(therefore)h(the)g(maxim)m(um)d(is)i (equal)h(to)f Fn(g)2395 1445 y Fm(l)2421 1430 y Fo(\()p Fn(\013)q Fo(\),)g(whic)m(h)h(is)f(precisely)h(\(11\).)28 1550 y(F)-8 b(or)p 203 1497 56 4 v 32 w Fn(x)258 1565 y Fm(\013)335 1550 y Fk(\025)28 b Fn(x)495 1565 y Fm(\013)590 1550 y Fk(,)f Fn(\013)h Fk(\025)g Fn(\013)974 1565 y Fm(c)1009 1550 y Fo(,)33 b(the)f(inequalit)m(y)f(\(25\))h(is)g (satis\014ed)h(for)e(all)f Fn(x)f Fk(2)f Fo([0)p Fn(;)17 b(x)3003 1565 y Fm(\013)3052 1550 y Fo(\))32 b(if)f(it)h(is)f (satis\014ed)i(for)-118 1671 y Fn(x)-63 1686 y Fm(\013)-13 1671 y Fo(,)f(whic)m(h)i(implies)c(that,)i(using)g(\(22\),)1253 1941 y Fn(\024)56 b Fk(\025)1507 1874 y Fn(\013)18 b Fo(log)e(\(\007\))22 b Fk(\000)h Fo(\003)2071 1838 y Fg(\003)2110 1874 y Fo(\(\003)2216 1889 y Fm(R)2273 1874 y Fn(;)17 b(\026)p Fo(\))p 1507 1918 907 4 v 1866 2010 a Fn(\013)q Fo(\003)1997 2025 y Fm(R)2456 1941 y Fo(.)-118 2156 y(This)29 b(b)s(ound)f(coincides)h(with)f(the)h(b)s(ound)g(\(24\)) f(at)g Fn(\013)g Fo(=)g Fn(\013)2058 2171 y Fm(c)2092 2156 y Fo(,)i(but)f(for)f Fn(\013)g(>)g(\013)2725 2171 y Fm(c)2788 2156 y Fo(it)f(is)h(less)h(restrictiv)m(e)g(so)g(that)-118 2277 y(\(24\))j(giv)m(es)h(\(12\).)3295 b Fa(2)-118 2538 y Fe(Remark)37 b(5)49 b Fo(Using)32 b(\(9\),)g(it)g(is)g(straigh)m (tforw)m(ard)g(to)g(v)m(erify)1366 2742 y(2\003)1483 2757 y Fm(R)p 1314 2786 278 4 v 1314 2878 a Fo(log\(\007\))1657 2809 y Fk(\024)56 b Fn(\013)1852 2824 y Fm(c)1942 2809 y Fk(\024)2098 2742 y Fo(2\003\()p Fn(\026)p Fo(\))p 2085 2786 V 2085 2878 a(log\(\007\))2405 2809 y(.)-118 3029 y(F)-8 b(or)32 b(real)f(Julia)g(sets,)j(this)e(allo)m(ws)g(to)g (deduce)i(the)f(b)s(eha)m(vior)f(giv)m(en)h(in)f(Remark)g(1.)-118 3243 y Fe(Remark)37 b(6)49 b Fo(F)-8 b(or)27 b(big)f Fn(\013)q Fo(,)j(the)f(main)e(con)m(tribution)g(to)i Fn(M)2019 3258 y Fm(\013)2069 3243 y Fo(\()p Fn(T)14 b Fo(\))27 b(comes)h(from)e(the)i(part)g(of)f(the)h(w)m(a)m(v)m(e)i (pac)m(k)m(et)-118 3364 y(far)46 b(from)f(the)i(origin.)83 b(In)47 b(order)g(to)f(ha)m(v)m(e)i(a)e(b)s(etter)h(than)g(ballistic)c (b)s(ound)k(for)f(arbitrarily)e(large)h Fn(\013)-118 3484 y Fo(\()p Fn(\024)-24 3499 y Fm(\013)57 3484 y Fn(<)31 b Fo(1\),)k(one)g(has)g(to)g(sho)m(w)h(that)e(the)h(presence)i (probabilit)m(y)c Fn(B)2387 3499 y Fm(k)2430 3484 y Fo(\()p Fn(T)14 b Fo(\))34 b(in)g(the)h(gro)m(wing)f(rings)g(decreases)-118 3604 y(faster)g(than)g(an)m(y)g(p)s(o)m(w)m(er)h(in)d(time)h(for)g(all) f(rings)h(\()p Fn(k)f 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Fm(c)2109 5431 y Fo(.)59 b(Since)38 b(\007)f(as)h(giv)m(en)g(in) f(\(33\))g(is)g(equal)h(to)f(\(5\))-118 5551 y(\(see)d([10)o(]\),)f (the)g(theorem)f(follo)m(ws)f(directly)h(from)g(Prop)s(osition)f(1.) 1841 5800 y(10)p eop %%Page: 11 11 11 10 bop 28 28 a Fo(Note)40 b(that)f(for)h(uniform)d(Can)m(tor)j(sets) h Fn(\025)1608 43 y Fm(c)1643 28 y Fo(\()p Fn(x)p Fo(\))f(=)g(\003\()p Fn(\026)p Fo(\))f(=)g(\003)2355 43 y Fm(R)2452 28 y Fo(for)g(all)f Fn(x)p Fo(,)k(and)d(that)h Fn(x)3345 43 y Fm(\013)3435 28 y Fo(=)f(0)h(for)f(all)-118 148 y Fn(\013)28 b(>)g Fo(0.)43 b(Therefore)34 b(the)f(pro)s(of)e(of)i(Prop)s(osition)d(1)j(w) m(ould)f(already)g(b)s(e)h(concluded)g(b)m(y)h(\(24\).)493 b Fa(2)-118 483 y Fd(3.2)135 b(Large)46 b(deviation)g(b)t(ound)e(for)h (Ly)l(apuno)l(v)g(exp)t(onen)l(ts)-118 668 y Fo(In)29 b(this)f(section,)i(w)m(e)g(determine)e(the)h(rate)g(function)f(\003) 1934 632 y Fg(\003)1973 668 y Fo(\()p Fn(:;)17 b(\026)p Fo(\),)29 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Fo(so)f(that)f(\003)2816 4707 y Fg(\003)2856 4744 y Fo(\()p Fn(:;)17 b(\026)p Fo(\))40 b(as)i(giv)m(en)g(in)e(\(27\))h(is) -118 4864 y(optimal)30 b(in)h(\(7\).)43 b(This)33 b(is)f(the)h(case)h (for)e(Gibbs)g(measures)h(discussed)i(in)c(the)i(next)h(section)f([16)o (].)-118 5085 y Fe(Remark)k(9)49 b Fo(Let)32 b(us)i(set)543 5326 y(\003)611 5341 y Fs(max)802 5326 y Fo(=)55 b(sup)933 5404 y Fm(E)t Fg(2)p Fm(J)1097 5326 y Fo(lim)17 b(sup)1144 5404 y Fm(N)7 b Fg(!1)1413 5326 y Fo(\003)1481 5341 y Fm(N)1548 5326 y Fo(\()p Fn(E)f Fo(\))32 b(,)228 b(\003)2057 5341 y Fs(min)2234 5326 y Fo(=)69 b(inf)2365 5385 y Fm(E)t Fg(2)p Fm(J)2529 5326 y Fo(lim)17 b(sup)2576 5404 y Fm(N)7 b Fg(!1)2844 5326 y Fo(\003)2912 5341 y Fm(N)2979 5326 y Fo(\()p Fn(E)f Fo(\))33 b(.)-118 5551 y(Then)h(\003\()p Fn(:;)17 b(\026)p Fo(\))31 b(is)h(asymptotically)f(a\016ne:)1841 5800 y(11)p eop %%Page: 12 12 12 11 bop 901 73 a Fo(\003\()p Fn(q)t(;)17 b(\026)p Fo(\))1296 22 y Fk(\030)1249 73 y Fj(q)r Fi(#\0001)55 b 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y Fo(\))p Fk(j)p Fo(\))-118 606 y(where)d Fn(E)236 621 y Fg(\000)327 606 y Fo(is)f(the)g(negativ)m(e)f(\014xed)i(p)s(oin)m(t)e(of)g Fn(S)6 b Fo(.)-118 1041 y Fd(3.3)135 b(Generalized)46 b(Ly)l(apuno)l(v)f(exp)t(onen)l(ts)g(of)h(Gibbs)e(measures)-118 1226 y Fo(Here)31 b(w)m(e)g(calculate)d(the)j(generalized)e(Ly)m(apuno) m(v)j(exp)s(onen)m(ts)g(and)e(its)f(Legendre)i(transform)e(for)g(the)i (one-)-118 1346 y(parameter)45 b(family)e(of)j(Gibbs)f(measures)h (constructed)i(b)s(elo)m(w.)83 b(As)46 b(it)f(will)e(turn)j(out,)j(the) e(Legendre)-118 1467 y(transform)41 b(is)g(determined)h(b)m(y)h(the)f (singularit)m(y)e(sp)s(ectrum)i(of)g(the)g(maximal)d(en)m(trop)m(y)k (measure.)72 b(The)-118 1587 y(argumen)m(t)32 b(presen)m(ted)j(here)e (com)m(bines)g(results)g(from)e([4])i(and)f([18].)28 1707 y(The)45 b(pressure)g Fn(P)14 b Fo(\()p Fn(b)p Fo(\))43 b(at)h(\\in)m(v)m(erse)g(temp)s(erature")g Fn(b)g Fo(and)f(H\177)-49 b(olderian)42 b(\\in)m(teraction")g(log\()p Fk(j)p Fn(S)3597 1671 y Fg(0)3620 1707 y Fo(\()p Fn(:)p Fo(\))p Fk(j)p Fo(\))h(is)-118 1828 y(de\014ned)34 b(b)m(y)f([5,)g(16)o(])809 2081 y Fn(P)14 b Fo(\()p Fn(b)p Fo(\))55 b(=)117 b(sup)1189 2164 y Fm(\027)t Fg(2M)p Fs(\()p Fm(J)6 b Fs(\))1476 1960 y Fl(\022)1537 2081 y Fk(E)j Fo(\()p Fn(\027)d Fo(\))22 b Fk(\000)g Fn(b)1908 1964 y Fl(Z)2008 2081 y Fn(d\027)6 b Fo(\()p Fn(E)g Fo(\))33 b(log\()p Fk(j)p Fn(S)2558 2040 y Fg(0)2581 2081 y Fo(\()p Fn(E)6 b Fo(\))p Fk(j)p Fo(\))2801 1960 y Fl(\023)2911 2081 y Fo(,)-118 2323 y(where)43 b Fk(E)9 b Fo(\()p Fn(\027)d Fo(\))41 b(is)g(the)h (measure-theoretic)f(en)m(trop)m(y)i(of)e Fn(S)48 b Fo(with)41 b(resp)s(ect)i(to)e Fn(\027)48 b Fo(and)42 b Fk(M)p Fo(\()p Fn(J)9 b Fo(\))41 b(is)g(the)h(set)g(of)-118 2443 y Fn(S)6 b Fo(-in)m(v)-5 b(arian)m(t)26 b(measures)j(on)e Fn(J)9 b Fo(.)42 b(The)29 b(pressure)h Fn(P)14 b Fo(\()p Fn(b)p Fo(\))28 b(is)f(an)h(analytic,)g(con)m(v)m(ex)i(and)e(decreasing)h (function)e(of)-118 2564 y Fn(b)j Fo([16].)42 b(The)31 b(maxim)m(um)d(of)h(the)h(functional)e(on)i(the)g(righ)m(t)f(hand)h (side)g(is)f(attained)g(b)m(y)i(a)f(unique)g(in)m(v)-5 b(arian)m(t)-118 2684 y(and)35 b(ergo)s(dic)f(measure)i Fn(\026)857 2699 y Fm(b)891 2684 y Fo(,)f(called)f(the)i(Gibbs)e (measure)i(of)e Fn(b)17 b Fo(log)q(\()p Fk(j)p Fn(S)2501 2648 y Fg(0)2523 2684 y Fo(\()p Fn(:)p Fo(\))p Fk(j)p Fo(\))35 b([5].)51 b(Let)35 b(us)h(note)f(that)g(in)f(the)-118 2805 y(presen)m(t)i(con)m(text,)f(Gibbs)f(measures)h(coincide)e(with)h (equilibrium)d(measures)j([5,)g(16].)48 b(W)-8 b(e)34 b(further)g(p)s(oin)m(t)-118 2925 y(out)28 b(that)f Fn(\026)316 2940 y Fs(0)383 2925 y Fo(is)g(clearly)g(the)h(measure)g(of)g(maximal)c (en)m(trop)m(y)-8 b(,)30 b(notably)e(the)g(balanced)g(Bernoulli)d (measure.)-118 3193 y Fe(Prop)s(osition)35 b(3)49 b Fo(\003)660 3157 y Fg(\003)699 3193 y Fo(\()p Fn(:;)17 b(\026)867 3208 y Fm(b)901 3193 y Fo(\))35 b Fb(is)g(c)-5 b(onvex,)33 b(analytic)i(in)f Fo([\003)2005 3208 y Fs(min)2127 3193 y Fn(;)17 b Fo(\003)2239 3208 y Fs(max)2374 3193 y Fo(])35 b Fb(and)937 3471 y Fo(\003)1005 3430 y Fg(\003)1044 3471 y Fo(\()p Fn(\025;)17 b(\026)1242 3486 y Fm(b)1276 3471 y Fo(\))55 b(=)g Fk(\000)p Fn(\025f)1682 3486 y Fm(\026)1724 3495 y Fh(0)1781 3325 y Fl( )1856 3403 y Fo(log\()p Fn(L)p Fo(\))p 1856 3448 269 4 v 1962 3539 a Fn(\025)2135 3325 y Fl(!)2222 3471 y Fo(+)22 b Fn(\025b)h Fo(+)f Fn(P)14 b Fo(\()p Fn(b)p Fo(\))35 b Fb(.)926 b Fo(\(29\))-118 3748 y Fe(Pro)s(of.)42 b Fo(F)-8 b(or)28 b(the)i(Gibbs)f(measure)g Fn(\026)1278 3763 y Fm(b)1312 3748 y Fo(,)h(the)g(generalized)f(Ly)m(apuno)m(v)h(exp)s(onen)m(ts)i (can)d(b)s(e)h(calculated)e(from)-118 3869 y(the)33 b(pressure)h(b)m(y) g(the)f(form)m(ula)e([18)o(])1242 4109 y(\003\()p Fn(q)t(;)17 b(\026)1498 4124 y Fm(b)1532 4109 y Fo(\))55 b(=)g Fn(P)14 b Fo(\()p Fn(b)22 b Fk(\000)h Fn(q)t Fo(\))f Fk(\000)g Fn(P)14 b Fo(\()p Fn(b)p Fo(\))33 b(.)1230 b(\(30\))-118 4281 y(In)33 b(order)g(to)f(calculate)f(the)i(Legendre)h(transform,)e (let)f(us)j(in)m(tro)s(duce)e(the)h(Ly)m(apuno)m(v)h(sp)s(ectrum)879 4534 y Fn(`)p Fo(\()p Fn(\025)p Fo(\))55 b(=)g(dim)1402 4549 y Fm(H)1486 4413 y Fl(\022\032)1609 4534 y Fn(E)34 b Fk(2)28 b Fn(J)1905 4409 y Fl(\014)1905 4459 y(\014)1905 4509 y(\014)1905 4559 y(\014)1984 4534 y Fo(lim)1950 4593 y Fm(N)7 b Fg(!1)2170 4534 y Fo(\003)2238 4549 y Fm(N)2305 4534 y Fo(\()p Fn(E)f Fo(\))28 b(=)g Fn(\025)2657 4413 y Fl(\033)q(\023)2830 4534 y Fo(.)-118 4743 y(Note)35 b(that,)h(for)e Fn(\025)44 b(=)-61 b Fk(2)33 b Fo([\003)796 4758 y Fs(min)917 4743 y Fn(;)17 b Fo(\003)1029 4758 y Fs(max)1164 4743 y Fo(],)36 b Fn(`)p Fo(\()p Fn(\025)p Fo(\))c(=)g Fk(\0001)j Fo(b)s(ecause)i(the)e(Hausdor\013)h(dimension)d (of)i(an)g(empt)m(y)h(set)-118 4863 y(is)c(set)h(to)f Fk(\0001)p Fo(.)44 b(No)m(w)33 b([4,)g(Theorems)g(1)f(and)h(2])1289 5104 y Fk(j)p Fn(\025)p Fk(j)p Fn(`)p Fo(\()p Fn(\025)p Fo(\))54 b(=)65 b(inf)1761 5164 y Fm(b)p Fg(2)p Fc(R)1899 5104 y Fo(\()p Fn(b\025)23 b Fo(+)f Fn(P)14 b Fo(\()p Fn(b)p Fo(\)\))32 b(.)1277 b(\(31\))-118 5311 y(Hence)34 b(w)m(e)f(obtain)f(from)f(\(30\))h(and)h(\(31\))f(that)1099 5551 y(\003)1167 5510 y Fg(\003)1206 5551 y Fo(\()p Fn(\025;)17 b(\026)1404 5566 y Fm(b)1438 5551 y Fo(\))55 b(=)g Fk(\000j)p Fn(\025)p Fk(j)p Fo(\()p Fn(`)p Fo(\()p Fn(\025)p Fo(\))22 b(+)g Fn(b)p Fo(\))h(+)f Fn(P)14 b Fo(\()p Fn(b)p Fo(\))32 b(.)1841 5800 y(12)p eop %%Page: 13 13 13 12 bop -118 22 a Fo(As)39 b(the)g(Ly)m(apuno)m(v)h(sp)s(ectrum)f(is) f(link)m(ed)g(to)g(the)h(singularit)m(y)e(sp)s(ectrum)i Fn(f)2760 37 y Fm(\026)2802 46 y Fh(0)2880 22 y Fo(of)f(the)h(maximal)c (en)m(trop)m(y)-118 142 y(measure)e Fn(\026)321 157 y Fs(0)392 142 y Fo(b)m(y)h Fn(`)p Fo(\()p Fn(\025)p Fo(\))27 b(=)h Fn(f)881 157 y Fm(\026)923 166 y Fh(0)962 142 y Fo(\(log\()p Fn(L)p Fo(\))p Fn(=\025)p Fo(\),)33 b(the)g(pro)s(of)e(is) h(concluded.)1382 b Fa(2)-118 479 y Fd(3.4)135 b(Pro)t(of)45 b(of)g(Theorem)g(1)h(and)e(generalizations)-118 663 y Fo(In)j(this)g(section,)k(w)m(e)d(state)f(and)g(pro)m(v)m(e)h(the)g (most)e(general)g(statemen)m(t)i(that)f(can)g(b)s(e)g(deduced)i(from) -118 784 y(Prop)s(osition)23 b(1)i(and)g(the)h(b)s(ound)f(in)g(Hyp)s (othesis)h(I)s(I)f(as)g(pro)m(v)m(en)i(in)d([10].)41 b(When)26 b(applied)e(to)h(Julia)e(matrices,)-118 904 y(this)32 b(then)h(also)f(pro)m(v)m(es)i(Theorem)f(1.)28 1024 y(In)f(order)f(to)g(state)h(the)g(general)e(result,)i(w)m(e)g (need)g(to)f(in)m(tro)s(duce)h(a)f(constan)m(t)h(dep)s(ending)f(only)g (on)g(the)-118 1145 y(DOS)e(of)g Fn(H)8 b Fo(.)42 b(F)-8 b(or)29 b(a)h(Jacobi)f(matrix)f Fn(H)37 b Fo(as)30 b(in)m(tro)s(duced)g (in)f(Section)h(2.2,)g(the)g(DOS)f Fk(N)44 b Fo(of)29 b Fn(H)38 b Fo(is)29 b(the)h(unique)-118 1265 y(w)m(eak)39 b(limit)34 b(p)s(oin)m(t)j(of)g(the)h(sequence)j(of)c(pure-p)s(oin)m(t) g(measures)h(\()2387 1199 y Fl(P)2475 1286 y Fm(E)t(;)11 b(P)2607 1294 y Fj(n)2649 1286 y Fs(\()p Fm(E)t Fs(\)=0)2887 1265 y Fn(\016)2930 1280 y Fm(E)2990 1265 y Fn(=n)p Fo(\))3135 1280 y Fm(n)p Fg(2)p Fc(N)3330 1265 y Fo([21)o(,)38 b(10].)59 b(The)-118 1386 y(capacit)m(y)33 b(of)f Fn(J)42 b Fo(can)34 b(b)s(e)f(calculated)f(as)h(cap\()p Fn(J)9 b Fo(\))28 b(=)h(exp)q(\(lim)2108 1401 y Fm(n)p Fg(!1)2313 1319 y Fl(P)2400 1345 y Fm(n)2400 1410 y(j)t Fs(=1)2544 1386 y Fo(log\()p Fn(t)2743 1401 y Fm(j)2779 1386 y Fo(\))p Fn(=n)p Fo(\))k([21,)g(10)o(].)45 b(The)34 b(Green's)-118 1506 y(function)e(of)g Fn(J)42 b Fo(de\014ned)34 b(b)m(y)911 1747 y Fn(g)958 1762 y Fm(J)1007 1747 y Fo(\()p Fn(z)t Fo(\))56 b(=)1319 1630 y Fl(Z)1418 1747 y Fn(d)p Fk(N)15 b Fo(\()p Fn(E)6 b Fo(\))33 b(log\()p Fk(j)p Fn(z)26 b Fk(\000)d Fn(E)6 b Fk(j)p Fo(\))22 b Fk(\000)g Fo(log\(cap)q(\()p Fn(J)9 b Fo(\)\))-118 1945 y(go)m(v)m(erns)32 b(the)e(asymptotic)g (prop)s(erties)g(of)f(the)i(orthogonal)d(p)s(olynomials)f(b)m(y)k (means)g(of)e(a)h(Herb)s(ert-Jones-)-118 2066 y(Thouless)i(t)m(yp)s(e)g (form)m(ula)d([21,)i(10)o(].)43 b(Both)31 b Fk(N)46 b Fo(and)31 b(cap\()p Fn(J)9 b Fo(\))31 b(and)h(hence)g Fn(g)2617 2081 y Fm(J)2697 2066 y Fo(do)f(not)g(dep)s(end)h(on)f(the)g (c)m(hoice)-118 2186 y(of)h Fn(\026)p Fo(,)g(but)h(only)f(on)h(its)f (supp)s(ort)h Fn(J)9 b Fo(.)28 2307 y(Let)34 b(no)m(w)g Fn(E)480 2322 y Fm(c)548 2307 y Fo(and)g Fn(R)813 2322 y Fm(c)877 2307 y Fo(=)29 b Fk(j)p Fn(I)1061 2270 y Fs(0)1100 2307 y Fk(j)p Fn(=)p Fo(2)j(b)s(e)i(resp)s(ectiv)m(ely)g(the)g(cen)m (ter)h(of)e(the)h(sp)s(ectrum)g(and)f(its)g(radius,)g(and)-118 2427 y(\001)g(the)g(size)g(of)f(the)h(smallest)e(gap)h(at)g(the)h (\014rst)g(generation.)43 b(If)33 b Fn(S)38 b Fo(is)32 b(an)g(analytic)g(map,)g(then)h(w)m(e)g(set)1204 2698 y(\007)55 b(=)93 b(inf)1467 2757 y Fm(R>R)1628 2765 y Fj(c)1703 2630 y Fo(max)1885 2645 y Fm(z)s Fg(2)p Fs(\000)2012 2656 y Fj(R)2110 2630 y Fn(g)2157 2645 y Fm(J)2206 2630 y Fo(\()p Fn(S)6 b Fo(\()p Fn(z)t Fo(\)\))p 1703 2675 770 4 v 1734 2766 a(log\()p Fn(R)q Fo(\))22 b Fk(\000)g Fo(log\()p Fn(R)2370 2781 y Fm(c)2405 2766 y Fo(\))2516 2698 y(,)1181 b(\(32\))-118 2922 y(where)34 b(\000)225 2937 y Fm(R)315 2922 y Fo(is)f(the)g(circle)f(of)g(radius)g Fn(R)i Fo(around)f Fn(E)1752 2937 y Fm(c)1787 2922 y Fo(.)44 b(If)33 b(all)d(branc)m(hes)35 b Fn(S)2556 2937 y Fm(l)2582 2922 y Fo(,)e Fn(l)d Fo(=)e(1)p Fn(;)17 b(:)g(:)g(:)e(;)i (L)p Fo(,)33 b(ha)m(v)m(e)h(an)f(analytic)-118 3043 y(con)m(tin)m (uation)462 3018 y(^)446 3043 y Fn(S)506 3058 y Fm(l)564 3043 y Fo(giv)m(en)g(b)m(y)g(a)g(p)s(olynomial)c(of)j(degree)h Fn(D)2035 3058 y Fm(l)2061 3043 y Fo(,)g(then)g(w)m(e)h(p)s(ose)711 3358 y(\007)55 b(=)983 3271 y(max)1165 3286 y Fm(l)q Fs(=1)p Fm(:::)o(L)1432 3271 y Fo(sup)1579 3295 y Fm(E)t Fg(2)p Fm(J)1775 3271 y Fn(g)1822 3286 y Fm(J)1870 3271 y Fo(\()1925 3246 y(^)1908 3271 y Fn(S)1968 3286 y Fm(l)1994 3271 y Fo(\()p Fn(E)6 b Fo(\)\))22 b(+)g Fn(D)d Fo(arcsinh)2727 3175 y Fl(\020)2817 3232 y Fs(\001)p 2787 3248 121 4 v 2787 3306 a(4)p Fm(R)2875 3314 y Fj(c)2917 3175 y Fl(\021)p 983 3334 1984 4 v 1695 3444 a Fo(arcsinh)2015 3348 y Fl(\020)2106 3405 y Fs(\001)p 2075 3421 121 4 v 2075 3478 a(4)p Fm(R)2163 3486 y Fj(c)2205 3348 y Fl(\021)3009 3358 y Fo(,)688 b(\(33\))-118 3625 y(where)34 b Fn(D)c Fo(=)e(max)560 3640 y Fm(l)603 3625 y Fn(D)684 3640 y Fm(l)710 3625 y Fo(.)-118 3853 y Fe(Theorem)37 b(3)49 b Fb(L)-5 b(et)47 b Fn(H)55 b Fb(b)-5 b(e)46 b(the)h(Jac)-5 b(obi)47 b(matrix)f(of)h(a)g(Gibbs)f(me)-5 b(asur)g(e)47 b Fn(\026)2674 3868 y Fm(b)2755 3853 y Fb(on)g(a)f(self-similar)g(fr)-5 b(actal)46 b Fn(J)-118 3974 y Fb(with)e Fn(L)g Fb(br)-5 b(anches)43 b(c)-5 b(onstructe)g(d)44 b(with)g(an)g(analytic)g(or)g (pie)-5 b(c)g(ewise)42 b(p)-5 b(olynomial)43 b(map)g Fn(S)6 b Fb(.)73 b(L)-5 b(et)44 b Fo(\007)g Fb(b)-5 b(e)44 b(the)-118 4094 y(c)-5 b(orr)g(esp)g(onding)33 b(c)-5 b(onstant)35 b(given)f(in)g Fo(\(32\))h Fb(or)f Fo(\(33\))h Fb(and)f(let)h(us)g(set)1054 4369 y Fn(\013)1116 4384 y Fm(c)1206 4369 y Fo(=)1347 4302 y Fn(P)14 b Fo(\()p Fn(b)p Fo(\))22 b Fk(\000)g Fo(log\()p Fn(L)p Fo(\))17 b(\()p Fn(\034)2027 4317 y Fm(\026)2069 4326 y Fh(0)2109 4302 y Fo(\))2147 4266 y Fg(\000)p Fs(1)2241 4302 y Fo(\()p Fk(\000)p Fo(2)22 b Fk(\000)h Fn(b)p Fo(\))p 1347 4346 1260 4 v 1838 4438 a(log\(\007\))2651 4369 y Fb(.)-118 4599 y(Then,)34 b(for)g Fn(\013)29 b Fk(2)f Fo(\(0)p Fn(;)17 b(\013)699 4614 y Fm(c)733 4599 y Fo(])p Fb(,)534 4878 y Fn(\014)595 4836 y Fs(+)589 4902 y Fm(\013)709 4878 y Fk(\024)852 4810 y Fo(log\()p Fn(L)p Fo(\))22 b Fk(\000)h Fn(P)14 b Fo(\()p Fn(b)p Fo(\))21 b(+)h Fn(\013)c Fo(log\(\007\))p 852 4854 1061 4 v 1208 4946 a Fn(\013)g Fo(log\()p Fn(L)p Fo(\))1939 4878 y Fn(D)2020 4893 y Fm(\026)2062 4902 y Fh(0)2118 4731 y Fl( )2194 4810 y Fn(P)c Fo(\()p Fn(b)p Fo(\))21 b Fk(\000)i Fn(\013)17 b Fo(log\(\007\))p 2194 4854 673 4 v 2396 4946 a(log\()p Fn(L)p Fo(\))2876 4731 y Fl(!)2964 4878 y Fk(\000)3084 4810 y Fn(b)p 3074 4854 63 4 v 3074 4946 a(\013)3181 4878 y Fb(,)513 b Fo(\(34\))-118 5109 y Fb(and,)34 b(for)h Fn(\013)28 b Fk(\025)g Fn(\013)514 5124 y Fm(c)549 5109 y Fb(,)1452 5369 y Fn(\014)1513 5328 y Fs(+)1507 5394 y Fm(\013)1627 5369 y Fk(\024)56 b Fo(1)22 b Fk(\000)1940 5302 y Fn(\013)2002 5317 y Fm(c)2059 5302 y Fk(\000)h Fo(2)p 1940 5346 268 4 v 2043 5437 a Fn(\013)2253 5369 y Fb(.)1841 5800 y Fo(13)p eop %%Page: 14 14 14 13 bop -118 22 a Fe(Remark)37 b(10)49 b Fo(The)f(b)s(ound)f(dep)s (ends)h(on)f(the)h(LDOS)e(through)h(the)h(parameter)e Fn(b)p Fo(,)51 b(on)c(the)h(maximal)-118 142 y(en)m(trop)m(y)34 b(measure)f Fn(\026)676 157 y Fs(0)748 142 y Fo(through)f(its)h (generalized)f(dimensions)f Fn(D)2337 157 y Fm(\026)2379 166 y Fh(0)2451 142 y Fo(and)i(on)g(the)g(DOS)f(through)h(the)g(con-) -118 263 y(stan)m(t)38 b(\007.)57 b(Note)37 b(that,)h(in)f(the)g(limit) d Fn(\013)i Fk(!)f(1)p Fo(,)j(our)f(upp)s(er)h(b)s(ound)g(con)m(v)m (erges)h(to)e(the)g Ff(a)g(priori)f Fo(ballistic)-118 383 y(b)s(ound)d Fn(\014)244 347 y Fs(+)238 408 y Fm(\013)330 383 y Fk(\024)28 b Fo(1.)43 b(In)33 b(the)g(limit)c Fn(\013)f Fk(!)f Fo(0,)33 b(w)m(e)g(reco)m(v)m(er)h(the)f(b)s(ound)g(obtained)f (in)f(ref.)44 b([10)o(].)g(As)33 b(in)e([10],)i(it)e(is)-118 504 y(not)i Ff(a)g(priori)e Fo(clear)i(that)g(the)h(b)s(ound)g(\(34\))e (is)h(actually)f(b)s(etter)i(than)f(ballistic.)43 b(This)33 b(has)h(to)f(b)s(e)g(c)m(hec)m(k)m(ed)-118 624 y(in)i(concrete)j (situations)d(just)i(as)g(w)m(as)g(done)g(in)e(Remarks)i(1)f(and)g(2.) 55 b(F)-8 b(urthermore,)37 b(one)f(has)h(to)f(con)m(trol)-118 744 y Fn(\013)-56 759 y Fm(c)11 744 y Fo(b)m(y)e(using)e(the)h (estimates)f(in)g(Remark)g(5.)-118 1007 y Fe(Pro)s(of)c Fo(of)g(Theorem)g(3.)42 b(First)27 b(of)h(all,)f(it)g(follo)m(ws)g (from)g(Section)h(3.4)g(of)f([10])h(that)g(Hyp)s(othesis)h(I)s(I)f (holds)g(for)-118 1127 y(all)33 b Fn(\021)i(>)c Fo(0)j(with)h(\007)f (as)h(giv)m(en)g(in)f(\(32\))g(and)h(\(33\).)50 b(Prop)s(osition)33 b(3)h(and)h(the)g(de\014nition)f(of)g Fn(\034)3337 1142 y Fm(\026)3379 1151 y Fh(0)3454 1127 y Fo(then)h(allo)m(w)-118 1248 y(to)d(calculate)g(\003)476 1263 y Fm(R)566 1248 y Fo(and)g(then)h Fn(\013)1039 1263 y Fm(c)1074 1248 y Fo(.)44 b(Similarly)-8 b(,)28 b(for)k(the)h(calculation)d(of)i Fn(\024)2555 1263 y Fm(\013)2605 1248 y Fo(,)h(one)g(obtains)f(from)f (\(10\))h(that)-11 1652 y Fn(f)48 1611 y Fg(0)37 1676 y Fm(\026)79 1685 y Fh(0)135 1506 y Fl( )211 1584 y Fo(log\()p Fn(L)p Fo(\))p 211 1628 269 4 v 298 1720 a Fn(z)343 1735 y Fm(\013)489 1506 y Fl(!)610 1652 y Fo(=)752 1584 y Fk(\000)p Fn(\013)17 b Fo(log)g(\(\007\))22 b(+)g Fn(P)14 b Fo(\()p Fn(b)p Fo(\))p 752 1628 766 4 v 1000 1720 a(log\()p Fn(L)p Fo(\))1559 1652 y(,)228 b Fn(\024)1870 1667 y Fm(\013)1975 1652 y Fo(=)2116 1564 y Fk(\000)2203 1518 y Fg(\000)p Fm(\013)12 b Fs(log)r(\(\007\)+)p Fm(P)e Fs(\()p Fm(b)p Fs(\))p 2203 1541 508 4 v 2360 1599 a(log)q(\()p Fm(L)p Fs(\))2731 1518 y(log)q(\()p Fm(L)p Fs(\))p 2731 1541 195 4 v 2789 1599 a Fm(z)2822 1607 y Fj(\013)2957 1564 y Fo(+)22 b Fn(f)3103 1579 y Fm(\026)3145 1588 y Fh(0)3201 1468 y Fl(\020)3260 1518 y Fs(log)q(\()p Fm(L)p Fs(\))p 3260 1541 V 3319 1599 a Fm(z)3352 1607 y Fj(\013)3464 1468 y Fl(\021)3536 1564 y Fk(\000)h Fn(b)p 2116 1628 1562 4 v 2865 1720 a(\013)3720 1652 y Fo(.)-118 1917 y(By)33 b(de\014nition)f(of)g(the)h(m)m(ultifractal)c(dimensions,)j (this)g(implies)e(directly)i(the)h(result.)707 b Fa(2)-118 2085 y Fe(Pro)s(of)31 b Fo(of)f(Theorem)h(1.)43 b(F)-8 b(or)30 b(Julia)g(sets,)i(log\(\007\))c(=)f(log\()p Fn(L)p Fo(\))k(as)h(follo)m(ws)d(directly)i(from)f(\(32\))g([10],)h(and)g(for) -118 2205 y(the)g(maximal)d(en)m(trop)m(y)k(measure,)f Fn(b)d Fo(=)g(0)i(and)h Fn(P)14 b Fo(\(0\))27 b(=)h Fk(E)9 b Fo(\()p Fn(\026)2135 2220 y Fs(0)2173 2205 y Fo(\))28 b(=)f(log\()p Fn(L)p Fo(\).)43 b(Therefore)32 b(Theorem)f(1)g(follo)m (ws)-118 2325 y(directly)h(from)f(Theorem)i(3.)2868 b Fa(2)-118 2727 y Fp(References)28 2946 y Fo([1])49 b(J.-M.)27 b(Barbaroux,)i Ff(Dynamique)d(Quan)m(tique)h(des)i(Milieux)c(D)m(\023) -46 b(esordonn)m(\023)g(es)p Fo(,)30 b(PhD)d(Thesis,)i(T)-8 b(oulon,)180 3067 y(1997.)28 3267 y([2])49 b(J.-M.)27 b(Barbaroux,)h(J.M.)g(Com)m(b)s(es,)g(R.)f(Mon)m(tc)m(ho,)i Ff(Remarks)e(on)g(the)g(relation)e(b)s(et)m(w)m(een)k(quan)m(tum)180 3387 y(dynamics)j(and)h(fractal)e(sp)s(ectra)p Fo(,)j(J.)e(Math.)h (Anal.)f(and)h(Appl.)f(213,)g(698-722)f(\(1997\).)28 3587 y([3])49 b(J.)41 b(Bellissard,)g Ff(Stabilit)m(y)e(and)i (instabilit)m(y)e(in)h(quan)m(tum)h(mec)m(hanics)p Fo(,)i(in)d Ff(T)-8 b(rends)42 b(and)f(dev)m(elop-)180 3708 y(men)m(ts)h(in)f(the)h (eigh)m(ties)p Fo(,)h(Eds.)g(S)f(Alb)s(ev)m(erio,)h(Ph.)f(Blanc)m (hard,)i(\(W)-8 b(orld)41 b(Scien)m(ti\014c,)j(Singap)s(ore,)180 3828 y(1985\).)28 4028 y([4])49 b(T.)41 b(Bohr,)h(D.)e(Rand,)j Ff(The)e(en)m(trop)m(y)h(function)e(for)g(c)m(haracteristic)g(exp)s (onen)m(ts)p Fo(,)45 b(Ph)m(ysica)c Fe(D)46 b(25)p Fo(,)180 4149 y(387-398,)31 b(\(1987\).)28 4349 y([5])49 b(R.)24 b(Bo)m(w)m(en,)k Ff(Equilibrium)21 b(States)k(and)g(the)g(Ergo)s(dic)e (Theory)j(of)e(Anoso)m(v)h(Di\013eomorphisms)p Fo(,)f(Lect.)180 4469 y(Notes)33 b(Math.)g Fe(470)p Fo(,)g(\(1975\).)28 4669 y([6])49 b(A.)32 b(Dem)m(b)s(o,)f(O.)g(Zeitouni,)f Ff(Large)h(Deviation)f(T)-8 b(ec)m(hniques)34 b(and)d(Applications)p Fo(,)f(\(Jones)j(and)e(Bar-)180 4790 y(lett,)h(Boston,)h(London,)f (1993\).)28 4990 y([7])49 b(I.)d(Guarneri,)i Ff(Sp)s(ectral)d(prop)s (erties)h(of)g(quan)m(tum)g(di\013usion)e(on)i(discrete)g(lattices)p Fo(,)j(Europh)m(ys.)180 5110 y(Lett.,)32 b Fe(10)p Fo(,)g(95-100,)f (\(1989\);)g Ff(On)g(an)h(estimate)f(concerning)h(quan)m(tum)g (di\013usion)e(in)h(the)h(presence)180 5231 y(of)g(a)g(fractal)g(sp)s (ectrum)p Fo(,)h(Europh)m(ys.)h(Lett.,)f Fe(21)p Fo(,)g(729-733,)e (\(1993\).)28 5431 y([8])49 b(I.)24 b(Guarneri,)g(G.)g(Man)m(tica,)h Ff(Multifractal)c(Energy)j(Sp)s(ectra)h(and)e(Their)h(Dynamical)d (Implications)p Fo(,)180 5551 y(Ph)m(ys.)34 b(Rev.)g(Lett.)e Fe(73)p Fo(,)h(3379-3383)d(\(1994\).)1841 5800 y(14)p eop %%Page: 15 15 15 14 bop 28 28 a Fo([9])49 b(I.)37 b(Guarneri,)h Ff(Singular)d(con)m (tin)m(uous)j(sp)s(ectra)g(and)f(discrete)h(w)m(a)m(v)m(e)h(pac)m(k)m (et)g(dynamics)p Fo(,)f(J.)f(Math.)180 148 y(Ph)m(ys.)d Fe(37)p Fo(,)f(5195-5206,)d(\(1996\).)-21 351 y([10])49 b(I.)29 b(Guarneri,)g(H.)g(Sc)m(h)m(ulz-Baldes,)h Ff(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) 180 472 y(matrices)j(with)g(self-similar)c(sp)s(ectra)p Fo(,)34 b(to)e(app)s(ear)g(in)g(Rev.)i(Math.)e(Ph)m(ys..)-21 675 y([11])49 b(H.)36 b(Hiramoto,)e(S.)i(Ab)s(e,)h Ff(Dynamics)e(of)g (an)h(Electron)g(in)f(Quasip)s(erio)s(dic)e(Systems.)k(I.)f(Fib)s (onacci)180 796 y(Mo)s(del)p Fo(,)31 b(J.)h(Ph)m(ys.)i(So)s(c.)d(Japan) h Fe(57)p Fo(,)g(230-240)e(\(1988\);)h(and)h Ff(Dynamics)e(of)i(an)f (Electron)h(in)f(Quasi-)180 916 y(p)s(erio)s(dic)g(Systems.)j(I)s(I.)e (Harp)s(er)h(Mo)s(del)p Fo(,)f(J.)h(Ph)m(ys.)h(So)s(c.)f(Japan)f Fe(57)p Fo(,)h(1365-1372)d(\(1988\).)-21 1119 y([12])49 b(Y.)25 b(Last,)i Ff(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 Fo(,)i(J.)d(F)-8 b(unct.)180 1240 y(Anal.)32 b Fe(142)p Fo(,)h(402-445)d(\(1996\).)-21 1443 y([13])49 b(G.)27 b(Man)m(tica,)i Ff(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 1563 y(tral)32 b(prop)s(erties)p Fo(,)h(Ph)m(ysica)g Fe(D)38 b(103)p Fo(,)33 b(576-589,)e(\(1997\);)i Ff(W)-8 b(a)m(v)m(e)34 b(Propagation)d(in)h(Almost-P)m(erio)s(dic)180 1684 y(Structures)p Fo(,)i(Ph)m(ysica)g Fe(D)j(109)p Fo(,)c(113-127)d(\(1997\).)-21 1887 y([14])49 b(D.)22 b(Ma)m(y)m(ou,)j Ff(In)m(tro)s(duction)d(to)g(the)h(theory)g(of)f (electronic)g(prop)s(erties)g(of)g(quasicrystals)p Fo(,)j(in)c Ff(Lectures)180 2008 y(on)40 b(quasicrystals)p Fo(,)k(edited)c(b)m(y)i (F.)e(Hipp)s(ert,)j(D.)d(Gratias)f(417-462,)i(Les)d(\023)-46 b(editions)40 b(de)h(ph)m(ysique,)180 2128 y(Les)33 b(Ulis,)f (\(1994\).)-21 2331 y([15])49 b(F.)33 b(Pi)m(\023)-46 b(ec)m(hon,)34 b Ff(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 Fo(,)180 2452 y(Ph)m(ys.)j(Rev.)g(Lett.)e Fe(76)p Fo(,)h(4372-4375)d (\(1996\).)-21 2655 y([16])49 b(D.)32 b(Ruelle,)g Ff(Thermo)s(dynamic)f (F)-8 b(ormalism)p Fo(,)29 b(\(Addison-W)-8 b(esley)g(,)33 b(Reading,)f(MA,)h(1978\).)-21 2859 y([17])49 b(H.)31 b(Sc)m(h)m(ulz-Baldes,)h(J.)g(Bellissard,)e Ff(Anomalous)g(transp)s (ort:)42 b(a)31 b(mathematical)d(framew)m(ork)p Fo(,)k(Rev.)180 2979 y(Math.)47 b(Ph)m(ys.)h Fe(10)p Fo(,)j(1-46)45 b(\(1998\);)53 b(H.)47 b(Sc)m(h)m(ulz-Baldes,)j(J.)d(Bellissard,)i Ff(A)e(Kinetic)e (Theory)j(for)180 3099 y(Quan)m(tum)32 b(T)-8 b(ransp)s(ort)33 b(in)f(Ap)s(erio)s(dic)f(Media)p Fo(,)h(J.)h(Stat.)f(Ph)m(ys.)j Fe(91)p Fo(,)e(991-1027,)d(\(1998\).)-21 3303 y([18])49 b(G.)32 b(Servizi,)f(G.)h(T)-8 b(urc)m(hetti,)33 b(S.)f(V)-8 b(aien)m(ti,)31 b Ff(Generalized)h(Dynamical)d(V)-8 b(ariables)31 b(and)h(Measures)i(for)180 3423 y(the)f(Julia)e(Sets)p Fo(,)i(Nuo)m(v)m(o)h(Cimen)m(to)e Fe(101)37 b(B)p Fo(,)32 b(285-307)f(\(1988\).)-21 3627 y([19])49 b(C.)31 b(Sire,)f(B.)g(P)m (assaro,)i(V.)e(Benza,)h Ff(Anomalous)e(di\013usion)g(and)h (conductivit)m(y)h(in)e(o)s(ctagonal)f(tiling)180 3747 y(mo)s(dels)p Fo(,)j(Ph)m(ys.)k(Rev.,)e Fe(B46)p Fo(,)g(13751-13754,)c (\(1992\).)-21 3950 y([20])49 b(A.)32 b(S)s(\177)-51 b(ut\023)i(o,)31 b Ff(The)i(sp)s(ectrum)f(of)f(a)h(quasip)s(erio)s(dic) e(Sc)m(hr\177)-49 b(odinger)32 b(op)s(erator)p Fo(,)f(Comm)m(un.)h (Math.)g(Ph)m(ys.)180 4071 y Fe(111)p Fo(,)d(409-415)e(\(1987\);)i(and) f Ff(Singular)f(Con)m(tin)m(uous)i(Sp)s(ectrum)g(on)f(a)g(Can)m(tor)h (Set)g(of)f(Zero)g(Leb)s(es-)180 4191 y(gue)33 b(Measure)h(for)e(the)h (Fib)s(onacci)e(Hamiltonian)p Fo(,)e(J.)j(Stat.)h(Ph)m(ys.)h Fe(56)p Fo(,)f(525-530)e(\(1989\).)-21 4395 y([21])49 b(H.)23 b(Stahl,)g(V.)g(T)-8 b(otik,)24 b Ff(General)e(Orthogonal)f(P)m (olynomials)p Fo(,)h(\(Cam)m(bridge)g(Univ)m(ersit)m(y)h(Press,)k(Cam-) 180 4515 y(bridge,)32 b(1992\))1841 5800 y(15)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF