Content-Type: multipart/mixed; boundary="-------------0106050706468" This is a multi-part message in MIME format. ---------------0106050706468 Content-Type: text/plain; name="01-208.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-208.keywords" conditionally invariant probability measure dynamical system with hole Hausdorff dimension ---------------0106050706468 Content-Type: application/postscript; name="holes0601.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="holes0601.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: holes0601.dvi %%Pages: 26 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips holes0601 -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.06.05:1330 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S 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b(.Collet)24 b(for)f(suggesting)i(the)g(relev)l (ance)g(of)f([KL])f(to)i(the)g(presen)n(t)g(setting)g(and)f(G.Keller,)f (B.)h(Sc)n(hmitt)g(and)456 5216 y(J-P)-6 b(.)23 b(Thouv)n(enot)i(for)e (useful)h(discussions.)1933 5315 y Fl(1)p eop %%Page: 2 2 2 1 bop 456 251 a Fl(2)388 b(CARLANGELO)23 b(LIVERANI)f(AND)g(V)1947 236 y(\023)1941 251 y(ER)n(ONIQUE)g(MA)n(UME-DESCHAMPS)456 450 y Fs(sub-in)n(terv)-5 b(al)27 b(of)i Fq(I)35 b Fs(and)28 b Fq(g)1281 420 y Fl(0)1371 450 y Fs(:)53 b Fq(I)60 b Fn(\000)-14 b(!)53 b Fo(R)1784 420 y Fl(+)1845 450 y Fs(,)29 b(inf)21 b Fq(g)2055 420 y Fl(0)2116 450 y Fq(>)j Fs(0,)k(b)r(e)h(a)f(p)r(oten)n(tial)g(whic)n(h)h(b)r(elongs)f(to)456 550 y(BV)f(and)h(admits)f(a)h(conformal)e(measure)g Fq(m)i Fs(\(see)g(de\014nition)g(and)f(assumptions)g(b)r(elo)n(w\).)3349 518 y Fl(1)555 649 y Fs(Some)d(results)g(ha)n(v)n(e)f(b)r(een)h (obtained)g(for)g(suc)n(h)g(maps)f(with)i(holes,)f(limited)h(to)f(the)h (case)e(in)456 749 y(whic)n(h)30 b(the)g(p)r(oten)n(tial)g(is)g(giv)n (en)f(b)n(y)h(the)h(Jacobian)d(of)i(the)h(map,)f(in)h([C])f(and)g([BC]) g(for)g(v)n(ery)456 849 y(small)g(holes)g(and)h(under)g(some)f (additional)h(geometrical)e(assumption)h(on)h(the)g(holes.)46 b(Our)456 948 y(goal)21 b(here)i(is)g(on)g(the)g(one)g(hand)g(to)g (\014nd)h(constructiv)n(e)e(conditions)g(allo)n(wing)g(not)h (necessarily)456 1048 y(small)k(holes)g(and)g(on)h(the)g(other)f(to)g (sho)n(w)g(that)h(a)f(smallness)g(condition)g(alone)g(su\016ces.)456 1147 y(The)i(plan)h(of)f(the)h(pap)r(er)f(is)h(as)f(follo)n(ws.)41 b(In)30 b(section)f(one)g(some)g(general)f(facts)i(are)e(recalled)456 1247 y(and)37 b(the)g(main)h(theorems)e(pro)n(v)n(ed)g(in)h(the)h(pap)r (er)f(are)f(stated.)66 b(Section)37 b(t)n(w)n(o)g(is)g(dev)n(oted)456 1347 y(to)d(obtaining)f(a)h(sp)r(ecial)g(t)n(yp)r(e)h(of)f(Lasota-Y)-7 b(ork)n(e)31 b(lik)n(e)j(inequalit)n(y)g(that)g(will)h(b)r(e)g(the)f (basis)456 1446 y(for)28 b(future)i(argumen)n(ts.)40 b(Section)29 b(three)g(uses)g(the)h(previous)e(results)h(to)g (establish)g(that)g(the)456 1546 y(transfer)e(op)r(erator)f(is)i(a)f (con)n(traction)g(in)h(an)f(appropriate)g(\(pro)5 b(jectiv)n(e\))27 b(metric.)38 b(F)-7 b(rom)27 b(this)456 1646 y(results)g(the)g(w)n(an)n (ted)g(statistical)g(prop)r(erties)g(readily)f(follo)n(ws)h(as)g(is)g (sho)n(wn)g(in)h(section)f(four.)456 1745 y(Section)f(\014v)n(e)h(in)n (v)n(estigate)e(the)j(Hausdor\013)e(dimension)h(of)f(the)i(set)e(of)h (the)h(p)r(oin)n(ts)e(that)h(nev)n(er)456 1845 y(visit)f(the)g(hole.)36 b(In)26 b(section)g(six)g(w)n(e)f(in)n(v)n(estigate)g(man)n(y)h (concrete)f(examples)g(and)h(sho)n(w)f(that)456 1944 y(the)k(theory)f(so)g(far)g(dev)n(elop)r(ed)h(do)r(es)f(apply)h(to)f (maps)h(with)g(fairly)f(large)g(holes)g(ev)n(en)g(in)h(the)456 2044 y(absence)38 b(of)g(a)h(Mark)n(o)n(v)d(structure.)70 b(Finally)-7 b(,)42 b(section)c(sev)n(en)g(p)r(oin)n(ts)h(out)g(that)g (if)g(one)g(is)456 2144 y(concerned)24 b(only)h(with)h(p)r(ertubativ)n 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b(\014nite)h(n)n(um)n(b)r(er)g(of)f(p) r(oin)n(ts\))h(of)f Fq(I)33 b Fs(on)25 b(subin)n(terv)-5 b(als)456 3064 y(suc)n(h)25 b(that)h Fq(T)37 b Fs(is)26 b Fq(C)1052 3034 y Fl(1)1115 3064 y Fs(on)g(eac)n(h)p 1414 2997 63 4 v 25 w Fq(Z)6 b Fs(,)26 b Fq(Z)j Fn(2)23 b(Z)33 b Fs(and)25 b(monotonic.)36 b(Let)26 b Fn(Z)2594 3034 y Fl(\()p Fp(n)p Fl(\))2717 3064 y Fs(b)r(e)g(the)g(monotonicit)n (y)456 3164 y(partition)h(of)g Fq(T)957 3134 y Fp(n)1002 3164 y Fs(.)555 3263 y(Let)j Fq(g)749 3233 y Fl(0)843 3263 y Fs(:)56 b Fq(I)64 b Fn(\000)-14 b(!)56 b Fo(R)1266 3233 y Fl(+)1327 3263 y Fs(,)31 b(b)r(e)f(a)f(strictly)h(p)r(ositiv)n (e)f(p)r(oten)n(tial)h(whic)n(h)g(b)r(elongs)f(to)h(BV)g(and)456 3363 y(admits)j(a)h(conformal)e(measure)h Fq(m)p Fs(.)55 b(By)33 b Fn(L)1875 3375 y Fl(0)1947 3363 y Fs(w)n(e)g(designate)g(the) h(usual)f(P)n(erron-F)-7 b(rob)r(enius)456 3463 y(op)r(erator)23 b(\(or)h(transfer)g(op)r(erator\))g(asso)r(ciated)f(to)i(the)g(dynamic) g(and)g Fq(g)2754 3432 y Fl(0)2791 3463 y Fs(.)36 b(The)25 b(op)r(erator)e Fn(L)3407 3475 y Fl(0)456 3562 y Fs(acts)k(on)g Fq(L)799 3532 y Fl(1)836 3562 y Fs(\()p Fq(m)p Fs(\))h(and)f Fq(B)t(V)19 b Fs(:)456 3714 y(\(1.1\))862 b Fn(L)1546 3726 y Fl(0)1583 3714 y Fq(f)9 b Fs(\()p Fq(x)p Fs(\))24 b(=)1882 3635 y Fk(X)1856 3813 y Fp(T)9 b(y)r Fl(=)p Fp(x)2042 3714 y Fq(f)g Fs(\()p Fq(y)s Fs(\))p Fq(g)2243 3679 y Fl(0)2280 3714 y Fs(\()p Fq(y)s Fs(\))p Fq(:)456 3943 y Fs(Recall)27 b(that)h(a)f(measure)f Fq(m)i Fs(is)f(called)h Fq(g)1741 3913 y Fl(0)1778 3943 y Fs(-conformal)e(if)i(it)g (satis\014es:)1419 4089 y Fn(L)1476 4055 y Fj(\003)1476 4109 y Fl(0)1515 4089 y Fq(m)23 b Fs(=)f Fq(cm)28 b Fs(where)f Fq(c)c Fs(:=)g Fq(e)2284 4055 y Fp(P)9 b Fl(\()p Fp(g)2395 4030 y Fi(0)2427 4055 y Fl(\))2457 4089 y Fq(;)456 4224 y Fs(and)1348 4342 y Fq(P)j Fs(\()p Fq(g)1488 4308 y Fl(0)1525 4342 y Fs(\))23 b(=)52 b(lim)1668 4392 y Fp(n)p Fj(!1)1869 4286 y Fs(1)p 1865 4323 50 4 v 1865 4399 a Fq(n)1938 4342 y Fs(log)2116 4263 y Fk(X)2060 4449 y Fp(Z)t Fj(2Z)2207 4432 y Fi(\()p Fh(n)p Fi(\))2306 4342 y Fs(sup)2343 4412 y Fp(Z)2444 4342 y Fq(g)2487 4308 y Fl(0)2484 4362 y Fp(n)2529 4342 y Fq(:)456 4597 y Fs(De\014ne)37 b(also)e(\002\()p Fq(g)1037 4567 y Fl(0)1074 4597 y Fs(\))i(to)f(b)r(e) h(suc)n(h)f(that)h(log)14 b(\002\()p Fq(g)2021 4563 y Fl(0)2057 4597 y Fs(\))39 b(:=)66 b(lim)2253 4647 y Fp(n)p Fj(!1)2454 4541 y Fs(1)p 2450 4578 V 2450 4654 a Fq(n)2524 4597 y Fs(log)14 b(sup)2690 4667 y Fp(I)2784 4597 y Fq(g)2827 4563 y Fl(0)2824 4618 y Fp(n)2869 4597 y Fs(.)63 b(Our)36 b(standing)456 4751 y(assumptions)26 b(on)i Fq(g)1086 4721 y Fl(0)1150 4751 y Fs(will)g(b)r(e)g(the)g(follo)n(wing:)456 4869 y FB(Condition)i(0.)247 b Fn(\017)41 b Fs(inf)21 b Fq(g)1464 4838 y Fl(0)1523 4869 y Fq(>)i Fs(0)p Fr(,)661 4975 y Fn(\017)41 b Fr(the)30 b(p)l(otential)h Fq(g)1265 4945 y Fl(0)1331 4975 y Fr(is)f Fs(con)n(tracting)e Fr(i.e.,)k Fs(\002\()p Fq(g)2168 4945 y Fl(0)2205 4975 y Fs(\))23 b Fq(<)g(e)2387 4945 y Fp(P)9 b Fl(\()p Fp(g)2498 4920 y Fi(0)2531 4945 y Fl(\))2561 4975 y Fr(,)p 456 5034 499 4 v 555 5107 a Fl(1)588 5133 y FA(In)23 b(fact,)f(all)g(the)h (follo)n(wing)e(can)i(b)r(e)f(easily)g(extended)i(to)f(the)g(case)g(in) e(whic)n(h)i Fy(Y)37 b FA(is)22 b(a)g(\014nite)h(collection)g(of)456 5216 y(sub-in)n(terv)l(als.)31 b(W)-6 b(e)24 b(c)n(ho)r(ose)h(not)f(to) h(do)f(so)f(to)h(k)n(eep)h(the)g(exp)r(osition)f(as)g(simple)e(as)i(p)r (ossible.)p eop %%Page: 3 3 3 2 bop 1339 251 a Fl(LASOT)-5 b(A-YORKE)29 b(MAPS)f(WITH)h(HOLES)850 b(3)661 450 y Fn(\017)41 b Fr(the)30 b(p)l(otential)h Fq(g)1265 420 y Fl(0)1331 450 y Fr(b)l(elongs)g(to)e(the)h(sp)l(ac)l(e) h(BV)e(of)i(functions)f(of)g(b)l(ounde)l(d)g(variation.)661 550 y Fn(\017)41 b Fr(ther)l(e)30 b(exists)g(a)g Fq(g)1294 520 y Fl(0)1330 550 y Fr(-c)l(onformal)h(pr)l(ob)l(ability)h(me)l(asur) l(e)e Fq(m)p Fr(.)456 669 y FB(Remark)39 b(1.1.)45 b Fr(It)36 b(is)h(known)f(\(se)l(e)h Fs([K])p Fr(,)h Fs([BK])p Fr(,)h Fs([LSV])p Fr(\))e(that)f(if)i Fq(g)2648 639 y Fl(0)2721 669 y Fr(b)l(elongs)f(to)g Fq(B)t(V)56 b Fr(then)456 769 y(so)33 b(do)l(es)i Fq(g)795 739 y Fl(0)792 790 y Fp(n)870 769 y Fr(for)f(al)t(l)h Fq(n)30 b Fn(2)g Fo(N)43 b Fr(and)34 b(that)g(this)f(to)l(gether)h(with)g(the)g(c)l(ontr)l (acting)f(c)l(ondition)i(ar)l(e)456 869 y(su\016cient)28 b(to)g(ensur)l(e)f(the)i(existenc)l(e)f(of)h(a)g Fq(g)1866 839 y Fl(0)1902 869 y Fr(-c)l(onformal)h(non)e(atomic)h(pr)l(ob)l (ability)i(me)l(asur)l(e)456 968 y(pr)l(ovide)l(d)g(the)f(p)l(artition) h(is)f(gener)l(ating.)555 1088 y Fs(It)g(is)e(kno)n(wn)h(\(see)f([K],)i ([BK)o(],)f([LSV)q(]\))g(that)g(if)h Fq(g)2118 1058 y Fl(0)2184 1088 y Fs(b)r(elongs)e(to)h Fq(B)t(V)48 b Fs(then)29 b(so)g(do)r(es)f Fq(g)3274 1058 y Fl(0)3271 1108 y Fp(n)3345 1088 y Fs(for)456 1188 y(all)34 b Fq(n)h Fn(2)g Fo(N)45 b Fs(and)35 b(that)g(this)g(together)f(with)h(the)g(con)n(tracting)e (condition)i(are)f(su\016cien)n(t)h(to)456 1287 y(ensure)27 b(the)h(existence)f(of)g(a)h Fq(g)1418 1257 y Fl(0)1454 1287 y Fs(-conformal)e(non)i(atomic)f(probabilit)n(y)g(measure.)555 1486 y(Next,)d(consider)d(a)h(sub-in)n(terv)-5 b(al)22 b Fq(Y)42 b Fn(\032)22 b Fq(I)7 b Fs(,)24 b(the)f(hole.)34 b(T)-7 b(o)22 b(a)n(v)n(oid)f(trivial)h(considerations,)g(w)n(e)456 1586 y(assume)28 b Fq(m)p Fs(\()p Fq(Y)19 b Fs(\))p Fq(m)p Fs(\()p Fq(Y)1120 1556 y Fp(c)1154 1586 y Fs(\))25 b Fn(6)p Fs(=)g(0,)k Fq(X)1464 1598 y Fl(0)1530 1586 y Fs(denotes)g(the)g(complemen)n(tary)f(of)h(the)h(hole:)39 b Fq(X)3079 1598 y Fl(0)3141 1586 y Fs(=)25 b Fq(I)i Fn(n)19 b Fq(Y)f Fs(.)456 1686 y Fq(X)525 1698 y Fp(n)611 1686 y Fs(will)41 b(denote)g(the)g(set)g(of)g(p)r(oin)n(ts)g(that)g(ha) n(v)n(e)f(not)h(fallen)g(in)n(to)f(the)i(hole)e(at)h(time)h Fq(n)p Fs(:)456 1785 y Fq(X)525 1797 y Fp(n)593 1785 y Fs(=)680 1723 y Fk(T)749 1744 y Fp(n)749 1810 y(i)p Fl(=0)875 1785 y Fq(T)936 1755 y Fj(\000)p Fp(i)1015 1785 y Fq(X)1084 1797 y Fl(0)1121 1785 y Fs(.)34 b(W)-7 b(e)20 b(will)f(also)g(denote)g(b)n(y)g Fq(g)26 b Fs(=)d Fq(g)2183 1755 y Fl(0)2220 1785 y FB(1)2268 1797 y Fp(X)2322 1805 y Fi(0)2358 1785 y Fs(,)f Fq(g)2443 1797 y Fp(n)2487 1785 y Fs(\()p Fq(x)p Fs(\))i(=)f Fq(g)s Fs(\()p Fq(x)p Fs(\))r Fn(\002)r(\001)14 b(\001)g(\001)s(\002)r Fq(g)s Fs(\()p Fq(T)3236 1755 y Fp(n)p Fj(\000)p Fl(1)3365 1785 y Fq(x)p Fs(\))456 1885 y(and)27 b(\002)c(=)f(\002\()p Fq(g)s Fs(\).)555 1985 y Fr(Conditional)t(ly)33 b(invariant)d(pr)l(ob)l (ability)h(me)l(asur)l(es)f(\(c.i.p.m.)40 b(for)31 b(short\))c Fs(are)g(probabilit)n(y)456 2084 y(measures)f Fq(\027)33 b Fs(satisfying:)1084 2224 y Fn(8)p Fq(A)22 b Fn(2)i(B)29 b(8)p Fq(n)22 b Fn(2)i Fo(Z)1637 2236 y Fl(+)1714 2224 y Fq(\027)5 b Fs(\()p Fq(T)1853 2190 y Fj(\000)p Fp(n)1950 2224 y Fq(A)18 b Fn(\\)h Fq(X)2173 2236 y Fp(n)2218 2224 y Fs(\))k(=)g Fq(\027)5 b Fs(\()p Fq(A)p Fs(\))p Fq(\027)g Fs(\()p Fq(X)2680 2236 y Fp(n)2727 2224 y Fs(\))p Fq(;)529 b Fs(\()p Fn(\003)p Fs(\))456 2364 y(where)32 b Fn(B)j Fs(is)d(the)i(Borel)d Fq(\033)s Fs(-algebra.)51 b(Condition)32 b(\()p Fn(\003)p Fs(\))h(implies)g(that)g Fq(\027)39 b Fs(m)n(ust)32 b(b)r(e)i(supp)r(orted)456 2463 y(in)c Fq(X)624 2475 y Fl(0)691 2463 y Fs(and,)g(if)h Fq(\027)5 b Fs(\()p Fq(T)1096 2433 y Fj(\000)p Fl(1)1184 2463 y Fq(X)1253 2475 y Fl(0)1290 2463 y Fs(\))28 b(=:)e Fq(\013)i Fn(2)p Fs(]0)p Fq(;)14 b Fs(1],)30 b(that)g Fq(\027)5 b Fs(\()p Fq(X)2149 2475 y Fp(n)2194 2463 y Fs(\))28 b(=)e Fq(\013)2398 2433 y Fp(n)2444 2463 y Fs(,)k(i.e.)44 b(with)31 b(resp)r(ect)e(to)h Fq(\027)5 b Fs(,)31 b(the)456 2563 y(en)n(trance)26 b(time)i(in)n(to)g Fq(Y)46 b Fs(has)27 b(exp)r(onen)n(tial)g(la)n(w.)456 2662 y(Of)d(course,)g(w)n(e)f(are)h (not)g(in)n(terested)g(in)g(all)g(c.i.p.m.,)i(but)e(on)g(those)g(that)h (ha)n(v)n(e)e(some)g(reason-)456 2762 y(able)30 b(prop)r(erties)g(with) h(resp)r(ect)f(to)g(the)h(p)r(oten)n(tial)g Fq(g)2154 2732 y Fl(0)2191 2762 y Fs(.)46 b(W)-7 b(e)31 b(will)g(consider)e(only) h(absolutely)456 2862 y(con)n(tin)n(uous)35 b(with)h(resp)r(ect)g(to)g Fq(m)g Fs(c.i.p.m.)63 b(\()p Fr(a.c.c.i.p.m.)68 b Fs(for)35 b(short\).)62 b(T)-7 b(o)36 b(this)h(aim,)h(an)456 2961 y(useful)28 b(to)r(ol)f(is)g(the)h(transfer)f(op)r(erator)f Fn(L)i Fs(de\014ned)g(b)n(y)456 3101 y(\(1.2\))990 b Fn(L)p Fs(\()p Fq(f)9 b Fs(\))23 b(=)g Fn(L)1956 3113 y Fl(0)1993 3101 y Fs(\()p Fq(f)9 b FB(1)2123 3113 y Fp(X)2177 3121 y Fi(0)2214 3101 y Fs(\))14 b Fq(:)456 3241 y Fs(The)27 b(usefulness)h(of)f Fn(L)h Fs(is)g(readily)e (clari\014ed.)456 3360 y FB(Lemma)j(1.1.)40 b Fr(The)31 b(fol)t(lowing)h(two)e(assertions)g(hold)i(true.)592 3480 y(\(1\))42 b(L)l(et)25 b Fq(\027)j Fs(=)23 b FB(1)1088 3492 y Fp(X)1142 3500 y Fi(0)1179 3480 y Fq(h)8 b Fn(\001)g Fq(m)25 b Fr(b)l(e)g(a)g(pr)l(ob)l(ability)i(me)l(asur)l(e)e (absolutely)h(c)l(ontinuous)e(with)i(r)l(esp)l(e)l(ct)744 3580 y(to)f Fq(m)p Fr(.)37 b(Then,)27 b Fq(\027)j Fr(is)25 b(an)g(a.c.c.i.p.m.)41 b(if)26 b(and)f(only)h(if)f Fn(L)p Fq(h)e Fs(=)g Fq(c\013h)i Fr(for)h(some)f Fq(\013)f Fn(2)p Fs(]0)p Fq(;)14 b Fs(1])p Fr(.)592 3679 y(\(2\))42 b(L)l(et)32 b Fq(\013)c Fn(2)p Fs(]0)p Fq(;)14 b Fs(1])31 b Fr(and)h Fq(h)27 b Fn(2)h Fq(L)1602 3649 y Fl(1)1639 3679 y Fs(\()p Fq(m)p Fs(\))k Fr(b)l(e)g(such)g(that)g Fn(L)p Fq(h)c Fs(=)e Fq(c\013h)p Fr(,)34 b(let)e Fq(\026)g Fr(b)l(e)g(a)g(pr)l(ob)l (ability)744 3779 y(me)l(asur)l(e)26 b(on)g Fq(I)32 b Fr(such)26 b(that)g Fn(L)1653 3749 y Fj(\003)1691 3779 y Fq(\026)d Fs(=)g Fq(c\013\026)p Fr(.)38 b(Then)26 b Fq(\026)p Fr(is)g(supp)l(orte)l(d)g(in)g Fq(X)2934 3791 y Fj(1)3030 3779 y Fr(and)g Fq(\025)e Fs(=)f Fq(h\026)744 3879 y Fr(is)30 b Fq(T)12 b Fr(-invariant.)456 4035 y(Pr)l(o)l(of.)43 b FB(1)p Fq(:)30 b Fs(Let)h Fq(\027)i Fs(=)27 b(\()p FB(1)1215 4047 y Fp(X)1269 4055 y Fi(0)1306 4035 y Fq(h)p Fs(\))p Fq(m)k Fs(and)f(assume)g Fn(L)p Fq(h)e Fs(=)g Fq(c\013h)p Fs(.)46 b(W)-7 b(e)30 b(will)h(mak)n(e)f(extensiv)n(ely)g (use)456 4135 y(of)d(the)h(follo)n(wing)f(t)n(w)n(o)f(easily)h (obtained)h(prop)r(erties)e(on)i(the)g(iterates)e(of)i Fn(L)p Fs(:)843 4275 y Fn(8)f Fq(f)32 b Fn(2)23 b Fq(L)1125 4240 y Fl(1)1162 4275 y Fs(\()p Fq(m)p Fs(\))p Fq(;)42 b Fn(8)27 b Fq(n)c Fn(2)g Fo(Z)1650 4287 y Fl(+)1727 4275 y Fn(L)1784 4240 y Fp(n)1830 4275 y Fs(\()p Fq(f)9 b Fs(\))23 b(=)f Fn(L)2111 4240 y Fp(n)2111 4295 y Fl(0)2157 4275 y Fs(\()p Fq(f)9 b FB(1)2287 4287 y Fp(X)2341 4295 y Fh(n)p Fg(\000)p Fi(1)2459 4275 y Fs(\))-2035 b(\(1.3\))843 4455 y Fn(8)27 b Fq(f)9 b(;)14 b(')23 b Fn(2)g Fq(L)1216 4421 y Fl(1)1253 4455 y Fs(\()p Fq(m)p Fs(\))p Fq(;)42 b Fn(8)27 b Fq(n)c Fn(2)g Fo(Z)1742 4467 y Fl(+)1834 4342 y Fk(Z)1814 4580 y Fp(X)1868 4588 y Fi(0)1933 4455 y Fq(')p Fn(L)2044 4421 y Fp(n)2090 4455 y Fq(f)9 b(dm)22 b Fs(=)h Fq(c)2402 4421 y Fp(n)2467 4342 y Fk(Z)2443 4580 y Fp(X)2497 4588 y Fh(n)2570 4455 y Fq(')c Fn(\016)f Fq(T)2764 4421 y Fp(n)2826 4455 y Fn(\001)h Fq(f)9 b(dm:)-2601 b Fs(\(1.4\))456 4707 y(Let)27 b Fq(A)d Fn(2)f(B)s Fs(,)k(\(1.3,)g (1.4\))g(giv)n(e:)1079 4889 y Fq(\027)5 b Fs(\()p Fq(T)1218 4855 y Fj(\000)p Fp(n)1314 4889 y Fq(A)19 b Fn(\\)g Fq(X)1538 4901 y Fp(n)1583 4889 y Fs(\))51 b(=)1781 4776 y Fk(Z)1878 4889 y FB(1)1926 4901 y Fp(A)1998 4889 y Fn(\016)18 b Fq(T)2119 4855 y Fp(n)2182 4889 y Fn(\001)h FB(1)2272 4901 y Fp(X)2326 4909 y Fh(n)2389 4889 y Fn(\001)g Fq(hdm)1666 5107 y Fs(=)1811 5051 y(1)p 1791 5088 82 4 v 1791 5164 a Fq(c)1827 5140 y Fp(n)1898 4994 y Fk(Z)1878 5232 y Fp(X)1932 5240 y Fi(0)1997 5107 y FB(1)2045 5119 y Fp(A)2099 5107 y Fs(\()p Fn(L)2188 5073 y Fp(n)2233 5107 y Fq(h)p Fs(\))p Fq(dm)24 b Fs(=)e Fq(\013)2593 5073 y Fp(n)2639 5107 y Fq(\027)5 b Fs(\()p Fq(A)p Fs(\))p Fq(:)p eop %%Page: 4 4 4 3 bop 456 251 a Fl(4)388 b(CARLANGELO)23 b(LIVERANI)f(AND)g(V)1947 236 y(\023)1941 251 y(ER)n(ONIQUE)g(MA)n(UME-DESCHAMPS)456 450 y Fs(In)k(particular,)f(for)g Fq(A)f Fs(=)e Fq(I)7 b Fs(,)27 b(w)n(e)e(get)h Fq(\027)5 b Fs(\()p Fq(X)1758 462 y Fp(n)1804 450 y Fs(\))23 b(=)g Fq(\013)2000 420 y Fp(n)2071 450 y Fs(th)n(us,)k(for)e(an)n(y)g Fq(A)f Fn(2)f(B)s Fs(,)j Fq(\027)5 b Fs(\()p Fq(T)2966 420 y Fj(\000)p Fp(n)3062 450 y Fq(A)16 b Fn(\\)g Fq(X)3280 462 y Fp(n)3324 450 y Fs(\))24 b(=)456 550 y Fq(\027)5 b Fs(\()p Fq(A)p Fs(\))p Fq(\027)g Fs(\()p Fq(X)775 562 y Fp(n)821 550 y Fs(\).)456 649 y(Con)n(v)n(ersely)-7 b(,)29 b(assume)g Fq(\027)k Fs(=)27 b(\()p FB(1)1431 661 y Fp(X)1485 669 y Fi(0)1522 649 y Fq(h)p Fs(\))p Fq(m)k Fs(is)f(a)g(a.c.c.i.p.m..)45 b(Then,)31 b(b)n(y)f(de\014nition)h (of)f(c.i.p.m.,)456 749 y(there)d(exists)g Fq(\013)d Fn(2)p Fs(]0)p Fq(;)14 b Fs(1])27 b(suc)n(h)g(that,)h(for)f(an)n(y)g Fq(A)c Fn(2)h(B)s Fs(,)i Fq(\027)5 b Fs(\()p Fq(T)2307 719 y Fj(\000)p Fp(n)2404 749 y Fq(A)19 b Fn(\\)g Fq(X)2628 761 y Fp(n)2673 749 y Fs(\))k(=)g Fq(\013)2869 719 y Fp(n)2914 749 y Fq(\027)5 b Fs(\()p Fq(A)p Fs(\).)38 b(So,)1160 931 y Fn(8)27 b Fq(A)c Fn(2)g(B)s Fq(;)1521 818 y Fk(Z)1500 1056 y Fp(X)1554 1064 y Fi(0)1620 931 y FB(1)1668 943 y Fp(A)1740 931 y Fn(\001)1791 875 y(L)1848 845 y Fp(n)1894 875 y Fq(h)p 1791 912 151 4 v 1826 988 a(c)1862 964 y Fp(n)1952 931 y Fq(dm)g Fs(=)f Fq(\013)2231 897 y Fp(n)2293 818 y Fk(Z)2272 1056 y Fp(X)2326 1064 y Fi(0)2391 931 y FB(1)2439 943 y Fp(A)2512 931 y Fn(\001)c Fq(hdm;)456 1183 y Fs(w)n(e)27 b(deduce)h(that)f Fn(L)1091 1152 y Fp(n)1137 1183 y Fq(h)c Fs(=)f(\()p Fq(c\013)p Fs(\))1448 1152 y Fp(n)1495 1183 y Fq(h)p Fs(.)456 1282 y FB(2)p Fq(:)27 b Fs(Let)h Fq(\026)g Fs(b)r(e)g(a)f(probabilit)n(y)f (measure)h(on)g Fq(I)7 b Fs(,)28 b(assume)f(that)h Fn(L)2444 1252 y Fj(\003)2482 1282 y Fq(\026)23 b Fs(=)g Fq(c\013\026)p Fs(,)28 b Fq(\013)23 b Fn(2)p Fs(]0)p Fq(;)14 b Fs(1],)28 b(then)1088 1463 y Fn(8)f Fq(n)c Fn(2)h Fo(Z)1375 1475 y Fl(+)1424 1463 y Fq(;)42 b Fn(8)26 b Fq(f)32 b Fn(2)23 b Fq(L)1770 1429 y Fl(1)1807 1463 y Fs(\()p Fq(m)p Fs(\))28 b(\()p Fq(c\013)p Fs(\))2125 1429 y Fp(n)2171 1463 y Fq(\026)p Fs(\()p Fq(f)9 b Fs(\))24 b(=)2446 1350 y Fk(Z)2452 1588 y Fp(I)2543 1463 y Fn(L)2600 1429 y Fp(n)2646 1463 y Fq(f)9 b(d\026:)456 1702 y Fs(Assume)27 b(that)h Fq(f)36 b Fs(is)28 b(zero)e(on)i Fq(X)1464 1714 y Fp(n)p Fj(\000)p Fl(1)1594 1702 y Fs(.)37 b(Then)1161 1841 y(\()p Fq(c\013)p Fs(\))1314 1807 y Fp(n)1360 1841 y Fq(\026)p Fs(\()p Fq(f)9 b Fs(\))23 b(=)g Fq(\026)p Fs(\()p Fn(L)1774 1807 y Fp(n)1820 1841 y Fq(f)9 b Fs(\))23 b(=)f Fq(\026)p Fs(\()p Fn(L)2151 1807 y Fp(n)2151 1861 y Fl(0)2197 1841 y Fs(\()p FB(1)2277 1853 y Fp(X)2331 1861 y Fh(n)p Fg(\000)p Fi(1)2450 1841 y Fq(f)9 b Fs(\)\))23 b(=)f(0)p Fq(;)456 1980 y Fs(th)n(us)27 b Fq(\026)p Fs(\()p Fq(f)9 b Fs(\))23 b(=)g(0.)37 b(W)-7 b(e)28 b(deduce)f(that)h Fq(\026)g Fs(has)f(its)h(supp)r(ort)f(con)n(tained)g(in)h Fq(X)2805 1992 y Fj(1)2875 1980 y Fs(.)456 2079 y(The)33 b(fact)h(that)h(for)e Fq(h)g Fs(suc)n(h)h(that)g Fn(L)p Fq(h)g Fs(=)f Fq(c\013h)h Fs(the)g(measure)f Fq(\025)g Fs(=)g Fq(h\026)h Fs(is)g Fq(T)12 b Fs(-in)n(v)-5 b(arian)n(t)31 b(is)j(a)456 2179 y(direct)27 b(computation.)2209 b Ff(\003)555 2335 y Fs(In)27 b(the)f(next)h(section)f(w)n(e)g(will)g(in)n(tro)r(duce)g(t)n (w)n(o)g(conditions)g(on)g(the)h(holes)e(\(see)i(Condition)456 2434 y(1)g(and)g(Condition)h(2\))f(under)h(whic)n(h)f(the)h(follo)n (wing)f(statemen)n(ts)g(hold.)456 2534 y(Our)f(main)i(result)f(is)h (the)g(follo)n(wing.)456 2653 y FB(Theorem)k(A.)43 b Fr(Assume)30 b(that)i(Conditions)h(0,)f(1)g(and)g(2)g(ar)l(e)g (satis\014e)l(d.)44 b(Then)32 b(ther)l(e)g(exists)456 2753 y(a)37 b(unique)g(c)l(onditional)t(ly)j(invariant)e(pr)l(ob)l (ability)i(me)l(asur)l(e)d Fq(\027)42 b Fs(=)37 b Fq(hm)g Fr(which)i(is)e(absolutely)456 2852 y(c)l(ontinuous)23 b(with)i(r)l(esp)l(e)l(ct)f(to)g Fq(m)p Fr(.)36 b(Ther)l(e)26 b(exists)d(a)i(unique)f(pr)l(ob)l(ability)i(me)l(asur)l(e)e Fq(\026)g Fr(supp)l(orte)l(d)456 2952 y(in)34 b Fq(X)631 2964 y Fj(1)736 2952 y Fr(and)g(which)j(satis\014es)d Fq(\026)p Fs(\()p Fn(L)p Fq(f)9 b Fs(\))32 b(=)f Fq(\032\026)p Fs(\()p Fq(f)9 b Fs(\))p Fr(,)36 b(with)f Fq(\032)d Fn(\024)f Fq(c)p Fr(,)36 b(for)f(any)g(b)l(ounde)l(d)f(function)456 3052 y Fq(f)9 b Fr(.)59 b(The)38 b(me)l(asur)l(e)e Fq(\025)g Fs(=)f Fq(h\026)i Fr(is)g(the)g(only)g Fq(T)48 b Fr(invariant)38 b(me)l(asur)l(e)e(supp)l(orte)l(d)h(on)g Fq(X)3206 3064 y Fj(1)3313 3052 y Fr(and)456 3151 y(absolutely)31 b(c)l(ontinuous)f (with)h(r)l(esp)l(e)l(ct)f(to)g Fq(\026)p Fr(.)40 b(Mor)l(e)l(over,)33 b(ther)l(e)d(exists)g Fq(\024)24 b(<)g Fs(1)30 b Fr(such)g(that)g(for) 456 3251 y(any)g Fq(f)h Fn(2)24 b Fq(B)t(V)49 b Fr(and)30 b(any)g Fq(A)23 b Fn(2)h(B)s Fr(:)1341 3315 y Fk(\015)1341 3364 y(\015)1341 3414 y(\015)1341 3464 y(\015)1397 3379 y Fn(L)1454 3349 y Fp(n)1499 3379 y Fq(f)p 1397 3416 153 4 v 1429 3492 a(\032)1472 3468 y Fp(n)1577 3435 y Fn(\000)18 b Fq(h\026)p Fs(\()p Fq(f)9 b Fs(\))1872 3315 y Fk(\015)1872 3364 y(\015)1872 3414 y(\015)1872 3464 y(\015)1918 3518 y Fj(1)2012 3435 y Fn(\024)22 b Fr(Ct)14 b Fq(\024)2248 3401 y Fp(n)2293 3435 y Fn(k)p Fq(f)9 b Fn(k)2427 3447 y Fp(B)s(V)2536 3435 y Fq(;)1348 3640 y Fn(j)p Fq(m)p Fs(\()p Fq(T)1537 3606 y Fj(\000)p Fp(n)1633 3640 y Fq(A)p Fn(j)p Fq(X)1787 3652 y Fp(n)p Fj(\000)p Fl(1)1918 3640 y Fs(\))18 b Fn(\000)g Fq(\027)5 b Fs(\()p Fq(A)p Fs(\))p Fn(j)25 b(\024)d Fr(Ct)14 b Fq(\024)2507 3606 y Fp(n)1353 3766 y Fr(and)31 b Fn(j)p Fq(\027)5 b Fs(\()p Fq(A)p Fn(j)p Fq(X)1770 3778 y Fp(n)p Fj(\000)p Fl(1)1901 3766 y Fs(\))18 b Fn(\000)g Fq(\025)p Fs(\()p Fq(A)p Fs(\))p Fn(j)25 b(\024)d Fr(Ct)p Fq(\024)2478 3732 y Fp(n)2523 3766 y Fq(:)555 3887 y Fs(A)32 b(sub)f(pro)r(duct)g (of)g(our)f(Theorem)g(A)h(will)h(b)r(e)f(the)g(follo)n(wing)f(result)h (on)g(the)g(Hausdor\013)456 3987 y(dimension)c(of)h(the)g(set)f Fq(X)1284 3999 y Fj(1)1382 3987 y Fs(of)h(surviv)n(ors.)34 b(F)-7 b(or)27 b(an)n(y)g(0)22 b Fn(\024)h Fq(t)g Fn(\024)g Fs(1,)k(de\014ne)1341 4138 y Fn(L)1398 4150 y Fp(t)1427 4138 y Fq(f)9 b Fs(\()p Fq(x)p Fs(\))24 b(=)1726 4059 y Fk(X)1700 4237 y Fp(T)9 b(y)r Fl(=)p Fp(x)1872 4138 y Fs(\()p Fq(g)1947 4104 y Fl(0)1984 4138 y Fs(\))2016 4104 y Fp(t)2046 4138 y Fs(\()p Fq(y)s Fs(\))p FB(1)2202 4150 y Fp(X)2256 4158 y Fi(0)2293 4138 y Fs(\()p Fq(y)s Fs(\))p Fq(f)g Fs(\()p Fq(y)s Fs(\))456 4366 y(and)25 b(b)n(y)h(\002)794 4378 y Fp(t)823 4366 y Fs(,)g Fq(\032)915 4378 y Fp(t)970 4366 y Fs(and)g Fq(P)12 b Fs(\()p Fq(t)p Fs(\))27 b(the)f(n)n(um)n(b)r(er)g(corresp)r(onding)e(to)i(\002,)g Fq(\032)p Fs(,)g Fq(P)38 b Fs(in)26 b(the)g(case)f Fq(t)e Fs(=)g(1)j(\(see)456 4466 y(De\014nition)i(2.1)f(for)g(the)h (de\014nition)g(of)f Fq(\032)p Fs(\).)456 4566 y(W)-7 b(e)25 b(will)g(sa)n(y)e(that)i Fq(g)1110 4536 y Fl(0)1172 4566 y Fs(has)f(the)h(Bounded)g(Distortion)f(prop)r(ert)n(y)f(if)j (there)e(exists)h Fq(C)k(>)23 b Fs(1)h(suc)n(h)456 4665 y(that)j(for)h(all)f Fq(n)c Fn(2)g Fo(N)t Fs(,)34 b Fq(Z)29 b Fn(2)23 b(Z)1371 4635 y Fl(\()p Fp(n)p Fl(\))1495 4665 y Fs(and)28 b Fq(x)p Fs(,)g Fq(y)e Fn(2)d Fq(Z)6 b Fs(,)456 4853 y(\(1.5\))1752 4797 y Fq(g)1795 4766 y Fl(0)1792 4817 y Fp(n)1837 4797 y Fs(\()p Fq(x)p Fs(\))p 1752 4834 197 4 v 1754 4910 a Fq(g)1797 4886 y Fl(0)1794 4930 y Fp(n)1839 4910 y Fs(\()p Fq(y)s Fs(\))1982 4853 y Fn(\024)23 b Fq(C)6 b(:)456 5036 y Fs(W)-7 b(e)28 b(will)f(sa)n(y)g(that)h Fq(T)38 b Fs(has)28 b(large)e(images)g(if)456 5174 y(\(1.6\))902 b(inf)1516 5228 y Fp(n)p Fj(2)p Fe(N)1723 5174 y Fs(inf)1657 5236 y Fp(Z)t Fj(2Z)1804 5219 y Fi(\()p Fh(n)p Fi(\))1904 5174 y Fq(m)p Fs(\()p Fq(T)2070 5140 y Fp(n)2114 5174 y Fq(Z)6 b Fs(\))23 b Fq(>)g Fs(0)p Fq(:)p eop %%Page: 5 5 5 4 bop 1339 251 a Fl(LASOT)-5 b(A-YORKE)29 b(MAPS)f(WITH)h(HOLES)850 b(5)456 450 y Fs(W)-7 b(e)37 b(will)g(sa)n(y)e(that)j Fq(T)48 b Fs(has)36 b(large)f(images)h(with)h(resp)r(ect)g(to)g Fq(Y)55 b Fs(if)37 b(for)g(all)f Fq(n)i Fn(2)h Fo(N)t Fs(,)45 b(for)37 b(all)456 550 y Fq(Z)28 b Fn(2)c(Z)687 520 y Fl(\()p Fp(n)p Fl(\))784 550 y Fs(,)j Fq(Z)d Fn(\\)19 b Fq(X)1058 562 y Fj(1)1151 550 y Fn(6)p Fs(=)k Fn(;)p Fs(,)k Fq(T)1392 520 y Fp(n)1436 550 y Fs(\()p Fq(Z)e Fn(\\)19 b Fq(X)1693 562 y Fp(n)p Fj(\000)p Fl(1)1823 550 y Fs(\))k Fn(\033)g Fq(X)2035 562 y Fj(1)2105 550 y Fs(.)456 672 y FB(Theorem)36 b(B.)44 b Fr(L)l(et)34 b Fq(g)1197 642 y Fl(0)1265 672 y Fs(=)1389 640 y Fl(1)p 1371 654 71 4 v 1371 701 a Fp(T)1419 684 y Fg(0)1451 672 y Fr(.)52 b(Assume)34 b(that)g(for)h(al)t(l)h Fs(0)31 b Fn(\024)g Fq(t)g Fn(\024)g Fs(1)p Fr(,)k(Conditions)h(0,)g(1)f(and) 456 772 y(2)f(ar)l(e)h(satis\014e)l(d.)53 b(Then,)36 b(ther)l(e)f(exists)f(a)h(unique)e Fs(0)e Fq(<)g(t)2280 784 y Fl(0)2349 772 y Fn(\024)g Fs(1)j Fr(such)g(that)g(for)i Fs(0)30 b Fn(\024)h Fq(t)h(<)f(t)3382 784 y Fl(0)3419 772 y Fr(,)456 872 y Fq(\032)499 884 y Fp(t)560 872 y Fq(>)g Fs(1)k Fr(and)g(for)g Fs(1)d Fn(\025)g Fq(t)g(>)f(t)1395 884 y Fl(0)1432 872 y Fr(,)37 b Fq(\032)1537 884 y Fp(t)1598 872 y Fq(<)32 b Fs(1)p Fr(.)53 b(If)35 b Fq(T)46 b Fr(has)35 b(lar)l(ge)h(images)f(and)h(lar)l(ge)f(images)h(with)456 971 y(r)l(esp)l(e)l(ct)29 b(to)h Fq(Y)48 b Fr(then,)30 b(HD)p Fs(\()p Fq(X)1358 983 y Fj(1)1428 971 y Fs(\))23 b(=)g Fq(t)1601 983 y Fl(0)1638 971 y Fr(.)555 1094 y Fs(The)28 b(t)n(w)n(o)f(theorems)g(ab)r(o)n(v)n(e)f(will)i(follo)n(w)e (from)i(Theorems)e(4.4)h(and)g(5.1.)555 1194 y(As)d(w)n(e)g(will)g(see) g(in)g(section)g(6,)g(Theorems)f(A,)i(B)f(apply)g(to)g(maps)f(with)i (fairly)e(large)g(holes,)456 1293 y(in)36 b(fact)f(this)h(is)g(the)g (case)f(in)h(whic)n(h)g(they)f(are)g(of)h(in)n(terest.)61 b(If,)38 b(on)d(the)h(con)n(trary)-7 b(,)36 b(one)g(is)456 1393 y(willing)23 b(to)h(settle)g(for)f(small)h(holes,)g(then)g(it)h (is)e(p)r(ossible)h(to)f(apply)h(a)f(p)r(erturbativ)n(e)g(approac)n(h) 456 1493 y(whic)n(h)k(yields)h(the)g(follo)n(wing)e(stronger)g(result.) 1971 1460 y Fl(2)456 1615 y FB(Theorem)39 b(C.)46 b Fr(Assume)36 b Fq(g)1372 1585 y Fl(0)1445 1615 y Fr(is)i(satis\014es)f(Condition)h (0.)61 b(If)37 b(the)g(L)l(asota-Y)-6 b(orke)38 b(map)f Fq(T)47 b Fs(:)456 1715 y Fq(I)42 b Fn(!)36 b Fq(I)43 b Fr(has)38 b(a)f(unique)f(invariant)i(me)l(asur)l(e)e Fq(\026)1986 1727 y Fl(0)2060 1715 y Fr(absolutely)i(c)l(ontinuous)e (with)h(r)l(esp)l(e)l(ct)f(the)456 1815 y(c)l(onformal)31 b(me)l(asur)l(e)e Fq(m)p Fr(,)h(and)g(the)f(systems)h Fs(\()p Fq(I)7 b(;)14 b(T)7 b(;)14 b(\026)2151 1827 y Fl(0)2188 1815 y Fs(\))29 b Fr(is)h(mixing,)g(then)g(ther)l(e)f(exists) g Fq(")23 b(>)g Fs(0)456 1914 y Fr(such)29 b(that,)i(for)f(e)l(ach)h (hole)g Fq(Y)19 b Fr(,)30 b Fq(m)p Fs(\()p Fq(Y)19 b Fs(\))k Fn(\024)g Fq(")p Fr(,)29 b(the)h(c)l(onclusions)h(of)f(The)l (or)l(em)h(A)e(apply.)555 2037 y Fs(Theorem)c(C)g(is)g(pro)n(v)n(en)f (in)i(section)f(7.)35 b(In)26 b(view)f(of)g(Lemma)g(1.1,)g(w)n(e)g(are) g(led)g(to)g(start)g(our)456 2137 y(in)n(v)n(estigation)30 b(b)n(y)i(constructing)g(eigen)n(v)-5 b(alues)31 b(and)h (eigenfunctions)h(for)e Fn(L)p Fs(.)52 b(As)32 b(usually)g(in)456 2236 y(these)27 b(topics,)h(a)f(Lasota-Y)-7 b(ork)n(e)24 b(inequalit)n(y)k(is)f(useful.)558 2438 y(2.)41 b Ft(Transfer)32 b(opera)-6 b(tor)32 b(and)f(Lasot)-6 b(a-Yorke)31 b(inequalities)g (with)h(holes.)555 2587 y Fs(As)22 b(already)f(men)n(tioned,)i(our)e(p) r(oin)n(t)h(of)g(view)g(is)g(to)g(consider)f(the)h(T)-7 b(ransfer)21 b(op)r(erator)f Fn(L)i Fs(as)456 2687 y(asso)r(ciated)28 b(to)h(the)g(p)r(oten)n(tial)h Fq(g)e Fs(=)d Fq(g)1654 2657 y Fl(0)1691 2687 y FB(1)1739 2699 y Fp(X)1793 2707 y Fi(0)1830 2687 y Fs(,)30 b(that)f(is)g(a)g(p)r(ositiv)n(e,)h(but)g (not)f(strictly)g(p)r(ositiv)n(e,)456 2787 y(w)n(eigh)n(t.)44 b(W)-7 b(eigh)n(ts)30 b(of)g(suc)n(h)g(t)n(yp)r(e,)h(and)f(more)g (general,)f(ha)n(v)n(e)h(b)r(een)g(studied)h(in)g(quite)f(some)456 2886 y(detail.)39 b(In)28 b(particular)f(the)i(existence)f(of)h(a)f (quasi-in)n(v)-5 b(arian)n(t)26 b(and)i(an)g(in)n(v)-5 b(arian)n(t)27 b(measure)h(is)456 2986 y(pro)n(v)n(en)c(in)h([BK])g (under)h(v)n(ery)e(mild)i(tec)n(hnical)f(assumptions)g(plus)g(the)h(h)n (yp)r(othesis)f(that)h(the)456 3086 y(standard)33 b(b)r(ound)h(\002)g (for)g(the)h(essen)n(tial)e(sp)r(ectral)g(radius)h(of)g Fn(L)g Fs(b)r(e)h(strictly)f(less)g(than)g(the)456 3185 y(sp)r(ectral)e(radius)g(of)h Fn(L)p Fs(.)54 b(Y)-7 b(et,)35 b(the)e(argumen)n(ts)f(used)h(there)f(are)g(non)h(constructiv)n(e)f (\(quasi-)456 3285 y(compactness\))e(and)g(b)r(oth)h(the)g(problem)g (of)f(when)h(suc)n(h)g(a)f(condition)h(is)f(satis\014ed)h(and)f(the)456 3384 y(problem)h(of)h(the)h(uniqueness)f(of)g(the)g(ab)r(o)n(v)n(e)f (measure)g(are)g(not)h(addressed.)50 b(Here)31 b(w)n(e)h(will)456 3484 y(restrict)26 b(ourselv)n(es)e(to)j(a)f(sligh)n(tly)h(less)f (general)f(setting)i(and)f(use)h(a)f(di\013eren)n(t,)h(constructiv)n (e,)456 3584 y(approac)n(h)35 b(patterned)j(after)f(some)g(previous)g (results)g(for)g(strictly)h(p)r(ositiv)n(e)f(w)n(eigh)n(ts)g(\(see)456 3683 y([LSV]\).)e(The)21 b(presen)n(t)e(approac)n(h)g(will)i(allo)n(w)e (us,)j(in)f(the)g(follo)n(wing)e(sections,)j(to)e(\014nd)h(explicit)456 3783 y(conditions)26 b(for)g(the)h(existence)f(and)h(the)g(prop)r (erties)f(of)h(the)g(quasi-in)n(v)-5 b(arian)n(t)24 b(and)j(in)n(v)-5 b(arian)n(t)456 3883 y(probabilit)n(y)26 b(measures.)555 3982 y(First)i(of)f(all)h(w)n(e)f(need)h(to)f(imp)r(ose)g(a)h (condition)f(on)g(our)g(system.)456 4105 y FB(Condition)37 b(1.)44 b Fr(L)l(et)35 b Fq(D)1245 4117 y Fp(n)1322 4105 y Fs(:=)d Fn(f)p Fq(x)h Fn(2)g Fq(I)40 b Fn(j)32 b(L)1840 4075 y Fp(n)1886 4105 y Fs(1\()p Fq(x)p Fs(\))h Fn(6)p Fs(=)f(0)p Fn(g)p Fr(.)54 b(We)35 b(wil)t(l)h(c)l(onsider)g(only)g (systems)456 4205 y(that)29 b(satisfy)645 4327 y FB(C1:)40 b Fq(D)898 4339 y Fj(1)992 4327 y Fs(:=)1132 4265 y Fk(T)1102 4402 y Fp(n)p Fj(2)p Fe(N)1244 4327 y Fq(D)1313 4339 y Fp(n)1381 4327 y Fn(6)p Fs(=)22 b Fn(;)p Fr(.)555 4508 y Fs(Notice)28 b(that)g(if)g Fq(x)23 b Fn(62)h Fq(D)1290 4520 y Fp(n)1362 4508 y Fs(then)k Fn(L)1608 4478 y Fp(n)1654 4508 y Fq(f)9 b Fs(\()p Fq(x)p Fs(\))24 b(=)e(0)27 b(for)g(eac)n(h)g Fq(f)32 b Fn(2)23 b Fq(L)2517 4478 y Fj(1)2587 4508 y Fs(\([0)p Fq(;)14 b Fs(1]\))27 b(since)1212 4654 y Fn(jL)1292 4620 y Fp(n)1338 4654 y Fq(f)9 b Fs(\()p Fq(x)p Fs(\))p Fn(j)23 b(\024)g(L)1690 4620 y Fp(n)1735 4654 y Fn(j)p Fq(f)9 b Fn(j)p Fs(\()p Fq(x)p Fs(\))24 b Fn(\024)f(k)p Fq(f)9 b Fn(k)2188 4666 y Fj(1)2257 4654 y Fn(L)2314 4620 y Fp(n)2359 4654 y Fs(1\()p Fq(x)p Fs(\))24 b(=)e(0)p Fq(:)456 4800 y Fs(Accordingly)-7 b(,)26 b(for)h(eac)n(h)g Fq(n)c Fn(2)g Fo(N)38 b Fs(holds)456 4946 y(\(2.1\))1033 b Fn(L)1717 4911 y Fp(n)1762 4946 y Fq(f)32 b Fs(=)23 b FB(1)1971 4958 y Fp(D)2025 4966 y Fh(n)2070 4946 y Fn(L)2127 4911 y Fp(n)2172 4946 y Fq(f)t(:)p 456 5034 499 4 v 555 5107 a Fl(2)588 5133 y FA(In)i(fact,)g(the)g(h)n(yp)r (othesis)h(that)f(\()p Fy(I)5 b(;)12 b(T)e(;)h(\026)1680 5142 y Fw(0)1715 5133 y FA(\))25 b(is)e(mixing)g(is)h(sup)r(er\015uous) h(and)g(here)g(is)e(used)i(only)g(to)g(mak)n(e)456 5216 y(an)19 b(easy)h(comparison)e(with)i(Theorem)e(A)h(whic)n(h)h (conditions)g(insure)f(that)h(the)g(in)n(v)l(arian)n(t)g(measure)e(is)h (unique.)p eop %%Page: 6 6 6 5 bop 456 251 a Fl(6)388 b(CARLANGELO)23 b(LIVERANI)f(AND)g(V)1947 236 y(\023)1941 251 y(ER)n(ONIQUE)g(MA)n(UME-DESCHAMPS)456 450 y Fs(Equation)k(\(2.1\))i(in)f(particular)g(means)g(that)h(if)g Fq(x)23 b Fn(62)h Fq(D)2221 462 y Fp(n)2266 450 y Fs(,)j(then)1448 602 y Fn(L)1505 567 y Fp(n)p Fl(+1)1635 602 y Fs(1\()p Fq(x)p Fs(\))c(=)g Fn(L)1956 567 y Fp(n)2001 602 y Fs(\()p Fn(L)p Fs(1\)\()p Fq(x)p Fs(\))i(=)d(0)p Fq(;)456 752 y Fs(hence)27 b Fq(x)d Fn(62)f Fq(D)904 764 y Fp(n)p Fl(+1)1033 752 y Fs(,)28 b(that)g(is)f Fq(D)1416 764 y Fp(n)p Fl(+1)1568 752 y Fn(\032)c Fq(D)1725 764 y Fp(n)1770 752 y Fs(.)555 852 y(W)-7 b(e)28 b(can)f(no)n(w)g(de\014ne)h(the)g (functional)456 1029 y(\(2.2\))827 b(\003\()p Fq(f)9 b Fs(\))23 b(:=)52 b(lim)1760 1079 y Fp(n)p Fj(!1)1985 1029 y Fs(inf)1947 1083 y Fp(x)p Fj(2)p Fp(D)2084 1091 y Fh(n)2148 973 y Fn(L)2205 943 y Fp(n)2251 973 y Fq(f)9 b Fs(\()p Fq(x)p Fs(\))p 2148 1010 265 4 v 2152 1086 a Fn(L)2209 1062 y Fp(n)2255 1086 y Fs(1\()p Fq(x)p Fs(\))2422 1029 y Fq(:)456 1224 y Fs(The)31 b(ab)r(o)n(v)n(e)f(de\014nition)i (needs)g(a)f(few)h(commen)n(ts)f(to)h(con)n(vince)e(the)i(reader)e (that)i(it)g(is)g(w)n(ell)456 1323 y(p)r(osed.)k(T)-7 b(o)27 b(start)g(with)g(notice)h(that)f(Condition)g(1)g(implies)h(that) f(the)h(ratio)e(is)h(w)n(ell)g(de\014ned.)456 1423 y(Second)18 b(the)h(existence)f(of)g(the)h(limit)g(is)f(assured)g(b)n(y)g(the)h (fact)f(that)h(the)g(sequence)f(is)g(increasing)456 1522 y(and)27 b(b)r(ounded,)h(indeed)456 1978 y(\(2.3\))1071 1780 y(inf)997 1833 y Fp(x)p Fj(2)p Fp(D)1134 1841 y Fh(n)p Fi(+1)1269 1723 y Fn(L)1326 1693 y Fp(n)p Fl(+1)1456 1723 y Fq(f)9 b Fs(\()p Fq(x)p Fs(\))p 1269 1760 349 4 v 1273 1836 a Fn(L)1330 1812 y Fp(n)p Fl(+1)1460 1836 y Fs(1\()p Fq(x)p Fs(\))1650 1780 y(=)96 b(inf)1738 1833 y Fp(x)p Fj(2)p Fp(D)1875 1841 y Fh(n)p Fi(+1)2010 1693 y Fn(L)p FB(1)2115 1705 y Fp(D)2169 1713 y Fh(n)2228 1601 y Fk(h)2267 1693 y Fn(L)2324 1663 y Fp(n)2370 1693 y Fs(1)2422 1656 y Fj(L)2468 1631 y Fh(n)2508 1656 y Fp(f)p 2421 1674 126 4 v 2424 1722 a Fj(L)2470 1705 y Fh(n)2511 1722 y Fl(1)2557 1601 y Fk(i)p 2010 1760 586 4 v 2189 1836 a Fn(L)2246 1812 y Fp(n)p Fl(+1)2375 1836 y Fs(1)1650 2009 y Fn(\025)61 b Fs(inf)1738 2062 y Fp(x)p Fj(2)p Fp(D)1875 2070 y Fh(n)1939 1953 y Fn(L)1996 1923 y Fp(n)2041 1953 y Fq(f)9 b Fs(\()p Fq(x)p Fs(\))p 1939 1990 265 4 v 1943 2066 a Fn(L)2000 2042 y Fp(n)2045 2066 y Fs(1\()p Fq(x)p Fs(\))2300 2009 y(inf)2227 2062 y Fp(x)p Fj(2)p Fp(D)2364 2070 y Fh(n)p Fi(+1)2499 1953 y Fn(L)p FB(1)2604 1965 y Fp(D)2658 1973 y Fh(n)2717 1953 y Fs([)p Fn(L)2797 1923 y Fp(n)2842 1953 y Fs(1])p 2499 1990 409 4 v 2589 2066 a Fn(L)2646 2042 y Fp(n)p Fl(+1)2775 2066 y Fs(1)1650 2238 y(=)61 b(inf)1738 2292 y Fp(x)p Fj(2)p Fp(D)1875 2300 y Fh(n)1939 2182 y Fn(L)1996 2152 y Fp(n)2041 2182 y Fq(f)9 b Fs(\()p Fq(x)p Fs(\))p 1939 2219 265 4 v 1943 2295 a Fn(L)2000 2271 y Fp(n)2045 2295 y Fs(1\()p Fq(x)p Fs(\))2213 2238 y(;)456 2433 y(and)1355 2575 y Fn(\000k)p Fq(f)g Fn(k)1554 2587 y Fj(1)1646 2575 y Fn(\024)60 b Fs(inf)1733 2629 y Fp(x)p Fj(2)p Fp(D)1870 2637 y Fh(n)1934 2519 y Fn(L)1991 2489 y Fp(n)2037 2519 y Fq(f)9 b Fs(\()p Fq(x)p Fs(\))p 1934 2556 V 1938 2632 a Fn(L)1995 2608 y Fp(n)2041 2632 y Fs(1\()p Fq(x)p Fs(\))2231 2575 y Fn(\024)23 b(k)p Fq(f)9 b Fn(k)2453 2587 y Fj(1)2522 2575 y Fq(:)555 2764 y Fs(The)28 b(relev)-5 b(an)n(t)27 b(prop)r(erties)f(of)i(the)g(ab)r(o)n(v)n(e)e(functional)i(are)f(the)h (follo)n(wing:)2922 2732 y Fl(3)661 2889 y Fn(\017)41 b Fs(\003\(1\))23 b(=)g(1)13 b(;)661 2989 y Fn(\017)41 b Fs(\003)28 b(is)f(con)n(tin)n(uous)g(in)h(the)g Fq(L)1626 2959 y Fj(1)1723 2989 y Fs(norm;)661 3088 y Fn(\017)41 b Fq(f)32 b Fn(\025)23 b Fq(g)30 b Fs(implies)e(\003\()p Fq(f)9 b Fs(\))23 b Fn(\025)f Fs(\003\()p Fq(g)s Fs(\))28 b(\(monotonicit)n(y\);)661 3188 y Fn(\017)41 b Fs(\003\()p Fq(\025f)9 b Fs(\))24 b(=)e Fq(\025)p Fs(\003\()p Fq(f)9 b Fs(\))28 b(\(homogeneit)n(y\);)661 3288 y Fn(\017)41 b Fs(\003\()p Fq(f)27 b Fs(+)18 b Fq(g)s Fs(\))23 b Fn(\025)g Fs(\003\()p Fq(f)9 b Fs(\))18 b(+)g(\003\()p Fq(g)s Fs(\))28 b(\(sup)r(er-additivit)n(y\);)661 3387 y Fn(\017)41 b(8)p Fq(b)22 b Fn(2)i Fo(R)p Fs(,)33 b(\003\()p Fq(f)27 b Fs(+)18 b Fq(b)p Fs(\))23 b(=)g(\003\()p Fq(f)9 b Fs(\))18 b(+)g Fq(b)c Fs(;)661 3487 y Fn(\017)41 b Fs(if)28 b(for)g Fq(p)22 b Fn(\032)h Fq(I)35 b Fs(there)27 b(exists)g Fq(n)c Fn(2)h Fo(N)37 b Fs(suc)n(h)27 b(that)h Fq(p)19 b Fn(\\)f Fq(X)2421 3499 y Fp(n)2489 3487 y Fs(=)23 b Fn(;)p Fs(,)k(then)h(\003\()p FB(1)2996 3499 y Fp(p)3034 3487 y Fs(\))c(=)e(0)14 b(.)3256 3455 y Fl(4)456 3612 y Fs(All)28 b(the)g(ab)r(o)n(v)n(e)e(follo)n(ws)g(immediately)i(from)f (the)h(de\014nition.)456 3737 y FB(Remark)33 b(2.1.)42 b Fr(Note)32 b(that,)h(at)f(the)g(moment,)h(it)f(is)h(not)e(cle)l(ar)i (if)g(the)f(functional)h(is)g(line)l(ar)456 3837 y(or)d(not,)g(yet)f (homo)l(geneity)j(and)e(sup)l(er-additivity)i(imply)f(at)e(le)l(ast)h (c)l(onvexity.)456 3962 y FB(De\014nition)g(2.1.)40 b Fr(Set)30 b Fq(\032)23 b Fs(=)f(\003\()p Fn(L)p Fs(1\))p Fr(.)456 4087 y FB(Lemma)29 b(2.2.)40 b Fr(Under)30 b(c)l(ondition)h FB(C1)e Fr(we)h(have)h Fq(\032)23 b Fn(\024)g Fq(c)p Fr(.)456 4259 y(Pr)l(o)l(of.)43 b Fs(Let)456 4458 y(\(2.4\))925 b Fq(\032)1595 4470 y Fp(n)1663 4458 y Fs(:=)61 b(inf)1774 4511 y Fp(x)p Fj(2)p Fp(D)1911 4519 y Fh(n)1975 4402 y Fn(L)2032 4371 y Fp(n)p Fl(+1)2162 4402 y Fs(1\()p Fq(x)p Fs(\))p 1975 4439 341 4 v 2017 4515 a Fn(L)2074 4491 y Fp(n)2119 4515 y Fs(1\()p Fq(x)p Fs(\))2325 4458 y Fq(;)456 4657 y Fs(then)28 b(b)n(y)f(\(2.2\))56 b(lim)958 4707 y Fp(n)p Fj(!1)1146 4657 y Fq(\032)1189 4669 y Fp(n)1257 4657 y Fs(=)22 b Fq(\032)p Fs(.)37 b(Accordingly)-7 b(,)1471 4846 y FB(1)1519 4858 y Fp(D)1573 4866 y Fh(n)1618 4846 y Fn(L)1675 4811 y Fp(n)1720 4846 y Fs(1)p Fq(\032)1805 4858 y Fp(n)1873 4846 y Fn(\024)22 b FB(1)2008 4858 y Fp(D)2062 4866 y Fh(n)p Fi(+1)2178 4846 y Fn(L)2235 4811 y Fp(n)p Fl(+1)2365 4846 y Fs(1)p Fq(:)p 456 4945 499 4 v 555 5019 a Fl(3)588 5044 y FA(Essen)n(tially)k(the)g(prop)r(erties) g(of)g(\003)f(are)h(similar)c(to)27 b(the)f(ones)g(of)g(an)g(inner)f (measure.)36 b(In)26 b(the)h(follo)n(wing)456 5127 y(w)n(e)c(will)g (see)h(that,)g(under)g(certain)h(conditions,)f(it)f(is)g(indeed)i(a)f (measure.)555 5190 y Fl(4)588 5216 y FA(This)f(follo)n(ws)g(remem)n(b)r (ering)f(equation)j(\(1.3\).)p eop %%Page: 7 7 7 6 bop 1339 251 a Fl(LASOT)-5 b(A-YORKE)29 b(MAPS)f(WITH)h(HOLES)850 b(7)456 450 y Fs(In)n(tegrating)26 b(the)i(ab)r(o)n(v)n(e)e(equation)h (with)h(resp)r(ect)f(to)h Fq(m)p Fs(,)g(and)f(remem)n(b)r(ering)g (\(2.1\),)g(yields)1523 640 y Fq(e)1562 606 y Fj(\000)p Fp(P)1669 640 y Fq(\032)1712 652 y Fp(n)1780 640 y Fn(\024)1921 584 y Fq(m)p Fs(\()p Fq(X)2095 596 y Fp(n)2140 584 y Fs(\))p 1878 621 337 4 v 1878 697 a Fq(m)p Fs(\()p Fq(X)2052 709 y Fp(n)p Fj(\000)p Fl(1)2182 697 y Fs(\))2247 640 y Fn(\024)c Fs(1)456 826 y(whic)n(h)k(pro)r(duces)g(the)h(w)n(an)n(ted) f(result)g(b)n(y)h(taking)f(the)h(limit)g Fq(n)23 b Fn(!)g(1)p Fs(.)680 b Ff(\003)555 986 y Fs(T)-7 b(o)28 b(con)n(tin)n(ue)f(w)n(e)h (need)g(to)g(imp)r(ose)g(one)f(extra)h(condition)f(on)h(the)g(system.) 38 b(T)-7 b(o)28 b(do)g(so)f(w)n(e)456 1085 y(need)35 b(some)g(notation.)60 b(Let)36 b Fn(Z)1485 1055 y Fl(\()p Fp(n)p Fl(\))1617 1085 y Fs(b)r(e)g(the)g(partition)f(of)g(smo)r (othness)g(\(or)g(monotonicit)n(y\))456 1185 y(in)n(terv)-5 b(als)31 b(of)h Fq(T)953 1155 y Fp(n)997 1185 y Fs(.)51 b(Next)32 b(let)h Fn(A)1469 1197 y Fp(n)1546 1185 y Fs(b)r(e)g(the)f (set)h(of)f(\014nite)g(partitions)g(in)g(in)n(terv)-5 b(als)31 b Fq(A)g Fs(=)f Fn(f)p Fq(A)3375 1197 y Fp(i)3403 1185 y Fn(g)456 1293 y Fs(suc)n(h)i(that)832 1230 y Fk(W)901 1318 y Fp(A)951 1326 y Fh(i)996 1293 y Fq(g)1036 1305 y Fp(n)1111 1293 y Fn(\024)f Fs(2)p Fn(k)p Fq(g)1331 1305 y Fp(n)1375 1293 y Fn(k)1417 1305 y Fj(1)1487 1293 y Fs(.)1510 1261 y Fl(5)1594 1293 y Fs(Giv)n(en)h Fq(n)f Fn(2)g Fo(N)43 b Fs(and)32 b Fq(A)f Fn(2)g(A)2510 1305 y Fp(i)2571 1293 y Fs(let)2720 1272 y(^)2696 1293 y Fn(Z)2763 1263 y Fl(\()p Fp(n)p Fl(\))2892 1293 y Fs(b)r(e)i(the)f(coarsest)456 1408 y(partition)c(in)h(in)n(terv)-5 b(als)28 b(among)g(all)g(the)h (ones)g(\014ner)f(than)h(b)r(oth)g Fq(A)g Fs(and)g Fn(Z)2854 1378 y Fl(\()p Fp(n)p Fl(\))2980 1408 y Fs(and)f(enjo)n(ying)456 1508 y(the)j(prop)r(ert)n(y)f(that)h(the)g(elemen)n(ts)g(of)g(the)h (partition)e(are)g(either)h(disjoin)n(t)g(or)f(con)n(tained)g(in)456 1607 y Fq(X)525 1619 y Fp(n)p Fj(\000)p Fl(1)655 1607 y Fs(.)37 b(Finally)-7 b(,)27 b(let)1375 1729 y Fn(Z)1442 1686 y Fl(\()p Fp(n)p Fl(\))1435 1741 y Fj(\003)1562 1729 y Fs(=)c Fn(f)p Fq(Z)28 b Fn(2)1879 1708 y Fs(^)1855 1729 y Fn(Z)1922 1695 y Fl(\()p Fp(n)p Fl(\))2042 1729 y Fn(j)23 b Fq(Z)29 b Fn(\032)22 b Fq(X)2330 1741 y Fp(n)p Fj(\000)p Fl(1)2460 1729 y Fn(g)p Fq(;)1104 1877 y Fn(Z)1171 1834 y Fl(\()p Fp(n)p Fl(\))1164 1902 y Fp(b)1291 1877 y Fs(=)h Fn(f)p Fq(Z)28 b Fn(2)1608 1856 y Fs(^)1584 1877 y Fn(Z)1651 1843 y Fl(\()p Fp(n)p Fl(\))1771 1877 y Fn(j)23 b Fq(Z)29 b Fn(\032)23 b Fq(X)2060 1889 y Fp(n)p Fj(\000)p Fl(1)2217 1877 y Fs(and)28 b(\003\()p FB(1)2517 1889 y Fp(Z)2570 1877 y Fs(\))23 b(=)g(0)p Fn(g)1012 2021 y Fs(and)k Fn(Z)1240 1987 y Fl(\()p Fp(n)p Fl(\))1233 2042 y Fp(g)1360 2021 y Fs(=)c Fn(f)p Fq(Z)28 b Fn(2)1678 2000 y Fs(^)1653 2021 y Fn(Z)1720 1987 y Fl(\()p Fp(n)p Fl(\))1840 2021 y Fn(j)c Fq(Z)k Fn(\032)23 b Fq(X)2129 2033 y Fp(n)p Fj(\000)p Fl(1)2286 2021 y Fs(and)28 b(\003\()p FB(1)2586 2033 y Fp(Z)2639 2021 y Fs(\))23 b Fq(>)g Fs(0)p Fn(g)p Fq(:)555 2173 y Fs(As)d(w)n(e)f(will)g(see)g(in)h(the)f(pro)r (of)g(of)g(Lemma)h(2.5,)g(the)f(elemen)n(ts)h(of)f Fn(Z)2634 2130 y Fl(\()p Fp(n)p Fl(\))2627 2198 y Fp(b)2750 2173 y Fs(are)g(the)g(problematic)456 2293 y(ones)k(and)h(those)f(of)h Fn(Z)1166 2250 y Fl(\()p Fp(n)p Fl(\))1159 2302 y Fp(g)1287 2293 y Fs(are)f(the)i(go)r(o)r(d)e(ones.)35 b(W)-7 b(e)25 b(allo)n(w)d Fn(Z)2394 2250 y Fl(\()p Fp(n)p Fl(\))2387 2318 y Fp(b)2515 2293 y Fs(to)i(b)r(e)h(non)e(empt)n(y)i(pro)n(vided) 456 2392 y(it)j(satis\014es)e(the)i(follo)n(wing)f(condition)g FB(C2)p Fs(.)456 2525 y FB(De\014nition)48 b(2.3.)h Fr(We)43 b(wil)t(l)i(c)l(al)t(l)53 b Fs(con)n(tiguous)42 b Fr(two)h(elements)h (of)g Fn(Z)2759 2482 y Fl(\()p Fp(n)p Fl(\))2752 2536 y Fj(\003)2899 2525 y Fr(that)g(ar)l(e)g(either)456 2624 y(c)l(ontiguous,)39 b(in)f(the)g(usual)f(sense,)j(or)f(sep)l(ar)l(ate)l (d)f(by)g(a)g(c)l(onne)l(cte)l(d)g(c)l(omp)l(onent)f(of)i Fq(Y)3274 2636 y Fp(n)3357 2624 y Fs(:=)456 2695 y Fp(n)p Fj(\000)p Fl(1)472 2719 y Fk([)465 2896 y Fp(i)p Fl(=0)595 2798 y Fq(T)656 2764 y Fj(\000)p Fp(i)735 2798 y Fq(Y)19 b Fr(.)456 2992 y FB(Condition)30 b(2.)41 b Fr(We)30 b(wil)t(l)h(c)l(onsider)g(only)f(systems)f(that)h(satisfy)h(the)f(fol)t (lowing)i(c)l(ondition:)645 3112 y FB(C2:)40 b Fr(Ther)l(e)31 b(exists)e(c)l(onstants)f Fq(K)h Fn(\025)22 b Fs(0)p Fr(,)30 b(and)f Fq(\030)f Fn(\025)22 b Fs(1)p Fr(,)30 b(such)f(that)g(for)h(e)l(ach)h Fq(n)23 b Fn(2)g Fo(N)39 b Fr(ther)l(e)744 3224 y(exists)31 b Fq(A)25 b Fn(2)h(A)1207 3236 y Fp(n)1283 3224 y Fr(such)31 b(that)g(at)g(most)g Fq(K)6 b(\030)2065 3193 y Fp(n)2140 3224 y Fr(elements)31 b(of)h Fn(Z)2649 3180 y Fl(\()p Fp(n)p Fl(\))2642 3249 y Fp(b)2777 3224 y Fr(ar)l(e)f(c)l(ontiguous.)42 b(In)744 3323 y(addition,)32 b Fq(\030)t Fs(\002)23 b Fq(<)g(\032)p Fr(.)456 3444 y(Note)29 b(that)h(this)g(implies,)i(in)e(p)l(articular)g Fs(\002)23 b Fq(<)g(\032)p Fr(.)456 3565 y FB(Remark)38 b(2.2.)45 b Fr(Note)36 b(that)g(c)l(ondition)h FB(C2)f Fr(implies)h(that)g(ther)l(e)f(exists)k Fs(\026)-46 b Fq(n)34 b Fn(2)h Fo(N)46 b Fr(such)36 b(that)456 3664 y Fq(D)525 3676 y Fp(n)593 3664 y Fs(=)22 b Fq(D)753 3676 y Fl(\026)-37 b Fp(n)824 3664 y Fr(for)30 b(al)t(l)h Fq(n)23 b Fn(\025)k Fs(\026)-46 b Fq(n)p Fr(,)30 b(sinc)l(e)g(if)g(the) g(latter)g(wer)l(e)g(false)h(it)f(would)g(fol)t(low)i Fq(\032)23 b Fs(=)g(0)p Fr(.)555 3785 y Fs(The)28 b(follo)n(wing)e(is)i (y)n(et)f(another)g(simple)h(consequence)e(of)i FB(C2)p Fs(.)456 3917 y FB(Lemma)h(2.4.)40 b Fr(Condition)31 b(2)g(implies)g(that)f(for)g(al)t(l)h Fq(n)23 b Fn(2)g Fo(N)t Fr(,)36 b Fn(Z)2483 3874 y Fl(\()p Fp(n)p Fl(\))2476 3927 y Fp(g)2603 3917 y Fn(6)p Fs(=)23 b Fn(;)p Fr(.)456 4094 y(Pr)l(o)l(of.)43 b Fs(Supp)r(ose)27 b(that)g Fn(Z)1287 4051 y Fl(\()p Fp(n)p Fl(\))1280 4103 y Fp(g)1407 4094 y Fs(=)c Fn(;)j Fs(for)h(some)f Fq(n)p Fs(,)h(then)h(it)f(m)n(ust)g(b)r (e)h Fn(Z)2653 4051 y Fl(\()p Fp(m)p Fl(\))2646 4103 y Fp(g)2790 4094 y Fs(=)23 b Fn(;)k Fs(for)f(all)h Fq(m)c Fn(\025)g Fq(n)p Fs(.)456 4210 y(Assume)30 b(that)g Fn(Z)1015 4167 y Fl(\()p Fp(n)p Fl(\))1008 4219 y Fp(g)1140 4210 y Fs(=)d Fn(;)p Fs(,)j(then)h Fn(Z)1586 4167 y Fl(\()p Fp(n)p Fl(\))1579 4221 y Fj(\003)1710 4210 y Fs(=)c Fn(Z)1869 4167 y Fl(\()p Fp(n)p Fl(\))1862 4235 y Fp(b)1966 4210 y Fs(,)k(th)n(us)f(the)h(n)n(um)n(b)r(er)f(of)g(elemen)n(ts)g(in)g Fn(Z)3261 4167 y Fl(\()p Fp(n)p Fl(\))3254 4221 y Fj(\003)3389 4210 y Fs(is)456 4330 y(smaller)c(than)i Fq(K)6 b(\030)1053 4300 y Fp(n)1125 4330 y Fs(\(the)29 b(elemen)n(ts)e(of)h Fn(Z)1802 4287 y Fl(\()p Fp(n)p Fl(\))1795 4355 y Fp(b)1926 4330 y Fs(m)n(ust)g(b)r(e)g(all)f(con)n(tiguous\).)36 b(Then,)1217 4493 y Fn(L)1274 4458 y Fp(n)1320 4493 y Fs(1\()p Fq(x)p Fs(\))23 b Fn(\024)1640 4414 y Fk(X)1584 4614 y Fp(Z)t Fj(2Z)1731 4583 y Fi(\()p Fh(n)p Fi(\))1726 4624 y Fg(\003)1830 4493 y Fs(sup)14 b Fq(g)2012 4458 y Fl(\()p Fp(n)p Fl(\))2132 4493 y Fn(\024)22 b Fs(sup)14 b Fq(g)2401 4458 y Fl(\()p Fp(n)p Fl(\))2498 4493 y Fq(K)6 b(\030)2615 4458 y Fp(n)2660 4493 y Fq(:)555 4745 y Fs(On)28 b(the)g(other)f(hand,)g(remem)n(b)r(ering)g(\(2.4\),)g(w)n(e)g(ha)n(v)n (e,)g(for)g(eac)n(h)g Fq(x)c Fn(2)h Fq(D)2873 4757 y Fj(1)2943 4745 y Fs(,)1272 4966 y Fn(j)p Fq(g)1338 4932 y Fl(\()p Fp(n)p Fl(\))1435 4966 y Fn(j)1458 4978 y Fj(1)1529 4966 y Fq(K)6 b(\030)1646 4932 y Fp(n)1713 4966 y Fn(\025)1801 4862 y Fp(n)p Fj(\000)p Fl(1)1811 4887 y Fk(Y)1810 5064 y Fp(i)p Fl(=0)1951 4910 y Fn(L)2008 4880 y Fp(i)p Fl(+1)2120 4910 y Fs(1\()p Fq(x)p Fs(\))p 1951 4947 323 4 v 1993 5023 a Fn(L)2050 4999 y Fp(i)2078 5023 y Fs(1\()p Fq(x)p Fs(\))2306 4966 y Fn(\025)2394 4862 y Fp(n)p Fj(\000)p Fl(1)2404 4887 y Fk(Y)2403 5064 y Fp(i)p Fl(=0)2534 4966 y Fq(\032)2577 4978 y Fp(i)2604 4966 y Fq(:)p 456 5117 499 4 v 555 5190 a Fl(5)588 5216 y FA(Suc)n(h)25 b(partitions)f(alw)n (a)n(ys)g(exist,)f(if)g(in)g(doubt)i(see)f([R])f(Lemma)f(6.)p eop %%Page: 8 8 8 7 bop 456 251 a Fl(8)388 b(CARLANGELO)23 b(LIVERANI)f(AND)g(V)1947 236 y(\023)1941 251 y(ER)n(ONIQUE)g(MA)n(UME-DESCHAMPS)456 450 y Fs(Next,)28 b(taking)f(the)h(logarithm)e(of)i(b)r(oth)g(sides)f (and)g(the)h(limit)h(for)e Fq(n)c Fn(!)g(1)p Fs(,)k(w)n(e)h(get)1308 672 y(ln)14 b Fq(\030)23 b Fs(+)18 b(ln)c(\002)22 b Fn(\025)52 b Fs(lim)1791 722 y Fp(n)p Fj(!1)1992 616 y Fs(1)p 1988 653 50 4 v 1988 729 a Fq(n)2062 568 y Fp(n)p Fj(\000)p Fl(1)2065 593 y Fk(X)2071 770 y Fp(i)p Fl(=0)2202 672 y Fs(ln)14 b Fq(\032)2328 684 y Fp(i)2378 672 y Fs(=)23 b(ln)14 b Fq(\032)456 897 y Fs(\(recall)27 b(that)g(b)n(y)h (de\014nition)g Fq(\032)23 b Fs(=)f(lim)14 b Fq(\032)1699 909 y Fp(i)1727 897 y Fs(\),)28 b(con)n(trary)e(to)h(condition)g FB(C2)p Fs(.)632 b Ff(\003)555 1065 y Fs(Under)28 b(Condition)f(2)g(w)n (e)h(will)g(sho)n(w)e(that)i(the)g(cone)456 1212 y(\(2.5\))504 b Fn(C)1175 1224 y Fp(a)1238 1212 y Fs(:=)22 b Fn(f)p Fq(h)h Fn(2)g Fs(BV)g Fn(j)g Fq(h)g Fn(6\021)g Fs(0;)36 b Fq(h)23 b Fn(\025)g Fs(0;)2249 1133 y Fk(_)2355 1212 y Fq(h)g Fn(\024)g Fq(a)p Fs(\003\()p Fq(h)p Fs(\))p Fn(g)456 1372 y Fs(is)k(strictly)g(in)n(v)-5 b(arian)n(t)27 b(for)g(the)h(T)-7 b(ransfer)26 b(op)r(erator)g Fn(L)p Fs(.)555 1471 y(The)i(\014rst)f(step)h(is)f(to)h(obtain)f(a)g(suitable) h(Lasota-Y)-7 b(ork)25 b(t)n(yp)r(e)j(inequalit)n(y)-7 b(.)456 1595 y FB(Lemma)29 b(2.5.)40 b Fr(F)-6 b(or)30 b(any)g Fq(\022)c Fn(\025)c Fs(\002)p Fq(\030)t Fr(,)30 b Fq(h)23 b Fn(2)g Fq(B)t(V)c Fr(,)31 b(we)f(have)1363 1675 y Fk(_)1469 1754 y Fn(L)1526 1720 y Fp(n)1571 1754 y Fq(h)23 b Fn(\024)g Fq(C)1789 1766 y Fp(\022)1827 1754 y Fq(\022)1868 1720 y Fp(n)1927 1675 y Fk(_)2033 1754 y Fq(h)18 b Fs(+)g Fq(K)2253 1766 y Fp(n)2298 1754 y Fs(\003\()p Fn(j)p Fq(h)p Fn(j)p Fs(\))p Fq(;)456 1914 y Fr(wher)l(e)30 b Fq(C)749 1926 y Fp(\022)817 1914 y Fr(and)g Fq(K)1049 1926 y Fp(n)1123 1914 y Fr(do)h(not)e(dep)l(end)i (on)f Fq(h)p Fr(.)456 2093 y(Pr)l(o)l(of.)43 b Fs(Notice)33 b(that,)i(if)e Fq(Z)38 b Fn(2)1480 2072 y Fs(^)1456 2093 y Fn(Z)1523 2063 y Fl(\()p Fp(n)p Fl(\))1620 2093 y Fn(nZ)1729 2050 y Fl(\()p Fp(n)p Fl(\))1722 2105 y Fj(\003)1825 2093 y Fs(,)d(then)e Fn(L)2134 2063 y Fp(n)2180 2093 y Fs(\()p Fq(h)p FB(1)2308 2105 y Fp(Z)2361 2093 y Fs(\))g(=)f(0)g(for) h(eac)n(h)f Fq(h)g Fn(2)h Fs(BV,)i(since)456 2193 y Fq(Z)24 b Fn(\\)18 b Fq(X)679 2205 y Fp(n)p Fj(\000)p Fl(1)832 2193 y Fs(=)23 b Fn(;)p Fs(.)555 2292 y(W)-7 b(e)28 b(can)f(then)i (write)1082 2440 y Fn(L)1139 2406 y Fp(n)1184 2440 y Fq(h)23 b Fs(=)1399 2361 y Fk(X)1343 2561 y Fp(Z)t Fj(2Z)1490 2530 y Fi(\()p Fh(n)p Fi(\))1485 2571 y Fg(\003)1589 2440 y Fn(L)1646 2406 y Fp(n)1691 2440 y Fs(\()p FB(1)1771 2452 y Fp(Z)1825 2440 y Fq(h)p Fs(\))g(=)2072 2361 y Fk(X)2015 2561 y Fp(Z)t Fj(2Z)2162 2530 y Fi(\()p Fh(n)p Fi(\))2157 2571 y Fg(\003)2248 2440 y Fs(\()p FB(1)2328 2452 y Fp(Z)2381 2440 y Fq(g)2421 2452 y Fp(n)2466 2440 y Fq(h)p Fs(\))18 b Fn(\016)g Fq(T)2685 2404 y Fj(\000)p Fp(n)2673 2464 y(Z)2795 2440 y Fq(:)456 2694 y Fs(Accordingly)-7 b(,)1267 2739 y Fk(_)1374 2818 y Fn(L)1431 2784 y Fp(n)1476 2818 y Fq(h)23 b Fn(\024)1691 2739 y Fk(X)1635 2939 y Fp(Z)t Fj(2Z)1782 2909 y Fi(\()p Fh(n)p Fi(\))1777 2950 y Fg(\003)1881 2739 y Fk(_)1987 2818 y FB(1)2035 2830 y Fp(T)2083 2814 y Fh(n)2123 2830 y Fp(Z)2177 2818 y Fs(\()p Fq(g)2249 2830 y Fp(n)2294 2818 y Fq(h)p Fs(\))18 b Fn(\016)g Fq(T)2513 2783 y Fj(\000)p Fp(n)2501 2843 y(Z)2609 2818 y Fq(:)456 3054 y Fs(W)-7 b(e)28 b(will)f(compute)h (separately)e(eac)n(h)h(term)h(of)f(the)h(sum.)456 3701 y(\(2.6\))914 3135 y Fk(_)1020 3214 y FB(1)1068 3226 y Fp(T)1116 3210 y Fh(n)1157 3226 y Fp(Z)1210 3214 y Fs(\()p Fq(g)1282 3226 y Fp(n)1327 3214 y Fq(h)p Fs(\))19 b Fn(\016)f Fq(T)1547 3178 y Fj(\000)p Fp(n)1535 3238 y(Z)1666 3214 y Fn(\024)1754 3135 y Fk(_)1775 3313 y Fp(Z)1860 3214 y Fq(hg)1948 3226 y Fp(n)2038 3214 y Fs(+)46 b(2)14 b(sup)2242 3284 y Fp(Z)2343 3214 y Fn(j)p Fq(h)19 b Fn(\001)f Fq(g)2514 3226 y Fp(n)2559 3214 y Fn(j)1233 3442 y(\024)23 b Fs(3)1377 3363 y Fk(_)1398 3541 y Fp(Z)1482 3442 y Fq(hg)1570 3454 y Fp(n)1661 3442 y Fs(+)18 b(2)c(inf)1825 3495 y Fp(Z)1914 3442 y Fn(j)p Fq(h)19 b Fn(\001)f Fq(g)2085 3454 y Fp(n)2130 3442 y Fn(j)1233 3670 y(\024)23 b Fs(3)p Fn(k)p Fq(g)1445 3682 y Fp(n)1489 3670 y Fn(k)1531 3682 y Fj(1)1614 3591 y Fk(_)1636 3769 y Fp(Z)1720 3670 y Fq(h)c Fs(+)f(3)c(sup)1963 3739 y Fp(Z)2064 3670 y Fn(j)p Fq(h)p Fn(j)2172 3591 y Fk(_)2193 3769 y Fp(Z)2278 3670 y Fq(g)2318 3682 y Fp(n)2381 3670 y Fs(+)k(2)c(inf)2545 3723 y Fp(Z)2634 3670 y Fn(j)p Fq(h)19 b Fn(\001)f Fq(g)2805 3682 y Fp(n)2850 3670 y Fn(j)1233 3898 y(\024)23 b Fs(3)p Fn(k)p Fq(g)1445 3910 y Fp(n)1489 3898 y Fn(k)1531 3910 y Fj(1)1614 3819 y Fk(_)1636 3997 y Fp(Z)1720 3898 y Fq(h)c Fs(+)f(6)p Fn(k)p Fq(g)1994 3910 y Fp(n)2038 3898 y Fn(k)2080 3910 y Fj(1)2163 3898 y Fs(sup)2201 3967 y Fp(Z)2302 3898 y Fn(j)p Fq(h)p Fn(j)g Fs(+)h(2)p Fn(k)p Fq(g)2622 3910 y Fp(n)2665 3898 y Fn(k)2707 3910 y Fj(1)2791 3898 y Fs(inf)2817 3951 y Fp(Z)2906 3898 y Fn(j)p Fq(h)p Fn(j)1233 4126 y(\024)k Fs(9)p Fn(k)p Fq(g)1445 4138 y Fp(n)1489 4126 y Fn(k)1531 4138 y Fj(1)1614 4047 y Fk(_)1636 4225 y Fp(Z)1720 4126 y Fq(h)c Fs(+)f(8)p Fn(k)p Fq(g)1994 4138 y Fp(n)2038 4126 y Fn(k)2080 4138 y Fj(1)2163 4126 y Fs(inf)2189 4179 y Fp(Z)2278 4126 y Fn(j)p Fq(h)p Fn(j)p Fq(:)456 4376 y Fs(Next,)41 b(note)d(that)g(if)g Fq(Z)46 b Fn(2)41 b(Z)1432 4333 y Fl(\()p Fp(n)p Fl(\))1425 4386 y Fp(g)1529 4376 y Fs(,)g(then)d(b)n(y)g(de\014nition,)j(there)d (exists)g Fq(")2825 4388 y Fp(n)2910 4376 y Fq(>)i Fs(0)e(suc)n(h)f (that)521 4476 y(inf)456 4551 y Fp(Z)t Fj(2Z)603 4520 y Fi(\()p Fh(n)p Fi(\))598 4560 y Fh(g)702 4476 y Fs(\003\()p FB(1)840 4488 y Fp(Z)893 4476 y Fs(\))44 b Fn(\025)g Fs(2)p Fq(")1159 4488 y Fp(n)1248 4476 y Fq(>)f Fs(0,)g(it)e(is)f(p)r (ossible)g(to)g(c)n(ho)r(ose)f Fq(N)2438 4488 y Fp(n)2527 4476 y Fn(2)45 b Fo(N)50 b Fs(suc)n(h)40 b(that,)k(for)39 b(eac)n(h)456 4641 y Fq(x)23 b Fn(2)h Fq(D)674 4653 y Fp(N)727 4661 y Fh(n)771 4641 y Fs(,)1590 4809 y(inf)1524 4885 y Fp(Z)t Fj(2Z)1671 4854 y Fi(\()p Fh(n)p Fi(\))1666 4894 y Fh(g)1780 4753 y Fn(L)1837 4723 y Fp(N)1890 4731 y Fh(n)1935 4753 y FB(1)1983 4765 y Fp(Z)2036 4753 y Fs(\()p Fq(x)p Fs(\))p 1780 4790 368 4 v 1810 4866 a Fn(L)1867 4842 y Fp(N)1920 4850 y Fh(n)1965 4866 y Fs(1\()p Fq(x)p Fs(\))2181 4809 y Fn(\025)f Fq(")2308 4821 y Fp(n)2353 4809 y Fq(:)456 5036 y Fs(Accordingly)-7 b(,)26 b(for)h(eac)n(h)g Fq(x)d Fn(2)f Fq(D)1465 5048 y Fp(N)1518 5056 y Fh(n)1562 5036 y Fs(,)28 b Fq(h)23 b Fn(2)g Fs(BV)28 b(and)f Fq(Z)i Fn(2)23 b(Z)2303 4993 y Fl(\()p Fp(n)p Fl(\))2296 5045 y Fp(g)2428 5036 y Fs(holds)987 5187 y Fn(L)1044 5153 y Fp(N)1097 5161 y Fh(n)1142 5187 y Fs(\()p Fn(j)p Fq(h)p Fn(j)p FB(1)1316 5199 y Fp(Z)1369 5187 y Fs(\()p Fq(x)p Fs(\)\))i Fn(\025)d Fs(inf)1650 5240 y Fp(Z)1739 5187 y Fn(j)p Fq(h)p Fn(jL)1890 5153 y Fp(N)1943 5161 y Fh(n)1988 5187 y FB(1)2036 5199 y Fp(Z)2089 5187 y Fs(\()p Fq(x)p Fs(\))i Fn(\025)f Fs(inf)2338 5240 y Fp(Z)2427 5187 y Fn(j)p Fq(h)p Fn(j)p Fq(")2560 5199 y Fp(n)2605 5187 y Fn(L)2662 5153 y Fp(N)2715 5161 y Fh(n)2759 5187 y Fs(1\()p Fq(x)p Fs(\))p eop %%Page: 9 9 9 8 bop 1339 251 a Fl(LASOT)-5 b(A-YORKE)29 b(MAPS)f(WITH)h(HOLES)850 b(9)456 454 y Fs(T)-7 b(o)30 b(deal)g(with)i(the)f Fq(Z)j Fn(2)29 b(Z)1340 411 y Fl(\()p Fp(n)p Fl(\))1333 479 y Fp(b)1467 454 y Fs(w)n(e)i(m)n(ust)g(use)f(condition)h FB(C2)p Fs(.)46 b(Note)31 b(that)g(the)g(elemen)n(ts)g(of)456 574 y Fn(Z)523 531 y Fl(\()p Fp(n)p Fl(\))516 584 y Fp(g)651 574 y Fs(can)f(b)r(e)i(separated)d(b)n(y)-7 b(,)32 b(at)f(most,)h Fq(K)6 b(\030)1889 544 y Fp(n)1964 574 y Fs(elemen)n(ts)31 b(of)g Fn(Z)2472 531 y Fl(\()p Fp(n)p Fl(\))2465 599 y Fp(b)2569 574 y Fs(.)47 b(F)-7 b(or)31 b(eac)n(h)f Fq(Z)k Fn(2)29 b(Z)3224 531 y Fl(\()p Fp(n)p Fl(\))3217 599 y Fp(b)3352 574 y Fs(let)456 694 y Fq(I)492 706 y Fj(\006)548 694 y Fs(\()p Fq(Z)6 b Fs(\))27 b(b)r(e)g(the)g(union)f(of) h(the)f(con)n(tiguous)g(elemen)n(ts)g(of)g Fn(Z)2331 651 y Fl(\()p Fp(n)p Fl(\))2324 719 y Fp(b)2455 694 y Fs(on)g(the)h(left)g(and)f(on)g(the)h(righ)n(t)456 794 y(of)g Fq(Z)6 b Fs(,)28 b(resp)r(ectiv)n(ely)-7 b(.)36 b(Clearly)-7 b(,)26 b(for)h(eac)n(h)g Fq(Z)1823 763 y Fj(0)1869 794 y Fn(\032)c Fq(I)1993 806 y Fj(\000)2049 794 y Fs(\()p Fq(Z)6 b Fs(\))28 b(\(or)f Fq(Z)2401 763 y Fj(0)2447 794 y Fn(\032)c Fq(I)2571 806 y Fl(+)2626 794 y Fs(\()p Fq(Z)6 b Fs(\)\),)28 b(holds)1503 947 y(inf)1518 1001 y Fp(Z)1567 984 y Fg(0)1618 947 y Fn(j)p Fq(h)p Fn(j)23 b(\024)f Fs(inf)1848 1001 y Fp(Z)1937 947 y Fn(j)p Fq(h)p Fn(j)d Fs(+)2176 868 y Fk(_)2133 1050 y Fp(I)2162 1058 y Fg(\000)2211 1050 y Fl(\()p Fp(Z)t Fl(\))2326 947 y Fq(h:)456 1177 y Fs(Accordingly)-7 b(,)1186 1320 y Fk(X)1130 1520 y Fp(Z)t Fj(2Z)1277 1489 y Fi(\()p Fh(n)p Fi(\))1272 1540 y Fh(b)1376 1399 y Fs(inf)1402 1452 y Fp(Z)1491 1399 y Fn(j)p Fq(h)p Fn(j)23 b(\024)f Fs(2)p Fq(K)6 b(\030)1854 1364 y Fp(n)1912 1207 y Fk(2)1912 1353 y(6)1912 1406 y(4)2024 1320 y(X)1968 1520 y Fp(Z)t Fj(2Z)2115 1489 y Fi(\()p Fh(n)p Fi(\))2110 1529 y Fh(g)2214 1399 y Fs(inf)2240 1452 y Fp(Z)2329 1399 y Fn(j)p Fq(h)p Fn(j)18 b Fs(+)2524 1320 y Fk(_)2630 1399 y Fq(h)2678 1207 y Fk(3)2678 1353 y(7)2678 1406 y(5)2747 1399 y Fq(:)555 1638 y Fs(W)-7 b(e)28 b(can)f(then)i(conclude)950 1696 y Fk(_)1056 1775 y Fn(L)1113 1740 y Fp(n)1158 1775 y Fq(h)23 b Fn(\024)g(k)p Fq(g)1399 1787 y Fp(n)1443 1775 y Fn(k)1485 1787 y Fj(1)1555 1775 y Fs(\(9)18 b(+)g(16)p Fq(K)6 b(\030)1931 1740 y Fp(n)1975 1775 y Fs(\))2021 1696 y Fk(_)2127 1775 y Fq(h)1307 1980 y Fs(+)18 b(8\(2)p Fq(K)6 b(\030)1623 1945 y Fp(n)1686 1980 y Fs(+)18 b(1\))p Fn(k)p Fq(g)1925 1992 y Fp(n)1969 1980 y Fn(k)2011 1992 y Fj(1)2081 1980 y Fq(")2120 1945 y Fj(\000)p Fl(1)2120 2000 y Fp(n)2279 1901 y Fk(X)2223 2101 y Fp(Z)t Fj(2Z)2370 2070 y Fi(\()p Fh(n)p Fi(\))2365 2111 y Fg(\003)2479 1923 y Fn(L)2536 1893 y Fp(N)2589 1901 y Fh(n)2633 1923 y Fn(j)p Fq(h)p Fn(j)p FB(1)2775 1935 y Fp(Z)2828 1923 y Fs(\()p Fq(x)p Fs(\))p 2479 1961 462 4 v 2555 2037 a Fn(L)2612 2013 y Fp(N)2665 2021 y Fh(n)2710 2037 y Fs(1\()p Fq(x)p Fs(\))1229 2240 y Fn(\024)23 b(k)p Fq(g)1399 2252 y Fp(n)1443 2240 y Fn(k)1485 2252 y Fj(1)1555 2240 y Fs(\(9)18 b(+)g(16)p Fq(K)6 b(\030)1931 2206 y Fp(n)1975 2240 y Fs(\))2021 2161 y Fk(_)2127 2240 y Fq(h)1307 2445 y Fs(+)18 b(8\(2)p Fq(K)6 b(\030)1623 2411 y Fp(n)1686 2445 y Fs(+)18 b(1\))p Fn(k)p Fq(g)1925 2457 y Fp(n)1969 2445 y Fn(k)2011 2457 y Fj(1)2081 2445 y Fq(")2120 2411 y Fj(\000)p Fl(1)2120 2465 y Fp(n)2219 2389 y Fn(L)2276 2359 y Fp(N)2329 2367 y Fh(n)2373 2389 y Fn(j)p Fq(h)p Fn(j)p Fs(\()p Fq(x)p Fs(\))p 2219 2426 361 4 v 2245 2502 a Fn(L)2302 2478 y Fp(N)2355 2486 y Fh(n)2400 2502 y Fs(1\()p Fq(x)p Fs(\))2589 2445 y Fq(:)456 2626 y Fs(T)-7 b(aking)33 b(the)h(inf)g(on)g Fq(x)h Fs(in)f(the)g(previous)f (expression)f(and)i(noticing)f(that,)j(b)n(y)e(h)n(yp)r(othesis,)456 2725 y(there)27 b(m)n(ust)h(exists)f Fq(C)1162 2737 y Fp(\022)1228 2725 y Fs(suc)n(h)g(that)h(\(9)18 b(+)g(16)p Fq(K)6 b(\030)1971 2695 y Fp(n)2015 2725 y Fs(\))p Fn(k)p Fq(g)2129 2737 y Fp(n)2173 2725 y Fn(k)2215 2737 y Fj(1)2308 2725 y Fn(\024)23 b Fq(C)2455 2737 y Fp(\022)2493 2725 y Fq(\022)2534 2695 y Fp(n)2607 2725 y Fs(yields)k(the)h(result.)170 b Ff(\003)1025 2901 y Fs(3.)41 b Ft(Transfer)33 b(Opera)-6 b(tor)32 b(and)f(Inv)-7 b(ariant)30 b(Cones)456 3051 y FB(Hilb)s(ert)i(metric.)40 b Fs(In)29 b(this)h(section,)f(w)n(e)g(in) n(tro)r(duce)g(a)f(theory)h(dev)n(elop)r(ed)g(b)n(y)g(G.)g(Birkho\013) 456 3150 y([Bi],)e(whic)n(h)h(is)f(highly)h(p)r(o)n(w)n(erful)f(to)g (analyzing)f(of)i(the)g(so)f(called)g(p)r(ositiv)n(e)g(op)r(erators.) 555 3250 y(W)-7 b(e)37 b(will)h(apply)e(it)h(to)g(study)g(the)h(P)n (erron-F)-7 b(rob)r(enius)34 b(op)r(erator)h(for)h(our)g(maps.)65 b(This)456 3350 y(strategy)31 b(has)i(b)r(een)h(\014rst)f(implemen)n (ted)g(in)h([FS])g(to)f(estimate)g(the)g(deca)n(y)g(of)g(correlations) 456 3449 y(for)c(some)f(random)h(dynamical)g(systems.)41 b(Then,)31 b(this)e(strategy)f(had)i(b)r(een)g(used)f(b)n(y)g(man)n(y) 456 3549 y(authors.)53 b(Let)34 b(us)g(men)n(tion)f(C.)h(Liv)n(erani)e ([L1])h(and)h(M.)g(Viana)f([V])h(for)f(Anoso)n(v)f(and)i(Ax-)456 3649 y(iom)c(A)h(di\013eomorphisms.)45 b(They)30 b(used)h(Birkho\013)f (cones)f(to)i(obtain)f(exp)r(onen)n(tial)g(deca)n(y)g(of)456 3748 y(correlations.)k(W)-7 b(e)28 b(use)g(this)g(tec)n(hnique)f(in)h (a)f(w)n(a)n(y)g(v)n(ery)f(close)h(to)g([L2])g(and)h([LSV].)456 3866 y FB(De\014nition)i(3.1.)40 b Fr(L)l(et)29 b Fn(V)37 b Fr(b)l(e)29 b(a)h(ve)l(ctor)g(sp)l(ac)l(e.)40 b(We)29 b(wil)t(l)i(c)l(al)t(l)g(c)l(onvex)f(c)l(one)f(a)h(subset)f Fn(C)f(\032)23 b(V)456 3966 y Fr(which)31 b(enjoys)g(the)f(fol)t (lowing)i(pr)l(op)l(erties)555 4066 y(\(i\))e Fn(C)23 b(\\)c(\000C)27 b Fs(=)c Fn(;)555 4165 y Fr(\(ii\))31 b Fn(8)p Fq(\025)22 b(>)h Fs(0)59 b Fq(\025)p Fn(C)28 b Fs(=)23 b Fn(C)555 4265 y Fr(\(iii\))31 b Fn(C)j Fr(is)d(a)f(c)l (onvex)f(set)555 4364 y(\(iv\))h Fn(8)p Fq(f)t(;)14 b(g)25 b Fn(2)f(C)j(8)p Fq(\013)1160 4376 y Fp(n)1228 4364 y Fn(2)c Fo(R)52 b Fq(\013)1465 4376 y Fp(n)1533 4364 y Fn(!)24 b Fq(\013;)60 b(g)21 b Fn(\000)d Fq(\013)1973 4376 y Fp(n)2018 4364 y Fq(f)32 b Fn(2)23 b(C)28 b(\))23 b Fq(g)e Fn(\000)d Fq(\013f)32 b Fn(2)24 b(C)f([)18 b(f)p Fs(0)p Fn(g)p Fr(.)555 4483 y Fs(W)-7 b(e)28 b(no)n(w)f(de\014ne)h(the) g(Hilb)r(ert)g(metric)g(on)f Fn(C)32 b Fs(:)456 4601 y FB(De\014nition)e(3.2.)40 b Fr(The)31 b(distanc)l(e)g Fq(d)1625 4613 y Fj(C)1668 4601 y Fs(\()p Fq(f)t(;)14 b(g)s Fs(\))30 b Fr(b)l(etwe)l(en)f(two)h(p)l(oints)g Fq(f)t(;)14 b(g)32 b Fr(in)e Fn(C)k Fr(is)c(given)h(by)1318 4738 y Fq(\013)p Fs(\()p Fq(f)t(;)14 b(g)s Fs(\))83 b(=)g(sup)p Fn(f)p Fq(\025)23 b(>)g Fs(0)p Fn(j)p Fq(g)d Fn(\000)e Fq(\025f)32 b Fn(2)24 b(C)5 b(g)1320 4862 y Fq(\014)t Fs(\()p Fq(f)t(;)14 b(g)s Fs(\))83 b(=)g(inf)7 b Fn(f)p Fq(\026)23 b(>)f Fs(0)p Fn(j)p Fq(\026f)27 b Fn(\000)18 b Fq(g)26 b Fn(2)d(C)5 b(g)1285 5035 y Fq(d)1328 5047 y Fj(C)1371 5035 y Fs(\()p Fq(f)t(;)14 b(g)s Fs(\))83 b(=)g(log)1923 4979 y Fq(\014)t Fs(\()p Fq(f)t(;)14 b(g)s Fs(\))p 1922 5016 243 4 v 1922 5092 a Fq(\013)p Fs(\()p Fq(f)t(;)g(g)s Fs(\))456 5216 y Fr(wher)l(e)30 b(we)g(take)g Fq(\013)24 b Fs(=)e(0)30 b Fr(or)g Fq(\014)d Fs(=)c Fn(1)29 b Fr(when)i(the)f(c)l(orr)l(esp)l(onding)h(sets)e(ar)l(e)h(empty.)p eop %%Page: 10 10 10 9 bop 456 251 a Fl(10)355 b(CARLANGELO)23 b(LIVERANI)f(AND)g(V)1947 236 y(\023)1941 251 y(ER)n(ONIQUE)g(MA)n(UME-DESCHAMPS)555 450 y Fs(The)30 b(distance)g Fq(d)1097 462 y Fj(C)1170 450 y Fs(is)g(a)f(pseudo-metric,)h(b)r(ecause)g(t)n(w)n(o)f(elemen)n (ts)g(can)h(b)r(e)g(at)g(an)g(in\014nite)456 550 y(distance)j(from)g (eac)n(h)g(others,)h(and)f(it)h(is)f(a)g(pro)5 b(jectiv)n(e)32 b(metric)i(b)r(ecause)f(an)n(y)f(t)n(w)n(o)h(prop)r(or-)456 649 y(tional)27 b(elemen)n(ts)g(ha)n(v)n(e)g(a)g(n)n(ull)h(distance.) 456 849 y(The)38 b(next)g(theorem,)h(due)g(to)e(G.)h(Birkho\013)g ([Bi],)i(will)e(sho)n(w)f(that)h(ev)n(ery)f(p)r(ositiv)n(e)g(linear)456 948 y(op)r(erator)25 b(is)j(a)f(con)n(traction,)f(pro)n(vided)h(that)h (the)g(diameter)f(of)g(the)h(image)f(is)h(\014nite.)456 1071 y FB(Theorem)21 b(3.3.)34 b Fr(L)l(et)22 b Fn(V)1223 1083 y Fl(1)1283 1071 y Fr(and)h Fn(V)1488 1083 y Fl(2)1548 1071 y Fr(b)l(e)g(two)g(ve)l(ctor)g(sp)l(ac)l(es,)i Fn(C)2343 1083 y Fl(1)2403 1071 y Fn(\032)e(V)2542 1083 y Fl(1)2602 1071 y Fr(and)g Fn(C)2800 1083 y Fl(2)2860 1071 y Fn(\032)g(V)2999 1083 y Fl(2)3059 1071 y Fr(two)f(c)l(onvex)456 1170 y(c)l(one)j(\(se)l (e)g(de\014nition)h(ab)l(ove\))h(and)f Fq(L)c Fs(:)h Fn(V)1762 1182 y Fl(1)1822 1170 y Fn(!)g(V)1979 1182 y Fl(2)2042 1170 y Fr(a)j(p)l(ositive)g(line)l(ar)g(op)l(er)l(ator)h (\(which)f(me)l(ans)456 1270 y Fq(L)p Fs(\()p Fn(C)589 1282 y Fl(1)625 1270 y Fs(\))e Fn(\032)e(C)812 1282 y Fl(2)849 1270 y Fr(\).)39 b(L)l(et)29 b Fq(d)1133 1282 y Fj(C)1169 1290 y Fh(i)1230 1270 y Fr(b)l(e)g(the)h(Hilb)l(ert)g (metric)g(asso)l(ciate)l(d)h(to)f(the)g(c)l(one)g Fn(C)2865 1282 y Fp(i)2892 1270 y Fr(.)39 b(If)30 b(we)g(denote)1505 1415 y Fs(\001)23 b(=)110 b(sup)1684 1489 y Fp(f)s(;g)r Fj(2)p Fp(L)p Fl(\()p Fj(C)1926 1497 y Fi(1)1959 1489 y Fl(\))1998 1415 y Fq(d)2041 1427 y Fj(C)2077 1435 y Fi(2)2114 1415 y Fs(\()p Fq(f)t(;)14 b(g)s Fs(\))69 b Fq(;)456 1623 y Fr(then)1138 1762 y Fq(d)1181 1774 y Fj(C)1217 1782 y Fi(2)1254 1762 y Fs(\()p Fq(Lf)t(;)14 b(Lg)s Fs(\))22 b Fn(\024)h Fs(tanh)1847 1645 y Fk(\022)1918 1706 y Fs(\001)p 1918 1743 70 4 v 1931 1819 a(4)1997 1645 y Fk(\023)2058 1762 y Fq(d)2101 1774 y Fj(C)2137 1782 y Fi(1)2174 1762 y Fs(\()p Fq(f)t(;)14 b(g)s Fs(\))46 b Fn(8)p Fq(f)t(;)14 b(g)24 b Fn(2)g(C)2725 1774 y Fl(1)456 1940 y Fs(\(tanh\()p Fn(1)p Fs(\))g(=)e(1\))p Fq(:)555 2062 y Fs(Theorem)27 b(3.3)g(alone)g(is)h(not)g(completely)g (satisfactory:)35 b(giv)n(en)27 b(a)h(cone)f Fn(C)33 b Fs(and)28 b(its)g(metric)456 2162 y Fq(d)499 2174 y Fj(C)542 2162 y Fs(,)c(w)n(e)f(do)f(not)i(kno)n(w)e(if)h(\()p Fn(C)5 b Fq(;)14 b(d)1405 2174 y Fj(C)1448 2162 y Fs(\))24 b(is)f(complete.)35 b(This)23 b(asp)r(ect)g(is)g(tak)n(en)f(care)g(b)n (y)h(the)h(follo)n(wing)456 2261 y(lemma,)j(whic)n(h)h(allo)n(ws)e(to)h (link)h(the)g(Hilb)r(ert)g(metric)g(to)f(a)h(suitable)f(norm)g (de\014ned)h(on)f Fn(V)7 b Fs(.)456 2384 y FB(Lemma)29 b(3.4.)40 b Fs([LSV)q(])29 b Fr(L)l(et)h Fn(k)18 b(\001)g(k)29 b Fr(b)l(e)h(a)g(norm)g(on)g Fn(V)36 b Fr(such)30 b(that)1211 2529 y Fn(8)p Fq(f)t(;)14 b(g)25 b Fn(2)e(V)53 b Fq(g)21 b Fn(\000)d Fq(f)9 b(;)43 b(g)21 b Fs(+)d Fq(f)32 b Fn(2)23 b(C)28 b(\))23 b(k)p Fq(g)s Fn(k)f(\024)g(k)p Fq(f)9 b Fn(k)456 2674 y Fr(and)30 b(let)g Fq(`)22 b Fs(:)h Fn(C)28 b(!)23 b Fo(R)1069 2644 y Fl(+)1160 2674 y Fr(b)l(e)30 b(a)g(homo)l(gene)l(ous)g(and)h(or)l(der)f(pr)l(eserving)h(function,)f (i.e.)1095 2819 y Fn(8)p Fq(f)h Fn(2)23 b(C)5 b Fq(;)14 b Fn(8)p Fq(\025)22 b Fn(2)i Fo(R)1628 2785 y Fl(+)1855 2819 y Fq(`)p Fs(\()p Fq(\025f)9 b Fs(\))23 b(=)g Fq(\025`)p Fs(\()p Fq(f)9 b Fs(\))1368 2944 y Fn(8)p Fq(f)t(;)14 b(g)25 b Fn(2)e(C)171 b Fq(g)21 b Fn(\000)d Fq(f)32 b Fn(2)23 b(C)57 b(\))23 b Fq(`)p Fs(\()p Fq(f)9 b Fs(\))23 b Fn(\024)g Fq(`)p Fs(\()p Fq(g)s Fs(\))g Fq(;)456 3089 y Fr(then)744 3234 y Fn(8)p Fq(f)t(;)14 b(g)25 b Fn(2)e(C)51 b Fq(`)p Fs(\()p Fq(f)9 b Fs(\))23 b(=)g Fq(`)p Fs(\()p Fq(g)s Fs(\))f Fq(>)h Fs(0)g Fn(\))g(k)p Fq(f)j Fn(\000)18 b Fq(g)s Fn(k)k(\024)h Fs(\(e)2250 3200 y Fp(d)2285 3208 y Fg(C)2323 3200 y Fl(\()p Fp(f)s(;g)r Fl(\))2487 3234 y Fn(\000)18 b Fs(1\))c(min\()p Fn(k)p Fq(f)9 b Fn(k)p Fq(;)14 b Fn(k)p Fq(g)s Fn(k)p Fs(\))456 3379 y FB(Remark)31 b(3.1.)40 b Fr(In)29 b(the)h(pr)l(evious)h(lemma,)g(one)f(c)l(an)g(cho) l(ose)i Fq(`)p Fs(\()p Fn(\001)p Fs(\))23 b(=)g Fn(k)18 b(\001)h(k)29 b Fr(which)j(ful\014l)t(ls)e(the)456 3479 y(hyp)l(othesis.)45 b(A)n(n)30 b(inter)l(esting)h(c)l(ase)h(is)g(also)g (when)f Fq(`)g Fr(is)h(a)g(line)l(ar)f(functional)h(p)l(ositive)h(on)e Fn(C)5 b Fr(.)456 3579 y(However,)31 b(we)f(ar)l(e)g(c)l(onc)l(erne)l (d)g(with)g(the)g(p)l(ossibly)h(nonline)l(ar)g Fq(`)22 b Fs(=)h(\003)p Fr(.)456 3749 y FB(In)m(v)-5 b(arian)m(t)33 b(cone.)456 3849 y Fs(F)-7 b(rom)27 b(no)n(w)g(on,)g(w)n(e)g(\014x)h Fq(\022)d Fn(2)f Fo(R)33 b Fs(suc)n(h)28 b(that)f(\002)p Fq(\030)g Fn(\024)c Fq(\022)i(<)e(\032)p Fs(.)456 3971 y FB(Prop)s(osition)33 b(3.5.)41 b Fr(Ther)l(e)33 b(exists)f Fq(n)1687 3983 y Fj(\003)1752 3971 y Fn(2)c Fo(N)42 b Fr(and)33 b Fq(a)2135 3983 y Fl(0)2199 3971 y Fq(>)27 b Fs(0)32 b Fr(such)g(that,)h(for)g(e)l(ach)g Fq(n)27 b Fn(\025)g Fq(n)3297 3983 y Fj(\003)3335 3971 y Fr(,)33 b(if)456 4071 y Fq(a)23 b Fn(\025)f Fq(a)654 4083 y Fl(0)691 4071 y Fr(,)30 b(then)g(the)g(c)l(one)g Fn(C)1304 4083 y Fp(a)1373 4071 y Fr(is)h(not)e(empty)h(and)1760 4216 y Fn(L)1817 4182 y Fp(n)1862 4216 y Fn(C)1906 4228 y Fp(a)1969 4216 y Fn(\032)23 b(C)2101 4228 y Fp(a)456 4361 y Fr(with)30 b(\014nite)f(diameter.)555 4484 y Fs(Before)e(pro)n (ving)f(the)i(ab)r(o)n(v)n(e)e(prop)r(osition)h(w)n(e)g(need)h(few)g (auxiliary)e(results.)456 4606 y FB(Lemma)j(3.6.)40 b Fr(F)-6 b(or)30 b(e)l(ach)h Fq(n)23 b Fn(2)g Fo(N)40 b Fr(holds)1706 4751 y Fs(\003\()p Fn(L)1853 4717 y Fp(n)1898 4751 y Fs(1\))23 b Fn(\025)g Fq(\032)2126 4717 y Fp(n)2171 4751 y Fq(:)456 4916 y Fr(Pr)l(o)l(of.)43 b Fs(F)-7 b(or)27 b(eac)n(h)g Fq(g)e Fn(2)f Fs(BV,)j Fq(g)f Fn(\025)d Fs(0)k(and)g Fq(x)d Fn(2)f Fq(D)1969 4928 y Fp(n)p Fl(+1)2098 4916 y Fs(,)28 b(holds)915 5115 y Fn(L)972 5085 y Fp(n)p Fl(+1)1101 5115 y Fq(g)s Fs(\()p Fq(x)p Fs(\))p 915 5152 342 4 v 957 5228 a Fn(L)1014 5204 y Fp(n)1060 5228 y Fs(1\()p Fq(x)p Fs(\))1289 5171 y Fn(\025)1387 5085 y(L)1458 4993 y Fk(h)1497 5085 y FB(1)1545 5097 y Fp(D)1599 5105 y Fh(n)1658 4993 y Fk(\020)1717 5048 y Fj(L)1763 5023 y Fh(n)1804 5048 y Fp(g)p 1717 5066 122 4 v 1718 5113 a Fj(L)1764 5097 y Fh(n)1805 5113 y Fl(1)1848 4993 y Fk(\021)1912 5085 y Fn(L)1969 5055 y Fp(n)2014 5085 y Fs(1)2056 4993 y Fk(i)2109 5085 y Fs(\()p Fq(x)p Fs(\))p 1387 5152 835 4 v 1676 5228 a Fn(L)1733 5204 y Fp(n)1778 5228 y Fs(1\()p Fq(x)p Fs(\))2254 5171 y Fn(\025)2351 5115 y(L)2408 5085 y Fp(n)p Fl(+1)2538 5115 y Fs(1\()p Fq(x)p Fs(\))p 2351 5152 341 4 v 2393 5228 a Fn(L)2450 5204 y Fp(n)2496 5228 y Fs(1\()p Fq(x)p Fs(\))2715 5171 y(inf)2718 5225 y Fp(D)2772 5233 y Fh(n)2840 5115 y Fn(L)2897 5085 y Fp(n)2943 5115 y Fq(g)p 2840 5152 146 4 v 2841 5228 a Fn(L)2898 5204 y Fp(n)2943 5228 y Fs(1)p eop %%Page: 11 11 11 10 bop 1339 251 a Fl(LASOT)-5 b(A-YORKE)29 b(MAPS)f(WITH)h(HOLES)817 b(11)456 450 y Fs(and,)27 b(taking)g(the)h(inf)g(on)g Fq(x)g Fs(and)f(the)h(limit)g Fq(n)23 b Fn(!)g(1)28 b Fs(w)n(e)f(ha)n(v)n(e)456 590 y(\(3.1\))952 b(\003\()p Fn(L)p Fq(g)s Fs(\))23 b Fn(\025)g Fs(\003\()p Fn(L)p Fs(1\)\003\()p Fq(g)s Fs(\))p Fq(:)456 730 y Fs(The)k(lemma)h(follo)n (ws)e(b)n(y)i(iterating)f(\(3.1\).)1567 b Ff(\003)456 888 y FB(Lemma)29 b(3.7.)40 b Fr(Ther)l(e)31 b(exists)e Fq(n)1500 900 y Fl(0)1560 888 y Fn(2)24 b Fo(N)39 b Fr(and)31 b Fq(a)1934 900 y Fl(0)1994 888 y Fn(2)23 b Fo(R)2126 858 y Fl(+)2217 888 y Fr(such)30 b(that)g(for)g(al)t(l)h Fq(a)23 b Fn(\025)g Fq(a)3026 900 y Fl(0)3092 888 y Fr(holds)1048 1028 y Fn(L)1105 994 y Fp(n)1150 1028 y Fn(C)1194 1040 y Fp(a)1257 1028 y Fn(\032)g(C)1389 1043 y Fp(a=)p Fl(2)1519 1028 y Fn(8)p Fq(n)f Fn(\025)g Fq(n)1775 1040 y Fl(0)1872 1028 y Fr(and)60 b Fn(L)2120 994 y Fp(n)2165 1028 y Fn(C)2209 1040 y Fp(a)2272 1028 y Fn(\032)23 b(C)2404 1040 y Fl(2)p Fp(aC)2521 1049 y Fh(\022)2581 1028 y Fn(8)p Fq(n)f Fn(\025)h Fs(0)p Fq(:)456 1186 y Fr(Pr)l(o)l(of.)43 b Fs(First)37 b(of)h(all,)i(it)f(is)e(ob)n(vious)g(that)h Fq(h)i Fn(\025)f Fs(0)f(implies)g Fn(L)2493 1156 y Fp(n)2538 1186 y Fq(h)i Fn(\025)g Fs(0.)67 b(Next)38 b(w)n(e)g(c)n(ho)r(ose)456 1285 y Fq(n)506 1297 y Fl(0)566 1285 y Fn(2)23 b Fo(N)t Fs(,)33 b(suc)n(h)26 b(that)g(for)g(all)g Fq(n)c Fn(\025)h Fq(n)1568 1297 y Fl(0)1605 1285 y Fs(,)k Fq(C)1720 1255 y Fl(2)1714 1309 y Fp(\022)1757 1285 y Fq(\022)1798 1255 y Fp(n)1844 1285 y Fq(\032)1887 1255 y Fj(\000)p Fp(n)2007 1285 y Fn(\024)2104 1253 y Fl(1)p 2104 1267 34 4 v 2104 1314 a(4)2147 1285 y Fs(.)37 b(Let)26 b Fq(h)d Fn(2)h(C)2548 1297 y Fp(a)2614 1285 y Fs(then)i(for)g(eac)n(h)f Fq(n)e Fn(2)h Fo(N)36 b Fs(w)n(e)456 1385 y(write)27 b Fq(n)c Fs(=)g Fq(k)s(n)925 1397 y Fl(0)980 1385 y Fs(+)18 b Fq(m)p Fs(,)28 b Fq(m)22 b(<)h(n)1420 1397 y Fl(0)1457 1385 y Fs(,)28 b(and)f(\(recall)g(Lemma)h(2.5\))456 1759 y(\(3.2\))880 1463 y Fk(_)986 1542 y Fn(L)1043 1508 y Fp(n)1088 1542 y Fq(h)23 b Fn(\024)g Fq(C)1306 1554 y Fp(\022)1344 1542 y Fq(\022)1385 1508 y Fp(n)1426 1516 y Fi(0)1476 1463 y Fk(_)1582 1542 y Fn(L)1639 1508 y Fl(\()p Fp(k)q Fj(\000)p Fl(1\))p Fp(n)1853 1516 y Fi(0)1887 1508 y Fl(+)p Fp(m)2001 1542 y Fq(h)18 b Fs(+)g Fq(K)2221 1554 y Fp(n)2262 1562 y Fi(0)2298 1542 y Fs(\003\()p Fn(L)2445 1508 y Fl(\()p Fp(k)q Fj(\000)p Fl(1\))p Fp(n)2659 1516 y Fi(0)2692 1508 y Fl(+)p Fp(m)2806 1542 y Fq(h)p Fs(\))1159 1773 y Fn(\024)23 b Fq(C)1312 1738 y Fp(k)q Fl(+1)1306 1798 y Fp(\022)1437 1773 y Fq(\022)1478 1739 y Fp(n)1537 1694 y Fk(_)1643 1773 y Fq(h)c Fs(+)1793 1669 y Fp(k)q Fj(\000)p Fl(1)1794 1694 y Fk(X)1800 1871 y Fp(i)p Fl(=0)1914 1773 y Fs(\()p Fq(C)2005 1785 y Fp(\022)2043 1773 y Fq(\022)2084 1739 y Fp(n)2125 1747 y Fi(0)2162 1773 y Fs(\))2194 1739 y Fp(i)2222 1773 y Fq(K)2293 1785 y Fp(n)2334 1793 y Fi(0)2370 1773 y Fs(\003\()p Fn(L)2517 1739 y Fl(\()p Fp(k)q Fj(\000)p Fp(i)p Fj(\000)p Fl(1\))p Fp(n)2806 1747 y Fi(0)2840 1739 y Fl(+)p Fp(m)2954 1773 y Fq(h)p Fs(\))1238 1987 y(+)f Fq(C)1386 1953 y Fp(k)1380 2008 y(\022)1427 1987 y Fq(\022)1468 1953 y Fp(k)q(n)1545 1961 y Fi(0)1582 1987 y Fq(K)1653 1999 y Fp(m)1716 1987 y Fs(\003\()p Fq(h)p Fs(\))p Fq(:)456 2132 y Fs(Th)n(us,)27 b(\(use)h(\(3.1\)\))762 2276 y Fk(_)868 2355 y Fn(L)925 2320 y Fp(n)970 2355 y Fq(h)23 b Fn(\024)1129 2213 y Fk(")1177 2237 y(\022)1238 2355 y Fq(a)18 b Fs(+)1427 2298 y Fq(K)1498 2310 y Fp(m)p 1393 2335 202 4 v 1393 2412 a Fq(C)1452 2424 y Fp(\022)1490 2412 y Fq(\022)1531 2388 y Fp(m)1605 2237 y Fk(\023)1690 2297 y Fq(C)1755 2262 y Fp(k)q Fl(+1)1749 2322 y Fp(\022)1880 2297 y Fq(\022)1921 2267 y Fp(n)p 1690 2335 277 4 v 1784 2412 a Fq(\032)1827 2388 y Fp(n)1995 2355 y Fs(+)2078 2251 y Fp(k)q Fj(\000)p Fl(1)2079 2276 y Fk(X)2085 2453 y Fp(i)p Fl(=0)2213 2237 y Fk(\022)2284 2298 y Fq(C)2343 2310 y Fp(\022)2381 2298 y Fq(\022)2422 2268 y Fp(n)2463 2276 y Fi(0)p 2284 2335 216 4 v 2332 2412 a Fq(\032)2375 2388 y Fp(n)2416 2396 y Fi(0)2510 2237 y Fk(\023)2571 2255 y Fp(i)2623 2298 y Fq(K)2694 2310 y Fp(n)2735 2318 y Fi(0)p 2623 2335 149 4 v 2636 2412 a Fq(\032)2679 2388 y Fp(n)2720 2396 y Fi(0)2781 2213 y Fk(#)2843 2355 y Fs(\003\()p Fn(L)2990 2320 y Fp(n)3035 2355 y Fq(h)p Fs(\))p Fq(:)456 2583 y Fs(Clearly)26 b(the)i(w)n(orst)e(case)h(is)h(when)f Fq(k)f Fs(=)d(0,)k(then)h(if)g Fq(a)2145 2595 y Fl(0)2206 2583 y Fn(\025)22 b Fs(max)2448 2595 y Fp(i)p Fj(\024)p Fp(n)2564 2603 y Fi(0)2655 2549 y Fp(K)2711 2557 y Fh(i)p 2625 2564 143 4 v 2625 2612 a Fp(C)2673 2621 y Fh(\022)2706 2612 y Fp(\032)2740 2595 y Fh(i)2804 2583 y Fs(w)n(e)27 b(ha)n(v)n(e)1528 2681 y Fk(_)1634 2759 y Fn(L)1691 2725 y Fp(n)1736 2759 y Fq(h)c Fn(\024)g Fs(2)p Fq(aC)2040 2771 y Fp(\022)2077 2759 y Fs(\003\()p Fn(L)2224 2725 y Fp(n)2269 2759 y Fq(h)p Fs(\))p Fq(:)456 2912 y Fs(When)28 b Fq(k)e(>)c Fs(0)27 b(instead)1054 3005 y Fk(_)1160 3084 y Fn(L)1217 3050 y Fp(n)1263 3084 y Fq(h)c Fn(\024)1421 2967 y Fk(\024)1475 3028 y Fs(1)p 1475 3065 42 4 v 1475 3141 a(4)1540 2967 y Fk(\022)1601 3084 y Fq(a)c Fs(+)1791 3028 y Fq(K)1862 3040 y Fp(m)p 1757 3065 202 4 v 1757 3141 a Fq(C)1816 3153 y Fp(\022)1854 3141 y Fq(\022)1895 3117 y Fp(m)1968 2967 y Fk(\023)2048 3084 y Fs(+)f(2)p Fq(K)2244 3096 y Fp(n)2285 3104 y Fi(0)2320 3084 y Fq(\032)2363 3050 y Fj(\000)p Fp(n)2456 3058 y Fi(0)2493 2967 y Fk(\025)2550 3084 y Fs(\003\()p Fn(L)2697 3050 y Fp(n)2743 3084 y Fq(h)p Fs(\))p Fq(:)456 3286 y Fs(Hence,)27 b(for)g(all)h Fq(n)23 b Fn(\025)f Fq(n)1178 3298 y Fl(0)1243 3286 y Fs(and)28 b Fq(a)22 b Fn(\025)h Fs(8)p Fq(K)1672 3298 y Fp(n)1713 3306 y Fi(0)1749 3286 y Fq(\032)1792 3255 y Fj(\000)p Fp(n)1885 3263 y Fi(0)1940 3286 y Fs(+)18 b(max)2177 3298 y Fp(i)p Fj(\024)p Fp(n)2293 3306 y Fi(0)2384 3252 y Fp(K)2440 3260 y Fh(i)p 2354 3267 143 4 v 2354 3314 a Fp(C)2402 3323 y Fh(\022)2435 3314 y Fp(\032)2469 3298 y Fh(i)2529 3286 y Fs(:=)23 b Fq(a)2684 3298 y Fl(0)2721 3286 y Fs(,)1587 3387 y Fk(_)1693 3466 y Fn(L)1750 3432 y Fp(n)1795 3466 y Fq(h)g Fn(\024)1964 3410 y Fq(a)p 1964 3447 44 4 v 1965 3523 a Fs(2)2018 3466 y(\003\()p Fn(L)2165 3432 y Fp(n)2210 3466 y Fq(h)p Fs(\))p Fq(:)3380 3628 y Ff(\003)555 3786 y Fs(The)36 b(ab)r(o)n(v)n(e)e(Lemma)i(sho)n (ws)e(the)i(in)n(v)-5 b(ariance)34 b(of)i(the)g(cone)f(but)h(has)f (also)g(man)n(y)g(other)456 3885 y(implications)27 b(the)h(\014rst)f (of)h(whic)n(h)f(b)r(eing)h(the)g(follo)n(wing.)456 4005 y FB(Lemma)35 b(3.8.)42 b Fr(Ther)l(e)35 b(exists)f(a)g(c)l(onstant)g Fq(B)h(>)30 b Fs(0)k Fr(such)g(that,)h(for)g(e)l(ach)g Fq(h)c Fn(2)g Fq(B)t(V)19 b Fr(,)36 b Fq(h)30 b Fn(\025)h Fs(0)456 4105 y Fr(and)f Fq(m)23 b Fn(2)g Fo(N)t Fr(,)1195 4210 y Fs(\003\()p Fn(L)1342 4176 y Fp(m)1405 4210 y Fs(1\)\003\()p Fq(h)p Fs(\))g Fn(\024)g Fs(\003\()p Fn(L)1907 4176 y Fp(m)1970 4210 y Fq(h)p Fs(\))g Fn(\024)g Fq(B)t Fs(\003\()p Fn(L)2375 4176 y Fp(m)2438 4210 y Fs(1\)\003\()p Fq(h)p Fs(\))p Fq(:)456 4368 y Fr(Pr)l(o)l(of.)43 b Fs(The)27 b(\014rst)g(inequalit)n(y)g(follo)n(ws)f(trivially)h(b)n(y)g(iterating) g(\(3.1\).)36 b(F)-7 b(or)27 b(the)g(second,)g(con-)456 4467 y(sider)g Fq(n;)14 b(m)22 b Fn(2)i Fo(N)37 b Fs(and)28 b Fq(x)23 b Fn(2)h Fq(D)1383 4479 y Fp(n)p Fl(+)p Fp(m)1538 4467 y Fs(,)j(then)1327 4602 y Fn(L)1384 4572 y Fp(m)p Fl(+)p Fp(n)1539 4602 y Fq(h)p Fs(\()p Fq(x)p Fs(\))p 1327 4639 373 4 v 1385 4715 a Fn(L)1442 4691 y Fp(n)1487 4715 y Fs(1\()p Fq(x)p Fs(\))1732 4658 y(=)1829 4602 y Fn(L)1886 4572 y Fp(m)p Fl(+)p Fp(n)2042 4602 y Fq(h)p Fs(\()p Fq(x)p Fs(\))p 1829 4639 V 1832 4715 a Fn(L)1889 4691 y Fp(n)p Fl(+)p Fp(m)2045 4715 y Fs(1\()p Fq(x)p Fs(\))2221 4602 y Fn(L)2278 4572 y Fp(n)p Fl(+)p Fp(m)2434 4602 y Fs(1\()p Fq(x)p Fs(\))p 2221 4639 366 4 v 2276 4715 a Fn(L)2333 4691 y Fp(n)2379 4715 y Fs(1\()p Fq(x)p Fs(\))1732 4891 y Fn(\024)1829 4835 y(L)1886 4805 y Fp(m)p Fl(+)p Fp(n)2042 4835 y Fq(h)p Fs(\()p Fq(x)p Fs(\))p 1829 4872 373 4 v 1832 4948 a Fn(L)1889 4924 y Fp(n)p Fl(+)p Fp(m)2045 4948 y Fs(1\()p Fq(x)p Fs(\))2211 4891 y Fn(kL)2310 4857 y Fp(m)2373 4891 y Fs(1)p Fn(k)2457 4903 y Fj(1)2527 4891 y Fq(;)456 5075 y Fs(whic)n(h,)g(b)n(y)h(taking)e (the)i(inf)h(on)e Fq(x)h Fs(and)g(the)g(limit)g Fq(n)23 b Fn(!)g(1)28 b Fs(yields)1495 5216 y(\003\()p Fn(L)1642 5181 y Fp(m)1706 5216 y Fq(h)p Fs(\))23 b Fn(\024)f(kL)1995 5181 y Fp(m)2058 5216 y Fs(1)p Fn(k)2142 5228 y Fj(1)2212 5216 y Fs(\003\()p Fq(h)p Fs(\))p Fq(:)p eop %%Page: 12 12 12 11 bop 456 251 a Fl(12)355 b(CARLANGELO)23 b(LIVERANI)f(AND)g(V)1947 236 y(\023)1941 251 y(ER)n(ONIQUE)g(MA)n(UME-DESCHAMPS)456 450 y Fs(Next,)28 b(since)f(1)c Fn(2)g(C)1072 462 y Fp(a)1112 450 y Fs(,)28 b(Lemma)f(3.7,)g(implies)1516 541 y Fk(_)1622 620 y Fn(L)1679 585 y Fp(m)1742 620 y Fs(1)c Fn(\024)g Fs(2)p Fq(aC)2040 632 y Fp(\022)2077 620 y Fs(\003\()p Fn(L)2224 585 y Fp(m)2287 620 y Fs(1\))p Fq(:)456 814 y FB(Sublemma)28 b(3.9.)40 b Fr(F)-6 b(or)30 b(e)l(ach)h Fq(f)h Fn(2)23 b Fq(B)t(V)c Fr(holds:)40 b(for)31 b(al)t(l)g Fq(x)1588 984 y(f)9 b Fs(\()p Fq(x)p Fs(\))24 b Fn(\024)e Fs(\003\()p Fq(f)9 b Fs(\))18 b(+)2133 905 y Fk(_)2240 984 y Fq(f)9 b(:)456 1178 y Fr(Pr)l(o)l(of.)43 b Fs(F)-7 b(or)27 b Fq(x)h Fs(and)f Fq(y)s Fs(,)1595 1313 y Fq(f)9 b Fs(\()p Fq(x)p Fs(\))23 b Fn(\024)g Fq(f)9 b Fs(\()p Fq(y)s Fs(\))18 b(+)2126 1234 y Fk(_)2233 1313 y Fq(f)9 b(;)456 1465 y Fs(\014x)27 b Fq(x)p Fs(,)h(using)g(the)g(prop)r(erties) e(of)i(\003)f(w)n(e)g(get:)1588 1634 y Fq(f)9 b Fs(\()p Fq(x)p Fs(\))24 b Fn(\024)e Fs(\003\()p Fq(f)9 b Fs(\))18 b(+)2133 1555 y Fk(_)2240 1634 y Fq(f)9 b(:)3380 1802 y Ff(\003)555 1985 y Fs(Th)n(us)1006 2107 y Fn(kL)1105 2073 y Fp(m)1168 2107 y Fs(1)p Fn(k)1252 2119 y Fj(1)1345 2107 y Fn(\024)22 b Fs(\003\()p Fn(L)1579 2073 y Fp(m)1643 2107 y Fs(1\))c(+)1818 2028 y Fk(_)1924 2107 y Fn(L)1981 2073 y Fp(m)2044 2107 y Fs(1)23 b Fn(\024)g Fs(\(2)p Fq(aC)2374 2119 y Fp(\022)2430 2107 y Fs(+)18 b(1\)\003\()p Fn(L)2734 2073 y Fp(m)2797 2107 y Fs(1\))p Fq(;)456 2259 y Fs(from)27 b(whic)n(h)g(the)h(result)g(follo)n(ws)e(with)i Fq(B)g Fs(:=)22 b(2)p Fq(aC)2071 2271 y Fp(\022)2127 2259 y Fs(+)c(1.)1105 b Ff(\003)456 2442 y FB(Lemma)42 b(3.10.)47 b Fr(F)-6 b(or)40 b(e)l(ach)h Fq(")g(>)h Fs(0)d Fr(ther)l(e)h(exists)g Fq(n)2187 2454 y Fl(0)2264 2442 y Fr(such)g(that)g(for)h(e)l(ach)g Fq(n)h Fn(\025)f Fq(n)3231 2454 y Fl(0)3268 2442 y Fr(,)i(the)456 2541 y(p)l(artition)30 b Fn(Z)863 2511 y Fl(\()p Fp(n)p Fl(\))990 2541 y Fr(has)g(the)g(pr)l (op)l(erty)1683 2699 y Fs(sup)1629 2777 y Fp(Z)t Fj(2Z)1776 2760 y Fi(\()p Fh(n)p Fi(\))1875 2699 y Fs(\003\()p FB(1)2013 2711 y Fp(Z)2066 2699 y Fs(\))24 b Fn(\024)e Fq(":)456 2947 y Fr(Pr)l(o)l(of.)43 b Fs(Cho)r(ose)29 b Fq(n)1057 2959 y Fl(0)1120 2947 y Fn(2)e Fo(N)40 b Fs(suc)n(h)29 b(that)h(for)g(all)f Fq(n)d Fn(\025)h Fq(n)2128 2959 y Fl(0)2165 2947 y Fs(,)j Fq(C)2277 2959 y Fp(\022)2315 2947 y Fq(\022)2356 2917 y Fp(n)2401 2947 y Fq(\032)2444 2917 y Fj(\000)p Fp(n)2568 2947 y Fn(\024)c Fq(")p Fs(,)k(this)g(is)g (p)r(ossible)f(due)456 3046 y(to)e(condition)g FB(C2)p Fs(.)37 b(Then,)28 b(for)f Fq(Z)i Fn(2)23 b(Z)1696 3016 y Fl(\()p Fp(n)p Fl(\))1070 3216 y Fn(L)1127 3182 y Fp(n)1172 3216 y FB(1)1220 3228 y Fp(Z)1273 3216 y Fs(\()p Fq(x)p Fs(\))h(=)1562 3137 y Fk(X)1496 3320 y Fp(y)r Fj(2)p Fp(T)1625 3303 y Fg(\000)p Fh(n)1710 3320 y Fp(x)1762 3216 y Fq(g)1802 3228 y Fp(n)1847 3216 y Fs(\()p Fq(y)s Fs(\))p FB(1)2003 3228 y Fp(Z)2056 3216 y Fs(\()p Fq(y)s Fs(\))f Fn(\024)g(k)p Fq(g)2357 3228 y Fp(n)2401 3216 y Fn(k)2443 3228 y Fj(1)2536 3216 y Fn(\024)g Fq(C)2683 3228 y Fp(\022)2721 3216 y Fq(\022)2762 3182 y Fp(n)2807 3216 y Fq(:)456 3463 y Fs(Accordingly)-7 b(,)26 b(for)h(eac)n(h)g Fq(x)d Fn(2)f Fq(D)1465 3475 y Fp(n)p Fl(+)p Fp(m)1643 3463 y Fn(\032)g Fq(D)1800 3475 y Fp(m)1862 3463 y Fs(,)1304 3615 y Fn(L)1361 3585 y Fp(n)p Fl(+)p Fp(m)1517 3615 y FB(1)1565 3627 y Fp(Z)1618 3615 y Fs(\()p Fq(x)p Fs(\))p 1304 3652 426 4 v 1334 3728 a Fn(L)1391 3704 y Fp(n)p Fl(+)p Fp(m)1546 3728 y Fs(1\()p Fq(x)p Fs(\))1763 3671 y Fn(\024)f Fq(C)1909 3683 y Fp(\022)1947 3671 y Fq(\022)1988 3637 y Fp(n)2188 3615 y Fs(1)p 2044 3652 331 4 v 2054 3705 a Fj(L)2100 3689 y Fh(m)2155 3705 y Fj(L)2201 3689 y Fh(n)2242 3705 y Fl(1\()p Fp(x)p Fl(\))p 2054 3727 311 4 v 2097 3774 a Fj(L)2143 3758 y Fh(m)2198 3774 y Fl(1\()p Fp(x)p Fl(\))1763 3932 y Fn(\024)g Fq(C)1909 3944 y Fp(\022)1947 3932 y Fq(\022)1988 3897 y Fp(n)2287 3876 y Fs(1)p 2044 3913 529 4 v 2087 4006 a(inf)2044 4060 y Fp(z)r Fj(2)p Fp(D)2177 4068 y Fh(m)2256 3966 y Fj(L)2302 3950 y Fh(m)2357 3966 y Fj(L)2403 3950 y Fh(n)2444 3966 y Fl(1\()p Fp(z)r Fl(\))p 2256 3987 307 4 v 2299 4035 a Fj(L)2345 4018 y Fh(m)2400 4035 y Fl(1\()p Fp(z)r Fl(\))2583 3932 y Fq(:)456 4200 y Fs(T)-7 b(aking)30 b(the)i(in\014m)n(um)h(with)f(resp)r(ect)f(to)h Fq(x)g Fs(and)f(the)i(limit)f Fq(m)e Fn(!)g(1)p Fs(,)i(the)g(ab)r(o)n(v)n(e)f (relations)456 4300 y(yields)456 4492 y(\(3.3\))618 b(\003\()p FB(1)1383 4504 y Fp(Z)1436 4492 y Fs(\))23 b Fn(\024)g Fq(C)1638 4504 y Fp(\022)1676 4492 y Fq(\022)1717 4458 y Fp(n)1885 4436 y Fs(1)p 1772 4473 267 4 v 1772 4549 a(\003\()p Fn(L)1919 4525 y Fp(n)1965 4549 y Fs(1\))2072 4492 y Fn(\024)f Fq(C)2218 4504 y Fp(\022)2256 4492 y Fq(\022)2297 4458 y Fp(n)2343 4492 y Fq(\032)2386 4458 y Fj(\000)p Fp(n)2506 4492 y Fn(\024)g Fq(";)456 4694 y Fs(where)27 b(w)n(e)g(ha)n(v)n(e)f(used)i(Lemma)f(3.6.)1755 b Ff(\003)456 4876 y FB(Lemma)28 b(3.11.)39 b Fr(F)-6 b(or)29 b(e)l(ach)g Fq(a)23 b Fn(\025)g Fq(a)1568 4888 y Fl(0)1634 4876 y Fr(ther)l(e)29 b(exists)f Fq(n)23 b Fn(2)g Fo(N)39 b Fr(such)29 b(that,)g(for)h(e)l(ach)f Fq(h)23 b Fn(2)h(C)3199 4888 y Fp(a)3267 4876 y Fr(ther)l(e)456 4987 y(exists)29 b Fq(Z)g Fn(2)23 b(Z)914 4944 y Fl(\()p Fp(n)p Fl(\))907 4997 y Fp(g)1041 4987 y Fr(such)29 b(that)1630 5149 y Fs(inf)1615 5203 y Fp(x)p Fj(2)p Fp(Z)1760 5149 y Fq(h)p Fs(\()p Fq(x)p Fs(\))24 b Fn(\025)2041 5093 y Fs(1)p 2041 5130 42 4 v 2041 5206 a(4)2092 5149 y(\003\()p Fq(h)p Fs(\))p Fq(:)p eop %%Page: 13 13 13 12 bop 1339 251 a Fl(LASOT)-5 b(A-YORKE)29 b(MAPS)f(WITH)h(HOLES)817 b(13)456 450 y Fr(Pr)l(o)l(of.)43 b Fs(F)-7 b(or)27 b(eac)n(h)g Fq(n;)14 b(m)22 b Fn(2)i Fo(N)t Fs(,)33 b Fq(n)23 b(<)g(m)p Fs(,)k(w)n(e)h(can)f(write)2167 418 y Fl(6)1007 611 y Fn(L)1064 577 y Fp(m)1128 611 y Fq(h)p Fs(\()p Fq(x)p Fs(\))c(=)1454 532 y Fk(X)1398 726 y Fp(Z)t Fj(2)1511 711 y Fl(^)1492 726 y Fj(Z)1545 709 y Fi(\()p Fh(n)p Fi(\))1644 611 y Fn(L)1701 577 y Fp(m)1764 611 y Fs(\()p Fq(h)p FB(1)1892 623 y Fp(Z)1946 611 y Fs(\)\()p Fq(x)p Fs(\))h(=)2257 532 y Fk(X)2201 732 y Fp(Z)t Fj(2Z)2348 702 y Fi(\()p Fh(n)p Fi(\))2343 743 y Fg(\003)2447 611 y Fn(L)2504 577 y Fp(m)2567 611 y Fs(\()p Fq(h)p FB(1)2695 623 y Fp(Z)2749 611 y Fs(\)\()p Fq(x)p Fs(\))456 866 y(W)-7 b(e)25 b(will)g(then)g(pro)n(v)n(e)e(the)i(Lemma)f(arguing)f(b)n (y)h(con)n(tradiction.)35 b(Supp)r(ose)25 b(that)g(the)g(Lemma)456 977 y(it)j(is)f(not)h(true)f(then,)h(since)g(b)n(y)f(Condition)h FB(C2)f Fs(and)g(Lemma)h(2.4,)e Fn(Z)2689 934 y Fl(\()p Fp(n)p Fl(\))2682 987 y Fp(g)2809 977 y Fn(6)p Fs(=)d Fn(;)p Fs(,)k(w)n(e)g(ha)n(v)n(e)837 1143 y Fn(L)894 1109 y Fp(m)957 1143 y Fq(h)p Fs(\()p Fq(x)p Fs(\))d(=)1275 1064 y Fk(X)1219 1264 y Fp(Z)t Fj(2Z)1366 1234 y Fi(\()p Fh(n)p Fi(\))1361 1273 y Fh(g)1465 1143 y Fn(L)1522 1109 y Fp(m)1585 1143 y Fs(\()p Fq(h)p FB(1)1713 1155 y Fp(Z)1766 1143 y Fs(\)\()p Fq(x)p Fs(\))c(+)2068 1064 y Fk(X)2012 1264 y Fp(Z)t Fj(2Z)2159 1234 y Fi(\()p Fh(n)p Fi(\))2154 1284 y Fh(b)2258 1143 y Fn(L)2315 1109 y Fp(m)2379 1143 y Fs(\()p Fq(h)p FB(1)2507 1155 y Fp(Z)2560 1143 y Fs(\)\()p Fq(x)p Fs(\))1140 1444 y Fn(\024)1275 1365 y Fk(X)1219 1565 y Fp(Z)t Fj(2Z)1366 1535 y Fi(\()p Fh(n)p Fi(\))1361 1574 y Fh(g)1465 1444 y Fn(L)1522 1410 y Fp(m)1585 1444 y FB(1)1633 1456 y Fp(Z)1686 1444 y Fs(\()p Fq(x)p Fs(\))1807 1388 y(\003\()p Fq(h)p Fs(\))p 1807 1425 171 4 v 1873 1501 a(4)2007 1444 y(+)2146 1365 y Fk(X)2090 1565 y Fp(Z)t Fj(2Z)2237 1535 y Fi(\()p Fh(n)p Fi(\))2232 1574 y Fh(g)2336 1444 y Fn(L)2393 1410 y Fp(m)2456 1444 y FB(1)2504 1456 y Fp(Z)2557 1444 y Fs(\()p Fq(x)p Fs(\))2682 1365 y Fk(_)2705 1543 y Fp(Z)2789 1444 y Fq(h)1279 1711 y Fs(+)e Fn(k)p Fq(h)p Fn(k)1494 1723 y Fj(1)1633 1632 y Fk(X)1577 1832 y Fp(Z)t Fj(2Z)1724 1802 y Fi(\()p Fh(n)p Fi(\))1719 1852 y Fh(b)1823 1711 y Fn(L)1880 1677 y Fp(m)1943 1711 y FB(1)1991 1723 y Fp(Z)2044 1711 y Fs(\()p Fq(x)p Fs(\))1140 2012 y Fn(\024)28 b(L)1290 1978 y Fp(m)1353 2012 y Fs(1\()p Fq(x)p Fs(\))1516 1956 y(\003\()p Fq(h)p Fs(\))p 1516 1993 V 1581 2069 a(4)1715 2012 y(+)1854 1933 y Fk(X)1798 2133 y Fp(Z)t Fj(2Z)1945 2103 y Fi(\()p Fh(n)p Fi(\))1940 2142 y Fh(g)2044 1920 y Fk(h)2083 2012 y Fs(\003\()p Fn(L)2230 1978 y Fp(m)2293 2012 y FB(1)2341 2024 y Fp(Z)2395 2012 y Fs(\))18 b(+)2528 1933 y Fk(_)2634 2012 y Fn(L)2691 1978 y Fp(m)2755 2012 y FB(1)2803 2024 y Fp(Z)2856 1920 y Fk(i)2909 1933 y(_)2930 2111 y Fp(Z)3015 2012 y Fq(h)1279 2279 y Fs(+)g Fn(k)p Fq(h)p Fn(k)1494 2291 y Fj(1)1633 2200 y Fk(X)1577 2400 y Fp(Z)t Fj(2Z)1724 2370 y Fi(\()p Fh(n)p Fi(\))1719 2420 y Fh(b)1823 2279 y Fn(L)1880 2245 y Fp(m)1943 2279 y FB(1)1991 2291 y Fp(Z)2044 2279 y Fs(\()p Fq(x)p Fs(\))p Fq(;)456 2573 y Fs(Where)37 b(w)n(e)h(ha)n(v)n (e)e(used)i(Sublemma)g(3.9.)66 b(T)-7 b(o)38 b(pro)r(ceed)f(notice)g (that)i(if)f Fq(Z)45 b Fn(2)c(Z)3123 2530 y Fl(\()p Fp(n)p Fl(\))3116 2598 y Fp(b)3220 2573 y Fs(,)f(then)456 2673 y(Lemma)27 b(3.8)g(implies)1348 2787 y(\003\()p Fn(L)1495 2752 y Fp(m)1558 2787 y FB(1)1606 2799 y Fp(Z)1659 2787 y Fs(\))d Fn(\024)e Fq(B)t Fs(\003\()p Fn(L)2016 2752 y Fp(m)2080 2787 y Fs(1\)\003\()p FB(1)2292 2799 y Fp(Z)2345 2787 y Fs(\))h(=)g(0)p Fq(:)456 2918 y Fs(Hence,)k(inequalit)n(y)h (\(3.2\))f(and)g(Sublemma)h(3.9)f(imply)793 3096 y Fn(L)850 3062 y Fp(m)913 3096 y FB(1)961 3108 y Fp(Z)1038 3096 y Fn(\024)1125 3018 y Fk(_)1231 3096 y Fn(L)1288 3062 y Fp(m)1352 3096 y FB(1)1400 3108 y Fp(Z)1476 3096 y Fn(\024)c Fs(2)p Fq(C)1671 3053 y Fp(m=n)1805 3061 y Fi(0)1837 3053 y Fl(+1)1665 3121 y Fp(\022)1925 3096 y Fq(\022)1966 3062 y Fp(m)2052 3096 y Fn(\024)g Fs(2)p Fq(C)2241 3108 y Fp(\022)2292 3004 y Fk(\020)2342 3096 y Fq(C)2437 3033 y Fi(1)p 2417 3042 69 3 v 2417 3075 a Fh(n)2454 3087 y Fi(0)2500 3096 y Fq(\022)r(\032)2584 3062 y Fj(\000)p Fl(1)2673 3004 y Fk(\021)2723 3021 y Fp(m)2800 3096 y Fs(\003\()p Fn(L)2947 3062 y Fp(m)3010 3096 y Fs(1\))p Fq(:)456 3295 y Fs(On)k(the)h(other)f(hand,)h(if)g Fq(Z)g Fn(2)c(Z)1492 3252 y Fl(\()p Fp(n)p Fl(\))1485 3304 y Fp(g)1589 3295 y Fs(,)j(b)n(y)h(the)g(same)f(argumen)n(ts)f(w)n (e)h(obtain)826 3382 y Fk(_)933 3460 y Fn(L)990 3426 y Fp(m)1053 3460 y FB(1)1101 3472 y Fp(Z)1177 3460 y Fn(\024)p Fs(2)p Fq(C)1349 3417 y Fl([)p Fp(m=n)1502 3425 y Fi(0)1534 3417 y Fl(]+1)1343 3486 y Fp(\022)1641 3460 y Fq(\022)1682 3426 y Fp(m)1763 3460 y Fs(+)18 b(2)p Fq(K)1959 3472 y Fp(n)2000 3480 y Fi(0)2036 3460 y Fq(\032)2079 3426 y Fj(\000)p Fp(n)2172 3434 y Fi(0)2208 3460 y Fs(\003\()p Fn(L)2355 3426 y Fp(m)2419 3460 y FB(1)2467 3472 y Fp(Z)2520 3460 y Fs(\))1177 3664 y Fn(\024)1256 3547 y Fk(\024)1299 3664 y Fs(2)p Fq(C)1400 3676 y Fp(\022)1452 3547 y Fk(\022)1513 3664 y Fq(C)1608 3581 y Fi(1)p 1588 3590 V 1588 3623 a Fh(n)1625 3635 y Fi(0)1572 3689 y Fp(\022)1671 3664 y Fq(\022)r(\032)1755 3630 y Fj(\000)p Fl(1)1844 3547 y Fk(\023)1905 3564 y Fp(m)1987 3664 y Fs(+)g(2)p Fq(K)2183 3676 y Fp(n)2224 3684 y Fi(0)2260 3664 y Fq(\032)2303 3630 y Fj(\000)p Fp(n)2396 3638 y Fi(0)2432 3664 y Fq(B)t Fs(\003\()p FB(1)2637 3676 y Fp(Z)2690 3664 y Fs(\))2722 3547 y Fk(\025)2780 3664 y Fs(\003\()p Fn(L)2927 3630 y Fp(m)2991 3664 y Fs(1\))p Fq(;)456 3858 y Fs(where)27 b(w)n(e)g(ha)n(v)n(e)f(used)i(Lemma)f(3.8.)555 3977 y(Accordingly)-7 b(,)27 b(setting)g Fq(\033)g Fs(:=)c Fq(C)1585 3893 y Fi(1)p 1565 3902 V 1565 3936 a Fh(n)1602 3948 y Fi(0)1549 4002 y Fp(\022)1648 3977 y Fq(\022)r(\032)1732 3947 y Fj(\000)p Fl(1)1844 3977 y Fn(\024)g Fs(4)1974 3935 y Fj(\000)2056 3913 y Fi(1)p 2035 3922 V 2035 3955 a Fh(n)2072 3967 y Fi(0)2118 3977 y Fs(,)1522 4126 y(\003\()p Fn(L)1669 4091 y Fp(m)1733 4126 y Fs(1\)\003\()p Fq(h)p Fs(\))g Fn(\024)f Fs(\003\()p Fn(L)2234 4091 y Fp(m)2297 4126 y Fq(h)p Fs(\))456 4275 y(and)719 4451 y(\003\()p Fn(L)866 4416 y Fp(m)929 4451 y Fq(h)p Fs(\))h Fn(\024)g Fs(\003\()p Fn(L)1267 4416 y Fp(m)1330 4451 y Fs(1\))1414 4395 y(\003\()p Fq(h)p Fs(\))p 1414 4432 171 4 v 1478 4508 a(4)947 4628 y(+)1086 4549 y Fk(X)1030 4749 y Fp(Z)t Fj(2Z)1177 4719 y Fi(\()p Fh(n)p Fi(\))1172 4758 y Fh(g)1276 4561 y Fk(\002)1311 4628 y Fq(B)t Fs(\003\()p FB(1)1516 4640 y Fp(Z)1569 4628 y Fs(\))c(+)f(2)p Fq(C)1804 4640 y Fp(\022)1841 4628 y Fq(\033)1891 4594 y Fp(m)1973 4628 y Fs(+)g(2)p Fq(K)2169 4640 y Fp(n)2210 4648 y Fi(0)2246 4628 y Fq(\032)2289 4594 y Fj(\000)p Fp(n)2382 4602 y Fi(0)2418 4628 y Fq(B)t Fs(\003\()p FB(1)2623 4640 y Fp(Z)2677 4628 y Fs(\))2709 4561 y Fk(\003)2757 4549 y(_)2779 4727 y Fp(Z)2863 4628 y Fq(h)p Fs(\003\()p Fn(L)3058 4594 y Fp(m)3122 4628 y Fs(1\))947 4895 y(+)1086 4816 y Fk(X)1030 5016 y Fp(Z)t Fj(2Z)1177 4986 y Fi(\()p Fh(n)p Fi(\))1172 5036 y Fh(b)1276 4895 y Fs(2)p Fq(C)1377 4907 y Fp(\022)1415 4895 y Fq(\033)1465 4861 y Fp(m)1528 4895 y Fn(k)p Fq(h)p Fn(k)1660 4907 y Fj(1)1729 4895 y Fs(\003\()p Fn(L)1876 4861 y Fp(m)1940 4895 y Fs(1\))p Fq(:)p 456 5107 499 4 v 555 5190 a Fl(6)588 5216 y FA(See)25 b(Lemma)d(2.5)h(for)g(the)i(de\014nition)f(of)1723 5199 y(^)1703 5216 y Fx(Z)1759 5192 y Fw(\()p Fd(n)p Fw(\))1873 5216 y FA(and)g Fx(Z)2066 5180 y Fw(\()p Fd(n)p Fw(\))2061 5228 y Fu(\003)2157 5216 y FA(.)p eop %%Page: 14 14 14 13 bop 456 251 a Fl(14)355 b(CARLANGELO)23 b(LIVERANI)f(AND)g(V)1947 236 y(\023)1941 251 y(ER)n(ONIQUE)g(MA)n(UME-DESCHAMPS)456 450 y Fs(Dividing)27 b(b)n(y)f(\003\()p Fn(L)1054 420 y Fp(m)1117 450 y Fs(1\))h(the)g(ab)r(o)n(v)n(e)e(inequalities)i(and)f (taking)g(the)h(limit)h Fq(m)23 b Fn(!)g(1)k Fs(yields)f(the)456 550 y(announced)h(con)n(tradiction)1017 722 y(\003\()p Fq(h)p Fs(\))c Fn(\024)1285 666 y Fs(\003\()p Fq(h)p Fs(\))p 1285 703 171 4 v 1349 779 a(4)1483 722 y(+)1623 643 y Fk(X)1566 843 y Fp(Z)t Fj(2Z)1713 813 y Fi(\()p Fh(n)p Fi(\))1708 852 y Fh(g)1813 722 y Fq(B)t Fs(\(2)p Fq(K)2025 734 y Fp(n)2066 742 y Fi(0)2102 722 y Fq(\032)2145 688 y Fj(\000)p Fp(n)2238 696 y Fi(0)2292 722 y Fs(+)c(1\)\003\()p FB(1)2588 734 y Fp(Z)2640 722 y Fs(\))2686 643 y Fk(_)2708 821 y Fp(Z)2793 722 y Fq(h)1210 1020 y Fn(\024)1285 964 y Fs(\003\()p Fq(h)p Fs(\))p 1285 1001 V 1349 1077 a(4)1483 1020 y(+)f Fq(B)t Fs(\(2)p Fq(K)1778 1032 y Fp(n)1819 1040 y Fi(0)1856 1020 y Fq(\032)1899 986 y Fj(\000)p Fp(n)1992 994 y Fi(0)2046 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Fh(f)2083 2152 y Fp(;)p Fj(L)2149 2127 y Fh(n)2190 2152 y Fp(h)2229 2160 y Fh(g)2263 2152 y Fl(\))2312 2187 y Fn(\000)g Fs(1)2437 2095 y Fk(\021)2500 2187 y Fn(k)p Fq(h)2590 2199 y Fp(f)2632 2187 y Fn(k)2674 2199 y Fj(1)456 2363 y Fs(whic)n(h)25 b(go)r(es)g(to)g(zero)g(when)g Fq(n)h Fs(go)r(es)e(to)i(in\014nit)n(y) -7 b(.)36 b(This)26 b(implies)g Fq(\025)2541 2375 y Fp(g)2603 2363 y Fs(=)c Fq(\025)2738 2375 y Fp(h)2805 2363 y Fs(:=)h Fq(\025)j Fs(and)f Fn(L)p Fs(\()p Fq(h)3286 2375 y Fj(\003)3324 2363 y Fs(\))f(=)456 2463 y Fq(\025h)552 2475 y Fj(\003)590 2463 y Fs(,)i(as)f(w)n(ell.)36 b(The)25 b(claimed)g(relation)g(from)g Fq(\032)g Fs(and)g Fq(\025)h Fs(follo)n(ws)f(from)g(the)g(follo)n(wing) g(c)n(hain)g(of)456 2562 y(inequalities)1056 2763 y(\003\()p Fn(L)p Fq(h)1251 2775 y Fj(\003)1289 2763 y Fs(\))f(=)42 b(lim)1423 2812 y Fp(n)p Fj(!1)1610 2763 y Fs(inf)1613 2816 y Fp(D)1667 2824 y Fh(n)1735 2706 y Fn(L)1792 2676 y Fp(n)p Fl(+1)1922 2706 y Fq(h)1970 2718 y Fj(\003)p 1735 2744 273 4 v 1799 2820 a Fn(L)1856 2796 y Fp(n)1902 2820 y Fs(1)2041 2763 y(=)51 b(lim)2128 2812 y Fp(n)p Fj(!1)2348 2763 y Fs(inf)2315 2816 y Fp(D)2369 2824 y Fh(n)p Fi(+1)2505 2706 y Fn(L)2562 2676 y Fp(n)p Fl(+1)2692 2706 y Fq(h)2740 2718 y Fj(\003)p 2505 2744 V 2569 2820 a Fn(L)2626 2796 y Fp(n)2672 2820 y Fs(1)1345 2997 y Fn(\025)42 b Fs(lim)1423 3046 y Fp(n)p Fj(!1)1643 2997 y Fs(inf)1610 3050 y Fp(D)1664 3058 y Fh(n)p Fi(+1)1800 2941 y Fn(L)1857 2910 y Fp(n)p Fl(+1)1986 2941 y Fq(h)2034 2953 y Fj(\003)p 1800 2978 V 1822 3054 a Fn(L)1879 3030 y Fp(n)p Fl(+1)2009 3054 y Fs(1)2096 2997 y(inf)2099 3050 y Fp(D)2153 3058 y Fh(n)2221 2941 y Fn(L)2278 2910 y Fp(n)p Fl(+1)2408 2941 y Fs(1)p 2221 2978 229 4 v 2263 3054 a Fn(L)2320 3030 y Fp(n)2366 3054 y Fs(1)2482 2997 y(=)23 b(\003\()p Fq(h)2708 3009 y Fj(\003)2746 2997 y Fs(\))p Fq(\032;)456 3196 y Fs(where)34 b(w)n(e)h(ha)n(v)n(e)e(used)i (t)n(wice)g(Remark)f(2.2.)58 b(Finally)-7 b(,)37 b(since)e(\003\()p Fq(h)2615 3208 y Fj(\003)2653 3196 y Fs(\))h Fq(>)e Fs(0,)j(it)e(follo) n(ws)f(that)456 3296 y Fn(L)513 3266 y Fp(n)558 3296 y Fq(h)606 3308 y Fj(\003)644 3296 y Fn(j)667 3308 y Fp(D)721 3316 y Fg(1)809 3296 y Fq(>)23 b Fs(0)k(whic)n(h)h(implies)f Fq(h)1533 3308 y Fj(\003)1571 3296 y Fn(j)1594 3308 y Fp(D)1648 3316 y Fg(1)1737 3296 y Fq(>)22 b Fs(0.)1491 b Ff(\003)456 3470 y FB(Lemma)35 b(4.2.)43 b Fr(The)35 b(functional)g Fs(\003)f Fr(\(r)l(estricte)l(d)g(to)h Fq(B)t(V)19 b Fr(\))34 b(is)h(line)l(ar,)h(p)l(ositive,)i(and)c(enjoys) 456 3569 y(the)c(pr)l(op)l(erty)g Fs(\003\()p Fn(L)p Fq(f)9 b Fs(\))23 b(=)g Fq(\025)p Fs(\003\()p Fq(f)9 b Fs(\))30 b Fr(for)g(al)t(l)h Fq(f)h Fn(2)23 b Fq(B)t(V)c Fr(.)39 b(Mor)l(e)l(over,)32 b Fq(\025)23 b Fs(=)g Fq(\032)p Fr(.)456 3743 y(Pr)l(o)l(of.)43 b Fs(Let)27 b Fq(f)32 b Fn(2)24 b(C)1060 3755 y Fp(a)1099 3743 y Fs(.)37 b(F)-7 b(or)27 b(all)h(in)n(tegers)e Fq(n;)14 b(k)30 b Fs(and)e Fq(x)23 b Fn(2)h Fq(D)2271 3755 y Fj(1)1145 3874 y(L)1191 3849 y Fh(n)p Fi(+)p Fh(k)1307 3874 y Fp(f)7 b Fl(\()p Fp(x)p Fl(\))p 1145 3895 290 4 v 1183 3943 a Fj(L)1229 3926 y Fh(n)1270 3943 y Fp(f)g Fl(\()p Fp(x)p Fl(\))1653 3914 y Fs(=)1814 3874 y Fj(L)1860 3849 y Fh(n)p Fi(+)p Fh(k)1976 3874 y Fp(f)g Fl(\()p Fp(x)p Fl(\))p 1810 3895 298 4 v 1810 3944 a(\003\()p Fj(L)1927 3927 y Fh(n)p Fi(+)p Fh(k)2043 3944 y Fp(f)g Fl(\))2211 3874 y(\003\()p Fj(L)2328 3849 y Fh(n)p Fi(+)p Fh(k)2444 3874 y Fp(f)g Fl(\))p 2211 3895 V 2249 3943 a(\003\()p Fj(L)2366 3926 y Fh(n)2407 3943 y Fp(f)g Fl(\))2634 3874 y(\003\()p Fj(L)2751 3849 y Fh(n)2792 3874 y Fp(f)g Fl(\))p 2634 3895 223 4 v 2638 3943 a Fj(L)2684 3926 y Fh(n)2725 3943 y Fp(f)g Fl(\()p Fp(x)p Fl(\))1270 4028 y Fn(#)627 b(#)358 b(#)344 b(#)1040 4184 y Fs(lim)1011 4234 y Fp(n)p Fj(!1)1208 4128 y Fn(L)1265 4098 y Fp(n)p Fl(+)p Fp(k)1398 4128 y Fq(f)9 b Fs(\()p Fq(x)p Fs(\))p 1208 4165 352 4 v 1252 4241 a Fn(L)1309 4217 y Fp(n)1354 4241 y Fq(f)g Fs(\()p Fq(x)p Fs(\))1653 4184 y(=)142 b Fq(h)1908 4196 y Fj(\003)1946 4184 y Fs(\()p Fq(x)p Fs(\))258 b Fq(\025)2363 4154 y Fp(k)2602 4184 y Fq(h)2650 4196 y Fj(\003)2688 4184 y Fs(\()p Fq(x)p Fs(\))2799 4154 y Fj(\000)p Fl(1)456 4358 y Fs(so)1420 4510 y(lim)1390 4560 y Fp(n)p Fj(!1)1578 4510 y Fs(sup)1583 4580 y Fp(D)1637 4588 y Fg(1)1717 4390 y Fk(\014)1717 4440 y(\014)1717 4490 y(\014)1717 4539 y(\014)1754 4454 y Fn(L)1811 4424 y Fp(n)p Fl(+)p Fp(k)1944 4454 y Fq(f)9 b Fs(\()p Fq(x)p Fs(\))p 1754 4491 V 1798 4567 a Fn(L)1855 4543 y Fp(n)1900 4567 y Fq(f)g Fs(\()p Fq(x)p Fs(\))2134 4510 y Fn(\000)18 b Fq(\025)2265 4476 y Fp(k)2307 4390 y Fk(\014)2307 4440 y(\014)2307 4490 y(\014)2307 4539 y(\014)2357 4510 y Fs(=)23 b(0)p Fq(:)456 4696 y Fs(But)970 4883 y(sup)975 4953 y Fp(D)1029 4961 y Fg(1)1109 4763 y Fk(\014)1109 4813 y(\014)1109 4862 y(\014)1109 4912 y(\014)1147 4827 y Fn(L)1204 4797 y Fp(n)1249 4827 y Fq(f)p 1147 4864 153 4 v 1151 4940 a Fn(L)1208 4916 y Fp(n)1253 4940 y Fs(1)1327 4883 y Fn(\000)1420 4827 y(L)1477 4797 y Fp(n)p Fl(+)p Fp(k)1610 4827 y Fq(f)p 1420 4864 240 4 v 1424 4940 a Fn(L)1481 4916 y Fp(n)p Fl(+)p Fp(k)1614 4940 y Fs(1)1670 4763 y Fk(\014)1670 4813 y(\014)1670 4862 y(\014)1670 4912 y(\014)1720 4883 y Fn(\024)g Fs(sup)1813 4953 y Fp(D)1867 4961 y Fg(1)1957 4827 y Fn(L)2014 4797 y Fp(n)p Fl(+)p Fp(k)2147 4827 y Fq(f)p 1957 4864 V 1961 4940 a Fn(L)2018 4916 y Fp(n)p Fl(+)p Fp(k)2151 4940 y Fs(1)2220 4763 y Fk(\014)2220 4813 y(\014)2220 4862 y(\014)2220 4912 y(\014)2302 4827 y Fn(L)2359 4797 y Fp(n)2404 4827 y Fq(f)p 2258 4864 V 2258 4940 a Fn(L)2315 4916 y Fp(n)p Fl(+)p Fp(k)2448 4940 y Fq(f)2518 4827 y Fn(L)2575 4797 y Fp(n)p Fl(+)p Fp(k)2708 4827 y Fs(1)p 2518 4864 232 4 v 2561 4940 a Fn(L)2618 4916 y Fp(n)2664 4940 y Fs(1)2778 4883 y Fn(\000)18 b Fs(1)2903 4763 y Fk(\014)2903 4813 y(\014)2903 4862 y(\014)2903 4912 y(\014)1720 5129 y Fn(\024)23 b(k)p Fq(f)9 b Fn(k)1941 5154 y Fj(1)2025 5129 y Fs(sup)2030 5199 y Fp(D)2084 5207 y Fg(1)2164 5009 y Fk(\014)2164 5059 y(\014)2164 5109 y(\014)2164 5158 y(\014)2245 5073 y Fn(L)2302 5043 y Fp(n)2348 5073 y Fq(f)p 2201 5110 240 4 v 2201 5186 a Fn(L)2258 5162 y Fp(n)p Fl(+)p Fp(k)2391 5186 y Fq(f)2461 5073 y Fn(L)2518 5043 y Fp(n)p Fl(+)p Fp(k)2651 5073 y Fs(1)p 2461 5110 232 4 v 2505 5186 a Fn(L)2562 5162 y Fp(n)2607 5186 y Fs(1)2721 5129 y Fn(\000)18 b Fs(1)2846 5009 y Fk(\014)2846 5059 y(\014)2846 5109 y(\014)2846 5158 y(\014)p eop %%Page: 16 16 16 15 bop 456 251 a Fl(16)355 b(CARLANGELO)23 b(LIVERANI)f(AND)g(V)1947 236 y(\023)1941 251 y(ER)n(ONIQUE)g(MA)n(UME-DESCHAMPS)456 458 y Fs(and)36 b(since)h(the)g(sequences)1387 421 y Fj(L)1433 396 y Fh(n)p Fi(+)p Fh(k)1549 421 y Fp(f)p 1387 439 201 4 v 1424 487 a Fj(L)1470 470 y Fh(n)1511 487 y Fp(f)1634 458 y Fs(and)1815 425 y Fj(L)1861 400 y Fh(n)p Fi(+)p Fh(k)1977 425 y Fl(1)p 1815 439 195 4 v 1852 487 a Fj(L)1898 470 y Fh(n)1939 487 y Fl(1)2056 458 y Fs(ha)n(v)n(e)f(the)h(same)g(limit)g Fq(\025)2882 428 y Fp(k)2923 458 y Fs(,)2996 421 y Fj(L)3042 396 y Fh(n)3082 421 y Fp(f)p 2996 439 126 4 v 2999 487 a Fj(L)3045 470 y Fh(n)3085 487 y Fl(1)3131 458 y Fn(j)3154 470 y Fp(D)3208 478 y Fg(1)3310 458 y Fs(is)g(a)456 564 y(Cauc)n(h)n(y)i (sequence,)k(hence)e(con)n(v)n(erges)c(to)k(a)f(function)h Fq(\027)2353 576 y Fp(f)2396 564 y Fs(.)75 b(Moreo)n(v)n(er,)41 b(if)g(w)n(e)g(tak)n(e)e(t)n(w)n(o)456 664 y(p)r(oin)n(ts)27 b Fq(x;)14 b(y)26 b Fn(2)e Fq(D)1004 676 y Fj(1)1074 664 y Fs(,)j(w)n(e)h(ha)n(v)n(e)980 859 y Fn(j)p Fq(\027)1044 871 y Fp(f)1087 859 y Fs(\()p Fq(x)p Fs(\))19 b Fn(\000)f Fq(\027)1341 871 y Fp(f)1384 859 y Fs(\()p Fq(y)s Fs(\))p Fn(j)24 b Fs(=)42 b(lim)1617 909 y Fp(n)p Fj(!1)1804 739 y Fk(\014)1804 789 y(\014)1804 839 y(\014)1804 888 y(\014)1842 803 y Fn(L)1899 773 y Fp(n)1944 803 y Fq(f)p 1842 840 153 4 v 1846 916 a Fn(L)1903 892 y Fp(n)1948 916 y Fs(1)2004 859 y(\()p Fq(x)p Fs(\))19 b Fn(\000)2227 803 y(L)2284 773 y Fp(n)2330 803 y Fq(f)p 2227 840 V 2231 916 a Fn(L)2288 892 y Fp(n)2334 916 y Fs(1)2389 859 y(\()p Fq(y)s Fs(\))2497 739 y Fk(\014)2497 789 y(\014)2497 839 y(\014)2497 888 y(\014)1539 1092 y Fs(=)42 b(lim)1617 1142 y Fp(n)p Fj(!1)1804 971 y Fk(\014)1804 1021 y(\014)1804 1071 y(\014)1804 1121 y(\014)1842 1036 y Fn(L)1899 1006 y Fp(n)1944 1036 y Fq(f)p 1842 1073 V 1846 1149 a Fn(L)1903 1125 y Fp(n)1948 1149 y Fs(1)2004 1092 y(\()p Fq(y)s Fs(\))2112 971 y Fk(\014)2112 1021 y(\014)2112 1071 y(\014)2112 1121 y(\014)2158 1092 y Fn(\001)2200 971 y Fk(\014)2200 1021 y(\014)2200 1071 y(\014)2200 1121 y(\014)2237 1036 y Fn(L)2294 1006 y Fp(n)2340 1036 y Fq(f)8 b Fs(\()p Fq(x)p Fs(\))p Fn(L)2557 1006 y Fp(n)2604 1036 y Fs(1)o(\()p Fq(y)s Fs(\))p 2237 1073 517 4 v 2237 1149 a Fn(L)2294 1125 y Fp(n)2340 1149 y Fs(1)o(\()p Fq(x)p Fs(\))p Fn(L)2549 1125 y Fp(n)2596 1149 y Fq(f)g Fs(\()p Fq(y)s Fs(\))2782 1092 y Fn(\000)18 b Fs(1)2907 971 y Fk(\014)2907 1021 y(\014)2907 1071 y(\014)2907 1121 y(\014)1539 1299 y Fn(\024)o(k)p Fq(f)9 b Fn(k)1736 1324 y Fj(1)1820 1299 y Fs(lim)14 b(sup)1860 1365 y Fp(n)p Fj(!1)2088 1207 y Fk(\020)2137 1299 y Fs(e)2174 1262 y Fp(d)2209 1270 y Fg(C)2241 1283 y Fi(+)2292 1262 y Fl(\()p Fj(L)2364 1237 y Fh(n)2405 1262 y Fp(f)s(;)p Fj(L)2506 1237 y Fh(n)2547 1262 y Fl(1\))2628 1299 y Fn(\000)k Fs(1)2753 1207 y Fk(\021)1539 1502 y Fn(\024)o(k)p Fq(f)9 b Fn(k)1736 1527 y Fj(1)1849 1502 y Fs(lim)1820 1552 y Fp(n)p Fj(!1)2007 1410 y Fk(\020)2057 1502 y Fs(e)2094 1468 y Fp(d)2129 1476 y Fg(C)2161 1484 y Fh(a)2201 1468 y Fl(\()p Fj(L)2273 1443 y Fh(n)2314 1468 y Fp(f)s(;)p Fj(L)2415 1443 y Fh(n)2456 1468 y Fl(1\))2537 1502 y Fn(\000)18 b Fs(1)2662 1410 y Fk(\021)2734 1502 y Fs(=)23 b(0)p Fq(;)456 1682 y Fs(where)39 b Fn(C)752 1694 y Fl(+)852 1682 y Fs(:=)44 b Fn(f)p Fq(h)f Fn(2)i Fq(B)t(V)64 b Fn(j)44 b Fq(h)g Fn(\025)g Fs(0)p Fn(g)p Fs(.)75 b(Therefore,)42 b Fq(\027)2302 1694 y Fp(f)2345 1682 y Fs(\()p Fq(x)p Fs(\))j(=)f(\003\()p Fq(f)9 b Fs(\))41 b(for)f(all)g Fq(x)45 b Fn(2)f Fq(D)3351 1694 y Fj(1)3421 1682 y Fs(.)456 1794 y(Hence,)31 b(\003\()p Fq(f)9 b Fs(\))28 b(=)g(lim)1137 1806 y Fp(n)p Fj(!1)1338 1757 y(L)1384 1732 y Fh(n)1425 1757 y Fp(f)p 1338 1775 126 4 v 1341 1823 a Fj(L)1387 1806 y Fh(n)1428 1823 y Fl(1)1505 1794 y Fs(for)i(all)g Fq(f)37 b Fn(2)28 b(C)1958 1806 y Fp(a)1998 1794 y Fs(.)46 b(Nev)n(ertheless,)30 b(if)h Fq(f)37 b Fn(2)28 b Fq(B)t(V)19 b Fs(,)32 b(the)f(function)456 1911 y(\()p Fq(f)d Fs(+)19 b Fq(a)685 1881 y Fj(\000)p Fl(1)788 1849 y Fk(W)871 1911 y Fq(f)28 b Fn(\000)19 b Fs(inf)i Fq(f)9 b Fs(\))26 b Fn(2)g(C)1372 1923 y Fp(a)1412 1911 y Fs(,)k(so)f(\003\()p Fq(f)9 b Fs(\))25 b(=)h(lim)1996 1874 y Fj(L)2042 1849 y Fh(n)2083 1874 y Fp(f)p 1996 1892 V 1999 1940 a Fj(L)2045 1923 y Fh(n)2085 1940 y Fl(1)2161 1911 y Fs(for)j(all)g Fq(f)34 b Fn(2)26 b Fq(B)t(V)19 b Fs(.)42 b(Clearly)-7 b(,)29 b(\003)g(is)g(linear)456 2011 y(b)n(y)e(the)h(linearit)n(y)f(of)g(the)h(limit.)456 2110 y(Next,)g(as)f Fn(L)p Fq(f)k Fn(2)24 b Fq(B)t(V)19 b Fs(,)28 b(w)n(e)f(kno)n(w)g(that)708 2309 y(\003\()p Fn(L)p Fq(f)9 b Fs(\))24 b(=)51 b(lim)1048 2359 y Fp(n)p Fj(!1)1245 2253 y Fn(L)1302 2223 y Fp(n)p Fl(+1)1432 2253 y Fq(f)p 1245 2290 237 4 v 1292 2366 a Fn(L)1349 2342 y Fp(n)1394 2366 y Fs(1)1515 2309 y(=)g(lim)1602 2359 y Fp(n)p Fj(!1)1799 2253 y Fn(L)1856 2223 y Fp(n)p Fl(+1)1986 2253 y Fq(f)p 1799 2290 V 1803 2366 a Fn(L)1860 2342 y Fp(n)p Fl(+1)1990 2366 y Fs(1)2055 2253 y Fn(L)2112 2223 y Fp(n)p Fl(+1)2242 2253 y Fs(1)p 2055 2290 229 4 v 2097 2366 a Fn(L)2154 2342 y Fp(n)2200 2366 y Fs(1)2316 2309 y(=)23 b(\003\()p Fq(f)9 b Fs(\)\003\()p Fn(L)p Fs(1\))23 b(=)g Fq(\032)p Fs(\003\()p Fq(f)9 b Fs(\))45 b Fq(:)456 2484 y Fs(But)24 b(then)h Fq(\032)e Fs(=)g Fq(\025)i Fs(is)f(obtained)g(b)n(y)g(taking)g Fq(f)31 b Fs(=)23 b Fq(h)2021 2496 y Fj(\003)2059 2484 y Fs(.)36 b(Notice)24 b(that)h(all)f(the)h(con)n(v)n(ergences)c(tak)n(e)456 2584 y(place)27 b(at)g(an)h(exp)r(onen)n(tial)f(rate.)1883 b Ff(\003)456 2757 y FB(Lemma)27 b(4.3.)39 b Fr(The)29 b(functional)g Fs(\003)f Fr(c)l(an)h(b)l(e)f(interpr)l(ete)l(d)h(as)f (a)h(non-atomic)g(me)l(asur)l(e)f Fq(\026)p Fr(,)h(i.e.)1357 2945 y Fs(\003\()p Fq(f)9 b Fs(\))23 b(=)1639 2832 y Fk(Z)1736 2945 y Fq(f)9 b(d\026)92 b Fn(8)p Fq(f)31 b Fn(2)24 b Fq(B)t(V)19 b Fs(\()p Fq(I)7 b(;)14 b(m)p Fs(\))p Fq(:)456 3134 y Fr(In)29 b(addition,)j Fs(supp)p Fq(\026)23 b Fn(\032)g Fq(X)1315 3146 y Fj(1)1415 3134 y Fr(and)30 b(the)g(me)l(asur)l(e)f Fq(h)2083 3146 y Fj(\003)2121 3134 y Fq(\026)h Fr(is)g Fq(T)12 b Fr({invariant.)456 3306 y(Pr)l(o)l(of.)43 b Fs(Clearly)-7 b(,)33 b(\003)g(can)f(b)r(e)i (extended)f(to)g(all)f(con)n(tin)n(uous)g(functions)h(since)g(it)g(is)g (con)n(tin)n(u-)456 3406 y(ous)d(in)h(the)g(sup)g(norm)f(and)h(con)n (tin)n(uous)f(functions)h(can)g(b)r(e)g(uniformly)f(appro)n(ximated)g (b)n(y)456 3505 y(b)r(ounded)22 b(v)-5 b(ariation)21 b(functions.)35 b(Hence)22 b(b)n(y)g(Riesz)g(theorem)f(there)h(exists)f (a)h(measure)f Fq(\026)h Fs(suc)n(h)456 3605 y(that)32 b(\003\()p Fq(f)9 b Fs(\))30 b(=)f Fq(\026)p Fs(\()p Fq(f)9 b Fs(\))33 b(on)e(eac)n(h)g(con)n(tin)n(uous)g(function.)50 b(Lemma)32 b(3.10)e(implies)i(immediately)456 3705 y(that)20 b(the)h(measure)e Fq(\026)h Fs(is)g(non)g(atomic.)34 b(Moreo)n(v)n(er)18 b(it)i(m)n(ust)g(agree)f(with)i(\003)f(on)f(the)i (c)n(haracteris-)456 3804 y(tic)h(function)g(of)f(eac)n(h)g(in)n(terv) -5 b(al.)35 b(Indeed,)23 b(let)f Fq(J)29 b Fs(b)r(e)22 b(an)g(in)n(terv)-5 b(al,)22 b(since)f Fq(\026)h Fs(is)g(a)f(Borel)f (measure,)456 3905 y(for)j(eac)n(h)g Fq(")g(>)f Fs(0)i(there)g(exists)f (a)h(larger)e(op)r(en)h(in)n(terv)-5 b(al)2222 3884 y(~)2202 3905 y Fq(J)32 b Fs(suc)n(h)23 b(that)h Fq(\026)p Fs(\()2742 3884 y(~)2721 3905 y Fq(J)9 b Fs(\))i Fn(\000)g Fq(\026)p Fs(\()p Fq(J)d Fs(\))23 b Fn(\024)g Fq(")h Fs(more-)456 4011 y(o)n(v)n(er)34 b(Lemma)h(3.10)g(implies)h(that)1629 3990 y(~)1609 4011 y Fq(J)44 b Fs(can)35 b(b)r(e)i(c)n(hosen)e(so)g (that)h(\003\()p FB(1)2706 4021 y Fl(~)2691 4036 y Fp(J)2761 4011 y Fn(\000)24 b FB(1)2898 4023 y Fp(J)2944 4011 y Fs(\))37 b Fn(\024)g Fq(")p Fs(.)62 b(Th)n(us,)456 4114 y(c)n(ho)r(osing)26 b(a)h(con)n(tin)n(uous)g(function)h Fq(f)36 b Fs(suc)n(h)27 b(that)h FB(1)2095 4126 y Fp(J)2164 4114 y Fn(\024)23 b Fq(f)32 b Fn(\024)22 b FB(1)2475 4124 y Fl(~)2460 4139 y Fp(J)2506 4114 y Fs(,)28 b(holds)2747 4081 y Fl(7)1074 4264 y Fs(\003\()p FB(1)1212 4276 y Fp(J)1258 4264 y Fs(\))18 b Fn(\000)g Fq(\026)p Fs(\()p FB(1)1521 4276 y Fp(J)1568 4264 y Fs(\))23 b Fn(\024)g Fs(\003\()p Fq(f)9 b Fs(\))18 b Fn(\000)g Fq(\026)p Fs(\()p Fq(f)9 b Fs(\))19 b(+)f(\003\()p FB(1)2403 4274 y Fl(~)2388 4289 y Fp(J)2452 4264 y Fn(\000)g FB(1)2583 4276 y Fp(J)2629 4264 y Fs(\))24 b Fn(\024)e Fq(")1074 4394 y(\026)p Fs(\()p FB(1)1204 4406 y Fp(J)1250 4394 y Fs(\))d Fn(\000)f Fs(\003\()p FB(1)1522 4406 y Fp(J)1568 4394 y Fs(\))23 b Fn(\024)g Fq(\026)p Fs(\()p Fq(f)9 b Fs(\))18 b Fn(\000)g Fs(\003\()p Fq(f)9 b Fs(\))19 b(+)f Fq(\026)p Fs(\()p FB(1)2395 4404 y Fl(~)2380 4419 y Fp(J)2445 4394 y Fn(\000)g FB(1)2576 4406 y Fp(J)2622 4394 y Fs(\))23 b Fn(\024)g Fq(":)456 4545 y Fs(Since)34 b(a)f(function)h(in)g Fq(B)t(V)53 b Fs(can)33 b(b)r(e)h(uniformly)g(appro)n(ximated)e(b)n(y)i(a)f (\014nite)h(linear)f(com)n(bi-)456 4644 y(nation)d(of)h(c)n (haracteristic)d(functions)j(of)g(in)n(terv)-5 b(als)30 b(it)h(follo)n(ws)e(that)i Fq(\026)p Fs(\()p Fq(f)9 b Fs(\))29 b(=)f(\003\()p Fq(f)9 b Fs(\))30 b(for)g(eac)n(h)456 4744 y(function)g(of)h(b)r(ounded)f(v)-5 b(ariation.)44 b(The)30 b(conclusion)g(of)g(the)g(lemma)h(follo)n(ws)e(from)h(Lemma) 456 4844 y(1.1)2817 b Ff(\003)555 5016 y Fs(In)28 b(conclusion,)f(w)n (e)g(ha)n(v)n(e)f(pro)n(v)n(ed)g(the)i(follo)n(wing)f(result.)p 456 5117 499 4 v 555 5190 a Fl(7)588 5216 y FA(The)d(existence)i(of)d (suc)n(h)h(a)g(function)g(is)f(insured)h(b)n(y)g(Urysohn's)f(Lemma.)p eop %%Page: 17 17 17 16 bop 1339 251 a Fl(LASOT)-5 b(A-YORKE)29 b(MAPS)f(WITH)h(HOLES)817 b(17)456 450 y FB(Theorem)43 b(4.4.)k Fr(Assume)40 b Fq(g)1435 420 y Fl(0)1512 450 y Fr(is)g(a)h(c)l(ontr)l(acting)f(p)l (otential)h(which)h(b)l(elongs)f(to)f(BV.)h(As-)456 550 y(sume)34 b(that)g(Condition)i(1)f(and)g(Condition)h(2)f(ar)l(e)g (satis\014e)l(d.)53 b(Then)35 b(ther)l(e)g(exists)f(a)h(unique)456 649 y(c)l(onditional)t(ly)e(invariant)f(pr)l(ob)l(ability)i(me)l(asur)l (e)d Fq(\027)g Fs(=)25 b Fq(hm)31 b Fr(which)i(is)f(absolutely)g(c)l (ontinuous)456 749 y(with)37 b(r)l(esp)l(e)l(ct)f(to)g Fq(m)p Fr(.)59 b(Ther)l(e)37 b(exists)f(a)h(unique)f(pr)l(ob)l(ability) i(me)l(asur)l(e)f Fq(\026)f Fr(whose)h(supp)l(ort)g(is)456 849 y(c)l(ontaine)l(d)d(in)g Fq(X)1006 861 y Fj(1)1110 849 y Fr(and)h(which)g(satis\014es)f Fq(\026)p Fs(\()p Fn(L)p Fq(f)9 b Fs(\))32 b(=)e Fq(\032\026)p Fs(\()p Fq(f)9 b Fs(\))34 b Fr(for)h(any)f(b)l(ounde)l(d)g(function)g Fq(f)9 b Fr(.)456 948 y(Mor)l(e)l(over,)31 b(ther)l(e)f(exists)f Fq(\024)23 b(<)g Fs(1)29 b Fr(such)h(that)g(for)g(any)g Fq(f)i Fn(2)23 b Fq(B)t(V)49 b Fr(and)30 b(any)h Fq(A)23 b Fn(\032)g Fq(I)7 b Fr(:)1352 1014 y Fk(\015)1352 1064 y(\015)1352 1114 y(\015)1352 1164 y(\015)1408 1078 y Fn(L)1465 1048 y Fp(n)1511 1078 y Fq(f)p 1408 1115 153 4 v 1440 1191 a(\032)1483 1167 y Fp(n)1589 1134 y Fn(\000)18 b Fq(h\026)p Fs(\()p Fq(f)9 b Fs(\))1884 1014 y Fk(\015)1884 1064 y(\015)1884 1114 y(\015)1884 1164 y(\015)1930 1217 y Fj(1)2023 1134 y Fn(\024)23 b Fr(Ct)14 b Fq(\024)2260 1100 y Fp(n)2305 1134 y Fn(k)p Fq(f)9 b Fn(k)2439 1146 y Fp(B)s(V)1256 1344 y Fr(and)30 b Fn(j)p Fq(m)p Fs(\()p Fq(T)1606 1310 y Fj(\000)p Fp(n)1702 1344 y Fq(A)p Fn(j)p Fq(X)1856 1356 y Fp(n)p Fj(\000)p Fl(1)1987 1344 y Fs(\))18 b Fn(\000)g Fq(\027)5 b Fs(\()p Fq(A)p Fs(\))p Fn(j)25 b(\024)d Fr(Ct)14 b Fq(\024)2576 1310 y Fp(n)2621 1344 y Fq(:)1209 1532 y Fs(5.)41 b Ft(The)32 b(Ha)n(usdorff)f(dimension)g (of)h Fq(X)2621 1544 y Fj(1)555 1681 y Fs(In)21 b(this)g(section)f(w)n (e)g(assume)g(that)g Fq(T)32 b Fs(is)20 b(uniformly)h(expanding)f(i.e.) 34 b(inf)21 b Fn(j)p Fq(T)2904 1651 y Fj(0)2926 1681 y Fn(j)j Fq(>)e Fs(1.)34 b(Remark)456 1784 y(that)28 b(this)h(implies)f(that)h(the)g(partition)e Fn(Z)1819 1754 y Fl(\()p Fp(n)p Fl(\))1945 1784 y Fs(is)h(generating.)37 b(F)-7 b(or)28 b Fq(g)2662 1754 y Fl(0)2723 1784 y Fs(=)2840 1752 y Fl(1)p 2822 1766 71 4 v 2822 1813 a Fp(T)2870 1796 y Fg(0)2902 1784 y Fs(,)h(and)f Fq(t)c Fn(\025)g Fs(0,)k(let)456 1889 y Fn(L)513 1901 y Fp(t)570 1889 y Fs(b)r(e)g(the)g(transfer)e(op)r(erator)g(with)i(hole)g(asso)r (ciated)e(to)h(\()p Fq(g)2404 1859 y Fl(0)2441 1889 y Fs(\))2473 1859 y Fp(t)2531 1889 y Fs(i.e.:)1329 2046 y Fn(L)1386 2058 y Fp(t)1416 2046 y Fq(f)9 b Fs(\()p Fq(x)p Fs(\))24 b(=)1715 1968 y Fk(X)1688 2146 y Fp(T)9 b(y)r Fl(=)p Fp(x)1861 2046 y Fs(\()p Fq(g)1936 2012 y Fl(0)1973 2046 y Fs(\))2005 2012 y Fp(t)2034 2046 y Fs(\()p Fq(y)s Fs(\))p FB(1)2190 2058 y Fp(X)2244 2066 y Fi(0)2281 2046 y Fs(\()p Fq(y)s Fs(\))p Fq(f)g Fs(\()p Fq(y)s Fs(\))p Fq(;)456 2277 y Fs(and)31 b(let)h(\002)810 2289 y Fp(t)839 2277 y Fs(,)h Fq(\032)938 2289 y Fp(t)999 2277 y Fs(and)f Fq(P)12 b Fs(\()p Fq(t)p Fs(\))32 b(b)r(e)g(the)h(n)n (um)n(b)r(ers)e(corresp)r(onding)f(to)i(\002,)g Fq(\032)p Fs(,)h Fq(P)12 b Fs(,)33 b(in)f(case)f Fq(t)f Fs(=)g(1.)456 2377 y(Recall)d(that)h(in)g(the)g(case)e Fq(g)1344 2347 y Fl(0)1404 2377 y Fs(=)1521 2344 y Fl(1)p 1502 2358 V 1502 2405 a Fp(T)1550 2389 y Fg(0)1582 2377 y Fs(,)i Fq(P)35 b Fs(=)23 b Fq(P)12 b Fs(\()p Fq(g)1949 2347 y Fl(0)1986 2377 y Fs(\))23 b(=)g(0.)456 2505 y FB(Theorem)33 b(5.1.)42 b Fr(L)l(et)31 b Fq(g)1244 2474 y Fl(0)1309 2505 y Fs(=)1429 2472 y Fl(1)p 1411 2486 V 1411 2533 a Fp(T)1459 2517 y Fg(0)1491 2505 y Fr(.)46 b(Assume)31 b(that)i(for)g(al)t(l)g Fs(0)27 b Fn(\024)g Fq(t)g Fn(\024)g Fs(1)p Fr(,)33 b(Conditions)h(0,)f(1)g(and)456 2604 y(2)h(ar)l(e)h (satis\014e)l(d.)53 b(Then,)36 b(ther)l(e)f(exists)f(a)h(unique)e Fs(0)e Fq(<)g(t)2280 2616 y Fl(0)2349 2604 y Fn(\024)g Fs(1)j Fr(such)g(that)g(for)i Fs(0)30 b Fn(\024)h Fq(t)h(<)f(t)3382 2616 y Fl(0)3419 2604 y Fr(,)456 2704 y Fq(\032)499 2716 y Fp(t)560 2704 y Fq(>)g Fs(1)k Fr(and)g(for)g Fs(1)d Fq(>)g(t)g(>)f(t)1395 2716 y Fl(0)1432 2704 y Fr(,)37 b Fq(\032)1537 2716 y Fp(t)1598 2704 y Fq(<)32 b Fs(1)p Fr(.)53 b(If)35 b Fq(T)46 b Fr(has)35 b(lar)l(ge)h(images)f(and)h(lar)l (ge)f(images)h(with)456 2803 y(r)l(esp)l(e)l(ct)29 b(to)h Fq(Y)48 b Fr(then,)30 b(HD)p Fs(\()p Fq(X)1358 2815 y Fj(1)1428 2803 y Fs(\))23 b(=)g Fq(t)1601 2815 y Fl(0)1638 2803 y Fr(.)456 2962 y(Pr)l(o)l(of.)43 b Fs(The)19 b(h)n(yp)r(othesis)g (of)h(Theorem)e(5.1)h(allo)n(w)f(us)h(to)h(apply)f(Theorem)g(A)g(to)h (the)f(op)r(erators)456 3062 y Fn(L)513 3074 y Fp(t)578 3062 y Fs(for)35 b(all)g(0)h Fn(\024)g Fq(t)g Fn(\024)g Fs(1.)60 b(Let)36 b(us)f(denote)h(b)n(y)f Fq(\026)2027 3074 y Fp(t)2092 3062 y Fs(the)h(conformal)e(measure)g(asso)r(ciated)h (to)456 3161 y Fq(g)499 3131 y Fp(t)550 3161 y Fs(=)23 b(\()p Fq(g)713 3131 y Fl(0)750 3161 y Fs(\))782 3131 y Fp(t)830 3161 y Fn(\001)c FB(1)920 3173 y Fp(X)974 3181 y Fi(0)1038 3161 y Fs(\(i.e.)37 b Fn(L)1270 3131 y Fj(\003)1270 3182 y Fp(t)1309 3161 y Fq(\026)1359 3173 y Fp(t)1411 3161 y Fs(=)23 b Fq(\032)1542 3173 y Fp(t)1571 3161 y Fq(\026)1621 3173 y Fp(t)1650 3161 y Fs(\).)456 3261 y(The)k(application)g Fq(t)c Fn(7!)g Fq(\032)1257 3273 y Fp(t)1314 3261 y Fs(is)28 b(strictly)f(decreasing.)35 b(Indeed,)28 b(remark)e(that:)37 b(for)27 b(all)h Fq(x)23 b Fn(2)h Fq(I)7 b Fs(,)1480 3409 y Fn(L)1537 3375 y Fp(n)1537 3430 y(t)1583 3409 y Fs(1\()p Fq(x)p Fs(\))23 b Fn(\024)g Fs(sup)14 b Fq(g)2029 3375 y Fp(t)p Fj(\000)p Fp(t)2131 3350 y Fg(0)2026 3430 y Fp(n)2157 3409 y Fn(L)2214 3375 y Fp(n)2214 3430 y(t)2239 3413 y Fg(0)2266 3409 y Fs(1\()p Fq(x)p Fs(\))456 3558 y(taking)30 b(the)g(p)r(o)n(w)n(er)1120 3525 y Fl(1)p 1116 3539 42 4 v 1116 3587 a Fp(n)1197 3558 y Fs(and)h(the)f(limit)i(giv)n(es:)41 b Fq(\032)1993 3570 y Fp(t)2050 3558 y Fn(\024)28 b Fs(\002)2208 3528 y Fp(t)p Fj(\000)p Fp(t)2310 3503 y Fg(0)2356 3558 y Fn(\001)21 b Fq(\032)2443 3570 y Fp(t)2468 3554 y Fg(0)2525 3558 y Fs(so)30 b(that)h Fq(\032)2856 3570 y Fp(t)2913 3558 y Fq(<)c(\032)3048 3570 y Fp(t)3073 3554 y Fg(0)3131 3558 y Fs(pro)n(vided)456 3686 y Fq(t)f(>)f(t)632 3656 y Fj(0)685 3686 y Fs(\(recall)k(that)g Fq(g)g(<)d Fs(1)j(and)g(remark)f (that)h(Theorem)g(A)h(implies:)40 b(lim)14 b(\()q Fn(L)2964 3656 y Fp(n)2964 3706 y(t)3009 3686 y Fs(1\()p Fq(x)p Fs(\)\))3209 3622 y Fi(1)p 3205 3631 37 3 v 3205 3664 a Fh(n)3282 3686 y Fs(=)25 b Fq(\032)3415 3698 y Fp(t)456 3789 y Fs(for)34 b(all)h Fq(x)p Fs(\).)61 b(Moreo)n(v)n(er,)34 b Fq(\032)1316 3801 y Fl(1)1389 3789 y Fn(\024)i Fq(e)1529 3758 y Fp(P)9 b Fl(\(1\))1705 3789 y Fs(=)35 b(1)g(\(see)g(Lemma)g (2.2\),)i(so)d(there)i(exists)e(a)h(unique)456 3888 y(n)n(um)n(b)r(er) 27 b(0)c Fn(\024)f Fq(t)940 3900 y Fl(0)1000 3888 y Fn(\024)h Fs(1)k(suc)n(h)h(that)f(for)g(1)c Fq(>)g(t)g(>)f(t)1974 3900 y Fl(0)2012 3888 y Fs(,)27 b Fq(\032)2105 3900 y Fp(t)2158 3888 y Fq(<)22 b Fs(1)27 b(and)h(for)f(0)22 b Fn(\024)h Fq(t)g(<)g(t)2926 3900 y Fl(0)2963 3888 y Fs(,)28 b Fq(\032)3057 3900 y Fp(t)3109 3888 y Fq(>)23 b Fs(1)k(.)456 3988 y(The)32 b(follo)n(wing)e(lemma)i(is)g(a)f(direct)h (consequence)f(of)h(the)g(b)r(ounded)g(distortion)f(and)h(large)456 4087 y(images)26 b(h)n(yp)r(othesis.)456 4246 y FB(Lemma)k(5.2.)41 b Fr(Assume)30 b(that)h Fq(g)1511 4216 y Fl(0)1573 4246 y Fs(=)1691 4214 y Fl(1)p 1672 4228 71 4 v 1672 4275 a Fp(T)1720 4258 y Fg(0)1753 4246 y Fr(.)42 b(Assume)30 b(that)g(for)i(al)t(l)g Fs(0)24 b Fn(\024)h Fq(t)g Fn(\024)f Fs(1)p Fr(,)31 b(Conditions)i(0,)456 4346 y(1)c(and)g(2)h(ar)l(e)f (satis\014e)l(d.)39 b(F)-6 b(or)29 b(al)t(l)h Fs(0)23 b Fn(\024)g Fq(t)g Fn(\024)f Fs(1)p Fr(,)29 b(ther)l(e)g(exists)g Fq(K)f(>)23 b Fs(0)p Fr(,)29 b(such)g(that)g(for)h(al)t(l)g Fq(n)23 b Fn(2)g Fo(N)456 4445 y Fr(and)30 b Fq(Z)f Fn(2)23 b(Z)848 4415 y Fl(\()p Fp(n)p Fl(\))945 4445 y Fr(,)30 b(if)h Fq(\026)1131 4457 y Fp(t)1160 4445 y Fs(\()p Fq(Z)6 b Fs(\))23 b Fq(>)g Fs(0)29 b Fr(then)g(for)i(al)t(l)g Fq(x)23 b Fn(2)h Fq(Z)6 b Fr(,)456 4640 y Fs(\(5.1\))658 b Fq(K)1362 4605 y Fj(\000)p Fl(1)1473 4640 y Fn(\024)1573 4584 y Fs(\()p Fq(g)1648 4553 y Fl(0)1645 4604 y Fp(n)1690 4584 y Fs(\))1722 4553 y Fp(t)1751 4584 y Fs(\()p Fq(x)p Fs(\))p 1571 4621 295 4 v 1571 4697 a Fq(\032)1614 4668 y Fp(n)1614 4717 y(t)1659 4697 y Fq(\026)1709 4709 y Fp(t)1738 4697 y Fs(\()p Fq(Z)6 b Fs(\))1905 4640 y Fr(and)60 b Fq(K)2173 4605 y Fj(\000)p Fl(1)2285 4640 y Fn(\024)2384 4584 y Fq(g)2427 4553 y Fl(0)2424 4604 y Fp(n)2469 4584 y Fs(\()p Fq(x)p Fs(\))p 2382 4621 201 4 v 2382 4697 a Fq(m)p Fs(\()p Fq(Z)6 b Fs(\))2592 4640 y Fq(:)456 4825 y Fr(If)30 b(mor)l(e)l(over)g Fq(T)41 b Fr(has)31 b(lar)l(ge)f(images)h(and)f(lar)l(ge)h(images)f(with)h(r)l(esp)l(e)l (ct)e(to)h Fq(Y)48 b Fr(then)456 5014 y Fs(\(5.2\))1412 4958 y(\()p Fq(g)1487 4928 y Fl(0)1484 4979 y Fp(n)1529 4958 y Fs(\))1561 4928 y Fp(t)1591 4958 y Fs(\()p Fq(x)p Fs(\))p 1410 4995 295 4 v 1410 5071 a Fq(\032)1453 5043 y Fp(n)1453 5092 y(t)1498 5071 y Fq(\026)1548 5083 y Fp(t)1577 5071 y Fs(\()p Fq(Z)6 b Fs(\))1738 5014 y Fn(\024)22 b Fq(K)35 b Fr(and)2104 4958 y Fq(g)2147 4928 y Fl(0)2144 4979 y Fp(n)2189 4958 y Fs(\()p Fq(x)p Fs(\))p 2103 4995 201 4 v 2103 5071 a Fq(m)p Fs(\()p Fq(Z)6 b Fs(\))2336 5014 y Fn(\024)22 b Fq(K)456 5199 y Fr(wher)l(e)30 b Fq(m)g Fr(is)g(the)g(L)l(eb)l(esgue)f(me)l(asur)l(e.)p eop %%Page: 18 18 18 17 bop 456 251 a Fl(18)355 b(CARLANGELO)23 b(LIVERANI)f(AND)g(V)1947 236 y(\023)1941 251 y(ER)n(ONIQUE)g(MA)n(UME-DESCHAMPS)456 450 y Fr(Pr)l(o)l(of.)43 b Fs(First)23 b(of)h(all,)g(remark)e(that)i (the)f(large)f(images)h(with)h(resp)r(ect)f(to)h Fq(Y)42 b Fs(prop)r(ert)n(y)22 b(implies)456 550 y(that)37 b(for)f(all)h(0)h Fn(\024)g Fq(t)g Fn(\024)g Fs(1,)h(the)e(supp)r(ort)g(of)g Fq(\026)1987 562 y Fp(t)2053 550 y Fs(is)f Fq(X)2214 562 y Fj(1)2285 550 y Fs(.)64 b(So,)39 b Fq(\026)2572 562 y Fp(t)2601 550 y Fs(\()p Fq(Z)6 b Fs(\))39 b Fq(>)f Fs(0)f(if)g(and)g(only)f(if)456 653 y Fq(Z)15 b Fn(\\)10 b Fq(X)662 665 y Fj(1)756 653 y Fn(6)p Fs(=)23 b Fn(;)p Fs(.)35 b(In)23 b(addition,)i Fq(Z)j Fn(2)c(Z)1622 622 y Fl(\()p Fp(n)p Fl(\))1742 653 y Fs(with)g Fq(Z)16 b Fn(\\)10 b Fq(X)2134 665 y Fj(1)2227 653 y Fn(6)p Fs(=)23 b Fn(;)p Fs(,)h Fq(\026)2454 665 y Fp(t)2483 653 y Fs(\()p Fq(T)2576 622 y Fp(n)2621 653 y Fq(Z)6 b Fs(\))23 b(=)f(1.)36 b(No)n(w,)23 b(compute)456 1007 y(\(5.3\))795 837 y Fq(\026)845 849 y Fp(t)874 837 y Fs(\()p Fq(Z)6 b Fs(\))23 b(=)1103 724 y Fk(Z)1200 837 y FB(1)1248 849 y Fp(Z)1301 837 y Fq(d\026)1394 849 y Fp(t)1446 837 y Fs(=)1567 780 y(1)p 1544 817 89 4 v 1544 893 a Fq(\032)1587 865 y Fp(n)1587 914 y(t)1656 724 y Fk(Z)1753 837 y Fn(L)1810 802 y Fp(n)1810 857 y(t)1855 837 y FB(1)1903 849 y Fp(Z)1956 837 y Fq(d\026)2049 849 y Fp(t)2102 837 y Fs(=)2223 780 y(1)p 2200 817 V 2200 893 a Fq(\032)2243 865 y Fp(n)2243 914 y(t)2339 724 y Fk(Z)2293 962 y Fp(T)2341 945 y Fh(n)2382 962 y Fp(Z)2463 769 y Fk(\002)2498 837 y Fs(\()p Fq(g)2573 802 y Fl(0)2570 857 y Fp(n)2615 837 y Fs(\))2647 802 y Fp(t)2677 837 y FB(1)2725 849 y Fp(X)2779 857 y Fh(n)p Fg(\000)p Fi(1)2897 769 y Fk(\003)2950 837 y Fn(\016)18 b Fq(T)3071 801 y Fj(\000)p Fp(n)3059 861 y(Z)3167 837 y Fq(d\026)3260 849 y Fp(t)1024 1116 y Fs(=)1122 1060 y(1)p 1099 1097 V 1099 1173 a Fq(\032)1142 1144 y Fp(n)1142 1193 y(t)1238 1003 y Fk(Z)1192 1241 y Fp(T)1240 1225 y Fh(n)1281 1241 y Fp(Z)1349 1116 y Fs(\()p Fq(g)1424 1082 y Fl(0)1421 1137 y Fp(n)1466 1116 y Fs(\))1498 1082 y Fp(t)1546 1116 y Fn(\016)g Fq(T)1667 1081 y Fj(\000)p Fp(n)1655 1140 y(Z)1763 1116 y Fq(d\026)1856 1128 y Fp(t)1886 1116 y Fq(;)456 1362 y Fs(\(recall)32 b(that)h(w)n(e)g(assume)f Fq(T)1381 1332 y Fp(n)1426 1362 y Fs(\()p Fq(Z)27 b Fn(\\)c Fq(X)1689 1374 y Fp(n)p Fj(\000)p Fl(1)1851 1362 y Fn(\033)32 b Fq(X)2017 1374 y Fj(1)2087 1362 y Fs(\)\).)54 b(The)33 b(b)r(ounded)g(distortion)g(prop)r(ert)n(y)456 1462 y(implies,)27 b(for)h Fq(x)23 b Fn(2)g Fq(Z)6 b Fs(,)767 1641 y Fq(K)844 1607 y Fj(\000)p Fl(1)933 1641 y Fq(\026)983 1653 y Fp(t)1012 1641 y Fs(\()p Fq(T)1105 1607 y Fp(n)1149 1641 y Fq(Z)g Fs(\)\()p Fq(g)1319 1607 y Fl(0)1316 1661 y Fp(n)1361 1641 y Fs(\))1393 1607 y Fp(t)1423 1641 y Fs(\()p Fq(x)p Fs(\))24 b Fn(\024)1673 1528 y Fk(Z)1628 1766 y Fp(T)1676 1749 y Fh(n)1716 1766 y Fp(Z)1784 1641 y Fs(\()p Fq(g)1859 1607 y Fl(0)1856 1661 y Fp(n)1901 1641 y Fs(\))1933 1607 y Fp(t)1981 1641 y Fn(\016)18 b Fq(T)2102 1605 y Fj(\000)p Fp(n)2090 1665 y(Z)2198 1641 y Fq(d\026)2291 1653 y Fp(t)2343 1641 y Fn(\024)23 b Fq(K)6 b(\026)2558 1653 y Fp(t)2587 1641 y Fs(\()p Fq(T)2680 1607 y Fp(n)2724 1641 y Fq(Z)g Fs(\)\()p Fq(g)2894 1607 y Fl(0)2891 1661 y Fp(n)2936 1641 y Fs(\))2968 1607 y Fp(t)2998 1641 y Fs(\()p Fq(x)p Fs(\))p Fq(:)456 1887 y Fs(This)38 b(giv)n(es)f(\(5.1\))i(for)e Fq(\026)1267 1899 y Fp(t)1335 1887 y Fs(and)i(\(5.2\))f(for)g Fq(\026)1905 1899 y Fp(t)1972 1887 y Fs(using)h(the)f(large)f(images)h (prop)r(ert)n(y)f(\(whic)n(h)456 1986 y(implies)29 b Fq(\026)789 1998 y Fp(t)819 1986 y Fs(\()p Fq(T)912 1956 y Fp(n)956 1986 y Fq(Z)6 b Fs(\))26 b(=)g(1\).)42 b(The)30 b(computation)f(is)g(the)h(same)f(for)g Fq(m)p Fs(,)h(recalling)e(that) i Fq(m)f Fs(is)h Fq(g)3408 1956 y Fl(0)456 2086 y Fs(conformal)c(with)i (eigen)n(v)-5 b(alue)27 b(1)g(and)g(using)h(the)g(large)e(image)h(prop) r(ert)n(y)-7 b(.)590 b Ff(\003)555 2246 y Fs(Fix)30 b Fq(")d(>)f Fs(0)k(and)f Fq(n)e Fn(2)g Fo(N)40 b Fs(suc)n(h)29 b(that)h(for)g(all)f Fq(Z)k Fn(2)27 b(Z)2204 2216 y Fl(\()p Fp(n)p Fl(\))2301 2246 y Fs(,)j(the)h(diameter)e(of)h Fq(Z)35 b Fs(is)30 b(less)f(than)456 2345 y Fq(")p Fs(.)37 b(Let)28 b Fn(F)k Fs(=)23 b Fn(f)p Fq(Z)28 b Fn(2)c(Z)1157 2315 y Fl(\()p Fp(n)p Fl(\))1277 2345 y Fn(j)g Fq(Z)g Fn(\\)19 b Fq(X)1548 2357 y Fj(1)1642 2345 y Fn(6)p Fs(=)k Fn(;g)p Fs(.)37 b(It)28 b(is)g(a)f(co)n(v)n(er)f(of)i Fq(X)2496 2357 y Fj(1)2594 2345 y Fs(of)g(diameter)f(less)h(than)g Fq(")p Fs(.)456 2445 y(In)f(what)h(follo)n(ws,)f Fq(x)1110 2457 y Fp(Z)1191 2445 y Fs(denotes)g(an)n(y)g(elemen)n(t)h(of)g Fq(Z)6 b Fs(.)1165 2520 y Fk(X)1151 2698 y Fp(Z)t Fj(2F)1312 2598 y Fs(\()q(diam)o Fq(Z)g Fs(\))1619 2557 y Fp(t)1672 2598 y Fn(\024)o Fq(K)1813 2564 y Fp(t)1870 2520 y Fk(X)1856 2698 y Fp(Z)t Fj(2F)2003 2598 y Fs(\()p Fq(g)2078 2564 y Fl(0)2075 2619 y Fp(n)2120 2598 y Fs(\))2152 2564 y Fp(t)2182 2598 y Fs(\()p Fq(x)2261 2610 y Fp(Z)2315 2598 y Fs(\))28 b(using)f(\(5.1\))1672 2831 y Fn(\024)o Fq(K)1813 2796 y Fl(2)p Fp(t)1875 2831 y Fq(\032)1918 2796 y Fp(n)1918 2851 y(t)1991 2752 y Fk(X)1977 2930 y Fp(Z)t Fj(2F)2138 2831 y Fq(\026)2188 2843 y Fp(t)2218 2831 y Fs(\()p Fq(Z)6 b Fs(\))27 b(using)h(\(5.2\))1672 3048 y(=)o Fq(K)1813 3013 y Fl(2)p Fp(t)1875 3048 y Fq(\032)1918 3013 y Fp(n)1918 3068 y(t)1963 3048 y Fq(\026)2013 3060 y Fp(t)2042 3048 y Fs(\()p Fq(X)2143 3060 y Fj(1)2214 3048 y Fs(\))23 b(=)g Fq(K)2434 3013 y Fl(2)p Fp(t)2496 3048 y Fq(\032)2539 3013 y Fp(n)2539 3068 y(t)2584 3048 y Fq(:)456 3190 y Fs(By)29 b(our)f(c)n(hoice)h(of)g Fq(n)p Fs(,)h(it)f(is)h(clear)e(that) i Fq(n)25 b Fn(!)h(1)k Fs(when)f Fq(")d Fn(!)g Fs(0.)41 b(If)30 b Fq(t)c(>)g(t)2794 3202 y Fl(0)2860 3190 y Fs(then)k Fq(\032)3094 3202 y Fp(t)3149 3190 y Fq(<)c Fs(1)j(and)456 3289 y(the)f(ab)r(o)n(v)n(e)e(expression)g(go)r(es)h(to)g(zero.)36 b(Hence)28 b(w)n(e)f(conclude)g Fq(H)7 b(D)r Fs(\()p Fq(X)2684 3301 y Fj(1)2754 3289 y Fs(\))24 b Fn(\024)e Fq(t)2927 3301 y Fl(0)2965 3289 y Fs(.)456 3389 y(Let)27 b(us)h(pro)n(v)n(e)e(the)i(con)n(v)n(erse)d(inequalit)n(y)-7 b(.)37 b(W)-7 b(e)28 b(use)f(the)h(follo)n(wing)f(Y)-7 b(oung's)27 b(result.)456 3548 y FB(Theorem)42 b(5.3.)k Fs([Y])39 b Fr(L)l(et)g Fq(X)45 b Fr(b)l(e)40 b(a)f(metric)h(sp)l(ac)l (e,)i(let)d Fq(Z)46 b Fn(\032)40 b Fq(X)45 b Fr(assume)39 b(ther)l(e)g(exists)g(a)456 3648 y(pr)l(ob)l(ability)32 b(me)l(asur)l(e)d Fq(\026)h Fr(such)g(that)f Fq(\026)p Fs(\()p Fq(Z)6 b Fs(\))24 b Fq(>)e Fs(0)p Fr(,)30 b(for)g(any)h Fq(x)23 b Fn(2)g Fq(Z)6 b Fr(,)30 b(de\014ne:)1406 3837 y Fq(d)1449 3849 y Fp(\026)p 1406 3874 88 4 v 1494 3837 a Fs(\()p Fq(x)p Fs(\))24 b(=)f(lim)14 b(inf)1766 3889 y Fp(")p Fj(!)p Fl(0)1970 3781 y Fs(log)h Fq(\026)p Fs(\()p Fq(B)t Fs(\()p Fq(x;)f(")p Fs(\)\))p 1970 3818 491 4 v 2136 3894 a(log)g Fq(")2471 3837 y(;)456 4023 y Fr(if)30 b(for)h(al)t(l)g Fq(x)23 b Fn(2)h Fq(Z)6 b Fr(,)29 b Fq(d)1097 4035 y Fp(\026)p 1054 4060 88 4 v 1142 4023 a Fs(\()p Fq(x)p Fs(\))24 b Fn(\025)f Fq(d)30 b Fr(then)f Fq(H)7 b(D)r Fs(\()p Fq(Z)f Fs(\))23 b Fn(\025)g Fq(d)p Fr(.)555 4189 y Fs(T)-7 b(ak)n(e)27 b Fq(x)c Fn(2)h Fq(X)974 4201 y Fj(1)1072 4189 y Fs(and)j Fq(")c(>)f Fs(0,)28 b(let)946 4331 y Fq(n)996 4343 y Fl(0)1056 4331 y Fs(=)23 b(inf)1259 4264 y Fk(\010)1307 4331 y Fq(n)g Fn(2)h Fo(N)1560 4260 y Fk(\014)1560 4310 y(\014)1615 4331 y Fn(9)k Fq(y)e Fn(2)d Fq(B)t Fs(\()p Fq(x;)14 b(")p Fs(\))52 b(:)78 b Fq(g)2284 4297 y Fl(0)2281 4351 y Fp(n)2326 4331 y Fs(\()p Fq(y)s Fs(\))23 b Fn(\024)g Fs(2)p Fq(K)33 b(")2739 4264 y Fk(\011)2806 4331 y Fn(\000)18 b Fs(1)p Fq(:)456 4472 y Fs(Accordingly)-7 b(,)26 b(there)i(exists)f Fq(y)1416 4484 y Fl(0)1476 4472 y Fn(2)c Fq(B)t Fs(\()p Fq(x;)14 b(")p Fs(\))29 b(suc)n(h)e(that,)1175 4615 y Fq(g)1218 4581 y Fl(0)1215 4636 y Fp(n)1256 4644 y Fi(0)1292 4615 y Fs(\()p Fq(y)1365 4627 y Fl(0)1403 4615 y Fs(\))p Fq(g)1478 4581 y Fl(0)1515 4615 y Fs(\()p Fq(T)1608 4581 y Fp(n)1649 4589 y Fi(0)1685 4615 y Fq(y)1726 4627 y Fl(0)1763 4615 y Fs(\))c(=)g Fq(g)1949 4581 y Fl(0)1946 4636 y Fp(n)1987 4644 y Fi(0)2019 4636 y Fl(+1)2107 4615 y Fs(\()p Fq(y)2180 4627 y Fl(0)2217 4615 y Fs(\))g Fn(\024)g Fs(2)p Fq(K)33 b(")55 b Fs(so,)1638 4795 y(2)p Fq(K)6 b(")22 b(<)50 b(g)1976 4761 y Fl(0)1973 4816 y Fp(n)2014 4824 y Fi(0)2050 4795 y Fs(\()p Fq(y)2123 4807 y Fl(0)2161 4795 y Fs(\))h Fn(\024)2360 4739 y Fs(2)p Fq(K)6 b(")p 2341 4776 195 4 v 2341 4852 a Fs(inf)21 b Fq(g)2499 4828 y Fl(0)2546 4795 y Fq(:)-2113 b Fs(\(5.4\))456 4981 y(Using)27 b(Lemma)g(5.2)g(and) h(\(5.4\))f(w)n(e)g(get:)1304 5176 y(2)p Fq(")22 b Fn(\024)50 b Fs(diam)p Fq(Z)1759 5188 y Fp(n)1800 5196 y Fi(0)1836 5176 y Fs(\()p Fq(y)1909 5188 y Fl(0)1947 5176 y Fs(\))23 b Fn(\024)2100 5120 y Fs(2)p Fq(K)2219 5089 y Fl(2)2255 5120 y Fq(")p 2100 5157 V 2100 5233 a Fs(inf)d Fq(g)2257 5209 y Fl(0)2327 5176 y Fs(:=)j Fq(C)2497 5188 y Fl(1)2535 5176 y Fq(":)p eop %%Page: 19 19 19 18 bop 1339 251 a Fl(LASOT)-5 b(A-YORKE)29 b(MAPS)f(WITH)h(HOLES)817 b(19)456 450 y Fs(Let)27 b Fq(B)667 462 y Fl(1)728 450 y Fs(=)22 b Fq(B)t Fs(\()p Fq(x;)14 b(")p Fs(\))19 b Fn(n)f Fq(Z)1205 462 y Fp(n)1246 470 y Fi(0)1283 450 y Fs(\()p Fq(y)1356 462 y Fl(0)1393 450 y Fs(\).)37 b(If)28 b Fq(B)1631 462 y Fl(1)1691 450 y Fn(6)p Fs(=)23 b Fn(;)p Fs(,)k(then)h(let)g(us)g(de\014ne:)1023 621 y Fq(n)1073 633 y Fl(1)1134 621 y Fs(=)22 b(inf)1336 554 y Fk(\010)1385 621 y Fq(n)g Fn(2)i Fo(N)1637 550 y Fk(\014)1637 600 y(\014)1693 621 y Fn(9)j Fq(y)f Fn(2)e Fq(B)1975 633 y Fl(1)2063 621 y Fs(:)78 b Fq(g)2207 587 y Fl(0)2204 641 y Fp(n)2249 621 y Fs(\()p Fq(y)s Fs(\))23 b Fn(\024)g Fs(2)p Fq(K)32 b(")2662 554 y Fk(\011)2729 621 y Fn(\000)18 b Fs(1)p Fq(:)456 790 y Fs(Hence,)27 b(there)h(exists)f Fq(y)1208 802 y Fl(1)1268 790 y Fn(2)c Fq(B)1409 802 y Fl(1)1474 790 y Fs(suc)n(h)28 b(that)1489 960 y(2)p Fq(")23 b Fn(\024)50 b Fs(diam)p Fq(Z)1945 972 y Fp(n)1986 980 y Fi(1)2022 960 y Fs(\()p Fq(y)2095 972 y Fl(1)2132 960 y Fs(\))24 b Fn(\024)e Fq(C)2334 972 y Fl(1)2372 960 y Fq(")456 1130 y Fs(b)n(y)41 b(the)i(same)f(argumen)n(ts)e(as)i(b) r(efore.)80 b(Remark)41 b(that)i Fq(n)2398 1142 y Fl(1)2482 1130 y Fn(\025)k Fq(n)2644 1142 y Fl(0)2723 1130 y Fs(b)n(y)42 b(construction,)j(so)456 1229 y Fq(Z)513 1241 y Fp(n)554 1249 y Fi(0)590 1229 y Fs(\()p Fq(y)663 1241 y Fl(0)700 1229 y Fs(\))25 b Fn(\\)g Fq(Z)894 1241 y Fp(n)935 1249 y Fi(1)971 1229 y Fs(\()p Fq(y)1044 1241 y Fl(1)1081 1229 y Fs(\))39 b(=)e Fn(;)p Fs(.)64 b(This)37 b(implies)g(that)g Fq(Z)2119 1241 y Fp(n)2160 1249 y Fi(0)2196 1229 y Fs(\()p Fq(y)2269 1241 y Fl(0)2306 1229 y Fs(\))25 b Fn([)g Fq(Z)2500 1241 y Fp(n)2541 1249 y Fi(1)2577 1229 y Fs(\()p Fq(y)2650 1241 y Fl(1)2687 1229 y Fs(\))39 b Fn(\033)e Fq(B)t Fs(\()p Fq(x;)14 b(")p Fs(\).)65 b(Indeed)456 1329 y(if)28 b(it)g(w)n(ere)e (not)i(the)g(case,)f(w)n(e)g(could)g(\014nd)h Fq(y)1846 1341 y Fl(2)1906 1329 y Fn(2)c Fq(B)2048 1341 y Fl(1)2103 1329 y Fn(n)18 b Fq(Z)2220 1341 y Fp(n)2261 1349 y Fi(1)2298 1329 y Fs(\()p Fq(y)2371 1341 y Fl(1)2408 1329 y Fs(\))28 b(and)f Fq(n)2679 1341 y Fl(2)2739 1329 y Fn(\025)c Fq(n)2877 1341 y Fl(1)2942 1329 y Fs(suc)n(h)k(that:)1478 1498 y(2)p Fq(")22 b Fn(\024)50 b Fs(diam)p Fq(Z)1933 1510 y Fp(n)1974 1518 y Fi(2)2011 1498 y Fs(\()p Fq(y)2084 1510 y Fl(2)2121 1498 y Fs(\))23 b Fn(\024)g Fq(C)2323 1510 y Fl(1)2360 1498 y Fq(":)456 1668 y Fs(By)e(construction,)h(w)n(e) f(w)n(ould)g(obtain)g(three)h(disjoin)n(t)f(in)n(terv)-5 b(als,)22 b(with)h(diameter)e(larger)e(than)456 1768 y(2)p Fq(")p Fs(,)27 b(all)g(in)n(tersecting)g Fq(B)t Fs(\()p Fq(x;)14 b(")p Fs(\),)28 b(but)g(this)g(is)g(clearly)e(imp)r (ossible.)456 1867 y(Therefore,)i(w)n(e)g(ha)n(v)n(e)g(sho)n(wn)g(that) h Fq(B)t Fs(\()p Fq(x;)14 b(")p Fs(\))26 b Fn(\033)f Fq(Z)2034 1879 y Fp(n)2075 1887 y Fi(0)2131 1867 y Fn([)19 b Fq(Z)2262 1879 y Fp(n)2303 1887 y Fi(1)2340 1867 y Fs(,)29 b(where)f(the)i(second)e(set)h(ma)n(y)f(b)r(e)456 1967 y(empt)n(y)-7 b(.)37 b(W)-7 b(e)28 b(ha)n(v)n(e)1405 2183 y Fq(\026)1455 2195 y Fp(t)1484 2183 y Fs(\()p Fq(Z)1573 2195 y Fp(n)1614 2203 y Fh(i)1645 2183 y Fs(\))83 b Fn(\024)1922 2127 y Fq(K)1999 2097 y Fp(t)p 1918 2164 115 4 v 1918 2240 a Fq(\032)1961 2203 y Fp(n)2002 2211 y Fh(i)1961 2261 y Fp(t)2056 2183 y Fs(\(diam)p Fq(Z)2325 2195 y Fp(n)2366 2203 y Fh(i)2396 2183 y Fs(\))2429 2142 y Fp(t)2472 2183 y Fq(;)456 2401 y Fs(b)n(y)27 b(\(5.3\))g(and)h(\(5.2\).)36 b(So,)647 2562 y(log)14 b Fq(\026)818 2574 y Fp(t)847 2562 y Fs(\()p Fq(B)t Fs(\()p Fq(x;)g(")p Fs(\)\))p 647 2600 520 4 v 827 2676 a(log)g Fq(")1260 2619 y Fn(\025)1417 2562 y Fs(log)g(\()p Fq(\026)1620 2574 y Fp(t)1650 2562 y Fs(\()p Fq(Z)1739 2574 y Fp(n)1780 2582 y Fi(0)1816 2562 y Fs(\))19 b(+)f Fq(\026)2000 2574 y Fp(t)2029 2562 y Fs(\()p Fq(Z)2118 2574 y Fp(n)2159 2582 y Fi(1)2196 2562 y Fs(\)\))p 1417 2600 844 4 v 1759 2676 a(log)c Fq(")1260 2859 y Fn(\025)82 b Fq(t)1475 2803 y Fs(log)14 b Fq(K)p 1475 2840 198 4 v 1494 2916 a Fs(log)g Fq(")1701 2859 y Fs(+)1794 2798 y(log)1915 2731 y Fk(\000)1953 2798 y Fq(\032)1996 2761 y Fj(\000)p Fp(n)2089 2769 y Fi(0)1996 2818 y Fp(t)2125 2798 y Fs(diam\()p Fq(Z)2394 2810 y Fp(n)2435 2818 y Fi(0)2472 2798 y Fs(\))2504 2768 y Fp(t)2552 2798 y Fs(+)k Fq(\032)2678 2761 y Fj(\000)p Fp(n)2771 2769 y Fi(1)2678 2818 y Fp(t)2807 2798 y Fs(diam\()p Fq(Z)3076 2810 y Fp(n)3117 2818 y Fi(1)3154 2798 y Fs(\))3186 2768 y Fp(t)3215 2731 y Fk(\001)p 1794 2840 1460 4 v 2444 2916 a Fs(log)c Fq(")1260 3095 y Fn(\025)82 b Fq(t)1475 3038 y Fs(log)14 b Fq(K)6 b(C)1732 3050 y Fl(1)p 1475 3076 295 4 v 1542 3152 a Fs(log)14 b Fq(")1797 3095 y Fs(+)1890 3038 y(log)q(\()p Fq(\032)2073 3002 y Fj(\000)p Fp(n)2166 3010 y Fi(0)2073 3059 y Fp(t)2221 3038 y Fs(+)k Fq(\032)2347 3002 y Fj(\000)p Fp(n)2440 3010 y Fi(1)2347 3059 y Fp(t)2476 3038 y Fs(\))p 1890 3076 619 4 v 2119 3152 a(log)d Fq(")2537 3095 y Fs(+)j Fq(t)456 3304 y Fs(Since,)26 b(for)g Fq(")g Fs(small)g(enough,)g Fq(n)1459 3316 y Fl(0)1522 3304 y Fs(and)g Fq(n)1732 3316 y Fl(1)1795 3304 y Fs(are)g(arbitrarily)e(large)h(and)h(for)f Fq(t)e(<)g(t)2989 3316 y Fl(0)3026 3304 y Fs(,)k Fq(\032)3119 3316 y Fp(t)3171 3304 y Fq(>)c Fs(1,)j(w)n(e)456 3431 y(can)i(assume)g Fq(\032)940 3394 y Fj(\000)p Fp(n)1033 3402 y Fi(0)940 3451 y Fp(t)1089 3431 y Fs(+)18 b Fq(\032)1215 3394 y Fj(\000)p Fp(n)1308 3402 y Fi(1)1215 3451 y Fp(t)1370 3431 y Fq(<)24 b Fs(1)29 b(so,)1666 3388 y Fl(log)q(\()p Fp(\032)1812 3356 y Fg(\000)p Fh(n)1894 3368 y Fi(0)1812 3405 y Fh(t)1930 3388 y Fl(+)p Fp(\032)2015 3356 y Fg(\000)p Fh(n)2097 3368 y Fi(1)2015 3405 y Fh(t)2134 3388 y Fl(\))p 1666 3412 494 4 v 1849 3459 a(log)12 b Fp(")2195 3431 y Fq(>)24 b Fs(0.)40 b(Therefore,)28 b(taking)g(the)h(lim)14 b(inf)7 b(,)456 3541 y Fq(d)499 3553 y Fp(\026)p 456 3578 88 4 v 543 3541 a Fs(\()p Fq(x)p Fs(\))24 b Fn(\025)f Fq(t)28 b Fs(for)f(all)g Fq(t)c(<)g(t)1237 3553 y Fl(0)1274 3541 y Fs(.)37 b(W)-7 b(e)28 b(conclude)f(that)h Fq(H)7 b(D)r Fs(\()p Fq(X)2246 3553 y Fj(1)2316 3541 y Fs(\))24 b Fn(\025)e Fq(t)2489 3553 y Fl(0)2527 3541 y Fs(.)830 b Ff(\003)1698 3817 y Fs(6.)41 b Ft(Examples)555 3966 y Fs(In)24 b(this)g(section)g(w)n(e)f(giv)n(e)g(v)n(eri\014able)g (criteria)g(to)g(insure)h(conditions)f FB(C1,)28 b(C2)23 b Fs(in)h(concrete)456 4066 y(situations)j(and)g(w)n(e)g(discuss)h (some)f(explicit)g(examples.)555 4165 y(Condition)21 b FB(C1)g Fs(is)g(rather)f(mild)i(and)f(in)g(most)g(cases)f(can)h(b)r (e)h(c)n(hec)n(k)n(ed)e(easily)g(\(for)h(example)456 4265 y(the)28 b(presence)e(of)i(a)f(full)h(branc)n(h)f(outside)g(the)h (hole)g(su\016ces\).)555 4365 y(Next)g(notice)g(that,)g(setting)1552 4583 y Fq(\032)1595 4595 y Fp(n)1663 4583 y Fs(:=)61 b(inf)1774 4636 y Fp(x)p Fj(2)p Fp(D)1911 4644 y Fh(n)1975 4527 y Fn(L)2032 4496 y Fp(n)p Fl(+1)2162 4527 y Fs(1\()p Fq(x)p Fs(\))p 1975 4564 341 4 v 2017 4640 a Fn(L)2074 4616 y Fp(n)2119 4640 y Fs(1\()p Fq(x)p Fs(\))2325 4583 y Fq(;)456 4801 y Fs(then)23 b Fq(\032)g Fn(\025)g Fq(\032)837 4813 y Fp(n)905 4801 y Fs(\(see)g(\(2.3\)\),)h(hence)f(one)g(can)f(v)n (erify)h(condition)f FB(C2)h Fs(b)n(y)g(using)g(some)f Fq(\032)3134 4813 y Fp(n)3202 4801 y Fs(\(whic)n(h)456 4901 y(is)30 b(explicitly)h(computable\))g(rather)f(than)h Fq(\032)p Fs(.)47 b(The)31 b(main)g(problem)f(is)h(then)h(to)e(con)n (trol)g(the)456 5016 y(n)n(um)n(b)r(er)j(of)g(con)n(tiguous)f(elemen)n (ts)h(in)h Fn(Z)1798 4973 y Fl(\()p Fp(n)p Fl(\))1791 5041 y Fp(b)1895 5016 y Fs(.)54 b(This)34 b(of)f(course)f(is)h(a)g (case)g(b)n(y)g(case)g(matter,)456 5116 y(y)n(et)h(it)i(is)f(p)r (ossible)g(to)g(mak)n(e)f(some)h(rather)f(general)f(statemen)n(ts.)59 b(Let)36 b(us)f(exemplify)g(the)456 5216 y(situation)27 b(b)n(y)g(lo)r(oking)g(at)g(few)h(relev)-5 b(an)n(t)27 b(examples.)p eop %%Page: 20 20 20 19 bop 456 251 a Fl(20)355 b(CARLANGELO)23 b(LIVERANI)f(AND)g(V)1947 236 y(\023)1941 251 y(ER)n(ONIQUE)g(MA)n(UME-DESCHAMPS)456 450 y FB(Mark)m(o)m(v)35 b(maps)e(with)h(non)g(Mark)m(o)m(v)i(hole.)k Fs(Let)30 b(us)g(giv)n(e)f(examples)g(of)g(Mark)n(o)n(v)f(maps)456 550 y(with)k(a)g(non)g(Mark)n(o)n(v)e(hole.)51 b(Recall)32 b(that)g Fq(T)44 b Fs(is)32 b(said)f(to)i(b)r(e)f Fr(Markov)i Fs(with)f(resp)r(ect)f(to)g(the)456 649 y(partition)i Fn(Z)41 b Fs(if)35 b(for)f(all)g Fq(Z)41 b Fn(2)35 b(Z)7 b Fs(,)36 b Fq(T)12 b(Z)39 b Fs(is)c(exactly)e(a)i(union)f(of)h(some)f (elemen)n(ts)g(of)h Fn(Z)7 b Fs(.)57 b(W)-7 b(e)456 749 y(call)32 b Fq(Y)52 b Fs(a)33 b Fr(Markov)k(hole)e Fs(if)e Fq(T)45 b Fs(is)33 b(Mark)n(o)n(v)e(and)i Fq(Y)51 b Fn(2)33 b(Z)2265 719 y Fl(\()p Fp(n)p Fl(\))2395 749 y Fs(for)g(some)g Fq(n)p Fs(;)2815 717 y Fl(8)2884 749 y Fs(up)h(to)f(replacing)456 849 y Fn(Z)41 b Fs(b)n(y)35 b Fn(Z)747 818 y Fl(\()p Fp(n)p Fl(\))844 849 y Fs(,)i(w)n(e)e(ma)n(y)f(alw)n(a)n(ys)f(assume)h (that)i(a)e(Mark)n(o)n(v)f(hole)i(is)g(an)f(elemen)n(t)h(of)g Fn(Z)7 b Fs(.)59 b(Let)456 952 y Fq(Y)50 b Fs(b)r(e)33 b(suc)n(h)e(a)h(Mark)n(o)n(v)e(hole,)j(let)1596 931 y(^)1572 952 y Fn(Z)39 b Fs(b)r(e)32 b(the)g(set)g(of)g(elemen)n(ts)g(of)g Fn(Z)39 b Fs(that)32 b(are)f(not)h Fq(Y)19 b Fs(.)50 b(W)-7 b(e)456 1051 y(call)34 b Fq(Y)54 b Fs(an)34 b Fr(ap)l(erio)l(dic)39 b(Markov)f(hole)f Fs(if)e(there)g(exists)f Fq(N)44 b Fn(2)36 b Fo(N)45 b Fs(suc)n(h)34 b(that)i(for)e(all)h Fq(n)g Fn(\025)g Fq(N)9 b Fs(,)456 1154 y(for)39 b(all)h Fq(Z)6 b Fs(,)43 b Fq(Z)915 1124 y Fj(0)982 1154 y Fn(2)h(Z)7 b Fs(,)43 b(there)d(are)f Fq(Z)1647 1166 y Fl(1)1684 1154 y Fs(,)44 b Fq(:)14 b(:)g(:)p Fs(,)43 b Fq(Z)1971 1166 y Fp(n)2100 1154 y Fn(2)2224 1133 y Fs(^)2199 1154 y Fn(Z)k Fs(suc)n(h)40 b(that)h(the)f(\()p Fq(n)27 b Fs(+)f(1\)-cylinder)456 1254 y Fq(Z)f Fn(\\)c Fq(T)675 1224 y Fj(\000)p Fl(1)763 1254 y Fq(Z)820 1266 y Fl(1)877 1254 y Fn(\\)g(\001)14 b(\001)g(\001)20 b(\\)h Fq(T)1207 1224 y Fj(\000)p Fp(n)1303 1254 y Fq(Z)1360 1266 y Fp(n)1425 1254 y Fn(\\)g Fq(T)1562 1224 y Fj(\000)p Fp(n)p Fj(\000)p Fl(1)1743 1254 y Fq(Z)1806 1224 y Fj(0)1859 1254 y Fs(is)30 b(non)g(empt)n(y)-7 b(.)45 b(F)-7 b(or)30 b(expanding)g(Mark)n(o)n(v)e (maps)456 1354 y(with)g(an)f(ap)r(erio)r(dic)g(Mark)n(o)n(v)e(hole,)j (Theorem)e(A)i(has)f(b)r(een)h(pro)n(v)n(ed)f(in)g([CMS2].)456 1453 y(W)-7 b(e)33 b(are)f(no)n(w)h(in)g(p)r(osition)g(to)g(giv)n(e)f (examples)g(of)h(Mark)n(o)n(v)e(maps)i(with)g(non)g(Mark)n(o)n(v)e (hole)456 1564 y(suc)n(h)c(that)h Fn(Z)890 1521 y Fl(\()p Fp(n)p Fl(\))883 1589 y Fp(b)1010 1564 y Fs(=)22 b Fn(;)p Fs(.)456 1698 y FB(Lemma)d(6.1.)32 b Fr(L)l(et)21 b Fq(T)32 b Fr(b)l(e)22 b(a)f(Markov)j(map)e(with)g(Lipschitz)h(derivative,)i (let)2841 1677 y Fs(~)2829 1698 y Fq(Y)40 b Fr(b)l(e)21 b(an)h(ap)l(erio)l(dic)456 1799 y(Markov)32 b(hole.)41 b(L)l(et)30 b Fq(Y)43 b Fn(\032)1302 1778 y Fs(~)1290 1799 y Fq(Y)49 b Fr(b)l(e)30 b(a)h(hole)g(such)g(that)f(ther)l(e)h (exists)f Fq(p)23 b Fn(2)i Fo(N)40 b Fr(and)31 b Fq(C)f Fn(2)25 b(Z)3165 1769 y Fl(\()p Fp(p)p Fl(\))3285 1799 y Fr(such)456 1900 y(that)32 b Fq(C)h Fn(\032)824 1879 y Fs(~)812 1900 y Fq(Y)38 b Fn(n)20 b Fq(Y)51 b Fr(and)32 b Fq(C)i Fn(\032)26 b Fq(X)1475 1912 y Fp(p)p Fj(\000)p Fl(1)1598 1900 y Fr(.)46 b(Then)33 b(for)f(the)h(map)f Fq(T)43 b Fr(with)33 b(hole)g Fq(Y)51 b Fr(one)32 b(c)l(an)g(cho)l(ose) 456 2019 y Fq(\030)e Fs(=)d(1)k Fr(in)h(c)l(ondition)h(2)f(\(inde)l(e)l (d,)h(for)g(al)t(l)g Fq(n)p Fr(,)f Fn(Z)1976 1976 y Fl(\()p Fp(n)p Fl(\))1969 2044 y Fp(b)2100 2019 y Fs(=)26 b Fn(;)p Fr(,)32 b(henc)l(e)g(one)g(c)l(an)g(cho)l(ose)h Fq(K)g Fs(=)26 b(0)31 b Fr(as)456 2119 y(wel)t(l\).)456 2287 y(Pr)l(o)l(of.)43 b Fs(First)28 b(of)g(all,)g(remark)e(that)j(since)f Fq(T)1883 2257 y Fj(0)1933 2287 y Fs(is)g(Lipsc)n(hitz,)h(there)e (exists)h(a)g(constan)n(t)f Fq(K)6 b Fs(\()p Fq(T)12 b Fs(\))456 2390 y(suc)n(h)29 b(that)h(for)g(all)f Fq(Z)k Fn(2)27 b(Z)1313 2360 y Fl(\()p Fp(n)p Fl(\))1410 2390 y Fs(,)1463 2328 y Fk(W)1533 2415 y Fp(Z)1600 2390 y Fq(g)1643 2360 y Fl(0)1640 2411 y Fp(n)1712 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1559 980 V 1561 976 V 1563 973 V 1566 970 V 1568 967 V 1570 964 V 1572 961 V 1574 957 V 1576 954 V 1578 951 V 1581 948 V 1583 945 V 1585 942 V 1587 938 V 1589 935 V 1591 932 V 1594 929 V 1596 926 V 1598 923 V 1600 919 V 1602 916 V 1605 913 V 1607 910 V 1609 907 V 1611 903 V 1613 900 V 1616 897 V 1618 894 V 1620 891 V 1622 887 V 1625 884 V 1627 881 V 1629 878 V 1631 875 V 1634 871 V 1636 868 V 1638 865 V 1640 862 V 1643 859 V 1645 855 V 1647 852 V 1650 849 V 1652 846 V 1654 842 V 1656 839 V 1659 836 V 1661 833 V 1663 830 V 1666 826 V 1668 823 V 1670 820 V 1673 817 V 1675 813 V 1677 810 V 1680 807 V 1682 804 V 1684 800 V 1687 797 V 1689 794 V 1692 791 V 1694 788 V 1696 784 V 1699 781 V 1701 778 V 1704 775 V 1706 771 V 1708 768 V 1711 765 V 1713 761 V 1716 758 V 1718 755 V 1720 752 V 1723 748 V 1725 745 V 1728 742 V 1730 739 V 1733 735 V 1735 732 V 1738 729 V 1740 726 V 1742 722 V 1745 719 V 1747 716 V 1750 712 V 1752 709 V 1755 706 V 1757 703 V 1760 699 V 1762 696 V 1765 693 V 1767 689 V 1770 686 V 1772 683 V 1775 680 V 1778 676 V 1780 673 V 1783 670 V 1785 666 V 1788 663 V 1790 660 V 1793 656 V 1795 653 V 1798 650 V 1801 646 V 1803 643 V 1806 640 V 1808 637 V 1811 633 V 1814 630 V 1816 627 V 1819 623 V 1821 620 V 1824 617 V 1827 613 V 1829 610 V 1832 607 V 1832 1551 V 1833 1548 V 1835 1545 V 1836 1541 V 1838 1538 V 1839 1535 V 1841 1532 V 1842 1528 V 1843 1525 V 1845 1522 V 1846 1518 V 1848 1515 V 1849 1512 V 1850 1509 V 1852 1505 V 1853 1502 V 1855 1499 V 1856 1495 V 1858 1492 V 1859 1489 V 1860 1486 V 1862 1482 V 1863 1479 V 1865 1476 V 1866 1473 V 1867 1469 V 1869 1466 V 1870 1463 V 1871 1459 V 1873 1456 V 1874 1453 V 1876 1450 V 1877 1447 V 1878 1443 V 1880 1440 V 1881 1437 V 1882 1434 V 1884 1430 V 1885 1427 V 1887 1424 V 1888 1421 V 1889 1417 V 1891 1414 V 1892 1411 V 1893 1408 V 1895 1405 V 1896 1401 V 1897 1398 V 1899 1395 V 1900 1392 V 1901 1389 V 1903 1385 V 1904 1382 V 1905 1379 V 1907 1376 V 1908 1373 V 1909 1370 V 1911 1366 V 1912 1363 V 1913 1360 V 1915 1357 V 1916 1354 V 1917 1351 V 1919 1347 V 1920 1344 V 1921 1341 V 1922 1338 V 1924 1335 V 1925 1332 V 1926 1329 V 1928 1325 V 1929 1322 V 1930 1319 V 1931 1316 V 1933 1313 V 1934 1310 V 1935 1307 V 1937 1303 V 1938 1300 V 1939 1297 V 1940 1294 V 1942 1291 V 1943 1288 V 1944 1285 V 1945 1282 V 1947 1279 V 1948 1276 V 1949 1272 V 1950 1269 V 1952 1266 V 1953 1263 V 1954 1260 V 1955 1257 V 1957 1254 V 1958 1251 V 1959 1248 V 1960 1245 V 1962 1242 V 1963 1239 V 1964 1236 V 1965 1232 V 1967 1229 V 1968 1226 V 1969 1223 V 1970 1220 V 1971 1217 V 1973 1214 V 1974 1211 V 1975 1208 V 1976 1205 V 1977 1202 V 1979 1199 V 1980 1196 V 1981 1193 V 1982 1190 V 1983 1187 V 1985 1184 V 1986 1181 V 1987 1178 V 1988 1175 V 1989 1172 V 1991 1169 V 1992 1166 V 1993 1163 V 1994 1160 V 1995 1157 V 1996 1154 V 1998 1151 V 1999 1148 V 2000 1145 V 2001 1142 V 2002 1139 V 2003 1136 V 2005 1133 V 2006 1130 V 2007 1127 V 2008 1124 V 2009 1121 V 2010 1118 V 2011 1115 V 2013 1112 V 2014 1109 V 2015 1106 V 2016 1103 V 2017 1100 V 2018 1098 V 2019 1095 V 2020 1092 V 2022 1089 V 2023 1086 V 2024 1083 V 2025 1080 V 2026 1077 V 2027 1074 V 2028 1071 V 2029 1068 V 2030 1065 V 2031 1062 V 2033 1060 V 2034 1057 V 2035 1054 V 2036 1051 V 2037 1048 V 2038 1045 V 2039 1042 V 2040 1039 V 2041 1036 V 2042 1034 V 2043 1031 V 2045 1028 V 2046 1025 V 2047 1022 V 2048 1019 V 2049 1016 V 2050 1013 V 2051 1011 V 2052 1008 V 2053 1005 V 2054 1002 V 2055 999 V 2056 996 V 2057 993 V 2058 991 V 2059 988 V 2060 985 V 2061 982 V 2062 979 V 2063 976 V 2064 974 V 2065 971 V 2067 968 V 2068 965 V 2069 962 V 2070 960 V 2071 957 V 2072 954 V 2073 951 V 2074 948 V 2075 946 V 2076 943 V 2077 940 V 2078 937 V 2079 934 V 2080 932 V 2081 929 V 2082 926 V 2083 923 V 2084 920 V 2085 918 V 2086 915 V 2087 912 V 2088 909 V 2089 907 V 2089 904 V 2090 901 V 2091 898 V 2092 896 V 2093 893 V 2094 890 V 2095 887 V 2096 885 V 2097 882 V 2098 879 V 2099 876 V 2100 874 V 2101 871 V 2102 868 V 2103 865 V 2104 863 V 2105 860 V 2106 857 V 2107 855 V 2108 852 V 2109 849 V 2109 847 V 2110 844 V 2111 841 V 2112 838 V 2113 836 V 2114 833 V 2115 830 V 2116 828 V 2117 825 V 2118 822 V 2119 820 V 2120 817 V 2120 814 V 2121 812 V 2122 809 V 2123 806 V 2124 804 V 2125 801 V 2126 798 V 2127 796 V 2128 793 V 2128 790 V 2129 788 V 2130 785 V 2131 782 V 2132 780 V 2133 777 V 2134 774 V 2135 772 V 2135 769 V 2136 767 V 2137 764 V 2138 761 V 2139 759 V 2140 756 V 2141 754 V 2141 751 V 2142 748 V 2143 746 V 2144 743 V 2145 741 V 2146 738 V 2146 735 V 2147 733 V 2148 730 V 2149 728 V 2150 725 V 2151 722 V 2151 720 V 2152 717 V 2153 715 V 2154 712 V 2155 710 V 2156 707 V 2156 704 V 2157 702 V 2158 699 V 2159 697 V 2160 694 V 2160 692 V 2161 689 V 2162 687 V 2163 684 V 2164 682 V 2164 679 V 2165 676 V 2166 674 V 2167 671 V 2167 669 V 2168 666 V 2169 664 V 2170 661 V 2171 659 V 2171 656 V 2172 654 V 2173 651 V 2174 649 V 2174 646 V 2175 644 V 2176 641 V 2177 639 V 2177 636 V 2178 634 V 2179 631 V 2180 629 V 2180 626 V 2181 624 V 2182 621 V 2183 619 V 2183 616 V 2184 614 V 2185 612 V 2185 609 V 2186 607 V 1808 959 24 4 v 1761 959 V 1714 959 V 1667 959 V 1619 959 V 1572 959 V 1570 1548 4 24 v 1570 1501 V 1570 1454 V 1570 1406 V 1570 1359 V 1570 1312 V 1570 1265 V 1570 1217 V 1570 1170 V 1570 1123 V 1570 1076 V 1570 1028 V 1570 981 V 1548 1219 24 4 v 1501 1219 V 1454 1219 V 1407 1219 V 1410 1548 4 24 v 1410 1501 V 1410 1454 V 1410 1406 V 1410 1359 V 1410 1312 V 1410 1265 V 1548 1619 a Fq(\015)-232 b(\015)1402 1631 y Fl(1)1241 1701 y Fk(|)p 1278 1701 103 10 v 103 w({z)p 1455 1701 V 103 w(})1396 1811 y Fs(~)1383 1832 y Fq(Y)1123 1076 y(I)1320 1992 y Ft(Figure)32 b(1.)41 b Fs(Ap)r(erio)r(dic)27 b(Mark)n(o)n(v)f(hole)456 2331 y(and)f Fq(\030)i Fs(=)c(1)h(\(the)i(elemen)n(ts)g(of)f Fn(Z)1502 2288 y Fl(\()p Fp(n)p Fl(\))1495 2356 y Fp(b)1624 2331 y Fs(are)f(those)h(made)h(up)f(with)h(the)g(in)n(terv)-5 b(al)24 b([0)p Fq(;)14 b(")p Fs(])25 b(and)g(they)456 2431 y(are)h(nev)n(er)h(con)n(tiguous\).)555 2531 y(In)h(this)g(last)f (example)g(w)n(e)h(ha)n(v)n(e)e(seen)i(that)f(some)h(sp)r(ecial)f (cases)f(can)i(b)r(e)g(easily)e(handled)456 2642 y(ev)n(en)h(if)h Fq(Z)784 2599 y Fl(\()p Fp(n)p Fl(\))778 2667 y Fp(b)904 2642 y Fn(6)p Fs(=)22 b Fn(;)p Fs(.)37 b(The)27 b(next)h(examples)f(go) g(further)g(in)h(this)g(direction.)456 2804 y FB(Non)j(Mark)m(o)m(v)i (maps.)456 2904 y Fs(Let)39 b Fq(I)50 b Fs(=)43 b([0)p Fq(;)14 b Fs(1],)41 b(for)e Fq(\014)47 b(>)c Fs(1)c(and)g(consider)g (the)h Fq(\014)t Fs(-map)f Fq(T)12 b Fs(\()p Fq(x)p Fs(\))43 b(=)g Fq(\014)t(x)p Fs(\(mo)r(d)28 b(1\))40 b(and)f(the)456 3003 y(p)r(oten)n(tial)31 b Fq(g)853 2973 y Fl(0)920 3003 y Fs(:=)f Fq(D)r(T)1170 2973 y Fj(\000)p Fl(1)1288 3003 y Fs(=)g Fq(\014)1434 2973 y Fj(\000)p Fl(1)1523 3003 y Fs(.)50 b(If)32 b Fq(\014)i Fn(62)c Fo(N)t Fs(,)39 b(then)33 b(the)f(map)f(it)i(is)e(not)h(Mark)n(o)n(v.)47 b(W)-7 b(e)32 b(will)456 3103 y(consider)h(only)h(suc)n(h)g(cases)g (and)g(w)n(e)g(will)h(designate)e(b)n(y)i([)p Fq(\014)t Fs(])g(the)f(in)n(teger)g(part)g(of)g Fq(\014)t Fs(.)58 b(Let)456 3216 y Fq(\015)39 b Fs(=)647 3175 y Fl([)p Fp(\014)s Fl(])p 647 3197 79 4 v 666 3244 a Fp(\014)770 3216 y Fs(and)34 b Fq(Y)53 b Fs(=)34 b([)p Fq(\015)1204 3228 y Fl(1)1242 3216 y Fq(;)14 b Fs(1])34 b(with)h Fq(\015)k(<)34 b(\015)1798 3228 y Fl(1)1870 3216 y Fq(<)g Fs(1.)57 b(Denote)35 b(the)g(elemen)n(t)f(of)h Fn(Z)41 b Fs(b)n(y)34 b Fq(Z)3228 3228 y Fl(1)3265 3216 y Fs(,)i Fq(:)14 b(:)g(:)p Fs(,)456 3343 y Fq(Z)513 3358 y Fl([)p Fp(\014)s Fl(+1])679 3343 y Fs(,)33 b(it)g(is)f(clear)f(that)h(for)g Fq(p)g Fs(large)f(enough,)i Fq(C)k Fs(:=)30 b Fq(Z)2298 3358 y Fl([)p Fp(\014)s Fl(])2401 3343 y Fn(\\)2478 3281 y Fk(T)2548 3301 y Fp(p)p Fj(\000)p Fl(1)2548 3368 y Fp(i)p Fl(=1)2685 3343 y Fq(T)2746 3313 y Fj(\000)p Fp(i)2824 3343 y Fq(Z)2881 3355 y Fl(1)2950 3343 y Fs(is)j(included)f(in)456 3468 y Fq(X)525 3480 y Fp(p)p Fj(\000)p Fl(1)648 3468 y Fs(,)g(this)g(leads)f(to)g(the)g 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Fl(\()p Fp(n)p Fl(\))3341 5067 y Fp(b)p 456 5117 499 4 v 555 5190 a Fl(9)588 5216 y FA(Note)25 b(that)g(this)e(is)g(a)h(general)g(b)r(ound,)g(b)r(etter)h(b)r(ounds)g (ma)n(y)e(b)r(e)h(a)n(v)l(ailable)g(for)f(sp)r(eci\014c)h(v)l(alues)g (of)g Fy(\014)p eop %%Page: 22 22 22 21 bop 456 251 a Fl(22)355 b(CARLANGELO)23 b(LIVERANI)f(AND)g(V)1947 236 y(\023)1941 251 y(ER)n(ONIQUE)g(MA)n(UME-DESCHAMPS)456 450 y Fs(is)29 b(still)h(2)747 420 y Fp(n)791 450 y Fs(,)h(th)n(us)e FB(C2)g Fs(is)h(satis\014ed.)42 b(Note)29 b(that)h(in)g(this)g(case)e (w)n(e)h(can)h(ha)n(v)n(e)e(holes)h(with)h(size)456 550 y(almost)731 517 y Fl(1)p 731 531 34 4 v 731 578 a(3)800 550 y Fs(whic)n(h)25 b(is)h(rather)e(large.)35 b(In)25 b(fact,)h(ev)n(en)f(more)g(dramatic)g(examples)f(can)h(b)r(e)h(easily) 456 649 y(pro)r(duced.)p 1241 1078 24 4 v 1289 1078 V 1336 1078 V 1383 1078 V 1430 1078 V 1478 1078 V 1525 1078 V 1572 1078 V 1619 1078 V 1667 1078 V 1714 1078 V 1761 1078 V 1808 1078 V 1856 1078 V 1903 1078 V 1950 1078 V 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V 1780 1265 V 1780 1218 V 1780 1171 V 1780 1124 V 2050 2021 V 2050 1974 V 2050 1927 V 2050 1879 V 2050 1832 V 2050 1785 V 2050 1738 V 2050 1690 V 2050 1643 V 2050 1596 V 2050 1549 V 2050 1501 V 2050 1454 V 2050 1407 V 2050 1360 V 2050 1312 V 2050 1265 V 2050 1218 V 2050 1171 V 2050 1124 V 1007 2294 a Ft(Figure)32 b(2.)41 b Fs(Non)27 b(Mark)n(o)n(v)f Fq(\014)t Fs(-map)h(with)h(a)f(hole)h(\()p Fq(\014)f Fs(=)2818 2261 y Fl(7)p 2818 2275 34 4 v 2818 2322 a(2)2861 2294 y Fs(\))456 2513 y(W)-7 b(e)27 b(ha)n(v)n(e)e(seen)i(that)g(it)g(is)g (p)r(ossible)f(to)h(insure)f(condition)h FB(C2)f Fs(b)n(y)h(using)f (the)h(com)n(binatorial)456 2613 y(prop)r(erties)g(of)g(a)h(Mark)n(o)n (v)d(map)j(or)f(the)h(sp)r(ecial)g(b)r(eha)n(vior)e(of)i Fq(\014)t Fs(-maps.)37 b(Some)28 b(of)g(the)g(ab)r(o)n(v)n(e)456 2712 y(discussion)22 b(can)h(b)r(e)g(generalized)f(b)n(y)h(requiring)f (the)h(existence)g(of)g(w)n(ell)g(b)r(eha)n(v)n(ed)f(elemen)n(ts)h(in) 456 2823 y(the)i(partition:)36 b(let)26 b Fn(Z)1159 2780 y Fl(\()p Fp(n)p Fl(\))1152 2848 y Fp(f)1281 2823 y Fs(b)r(e)g(the)g (collection)f(of)g(elemen)n(ts)h(in)f Fn(Z)2491 2780 y Fl(\()p Fp(n)p Fl(\))2484 2835 y Fj(\003)2614 2823 y Fs(suc)n(h)g(that)h Fq(T)3038 2793 y Fp(n)3082 2823 y Fq(Z)j Fs(=)22 b([0)p Fq(;)14 b Fs(1].)456 2955 y(Call)27 b Fn(Z)698 2912 y Fl(\()p Fp(n)p Fl(\))691 2965 y Fp(u)822 2955 y Fs(the)h(collection)f(of)h(the)g(others.)456 3072 y FB(De\014nition)f(6.2.)37 b Fr(F)-6 b(or)26 b Fq(\030)i(>)22 b Fs(0)p Fr(,)28 b(we)f(c)l(al)t(l)g(a)g(map)h Fq(\030)g Fs(full)d(branc)n(hed)h Fr(\()p Fq(\030)t Fr(-f.b.)38 b(for)28 b(short\))f(if)g(ther)l(e)456 3188 y(exists)j Fq(K)g(>)25 b Fs(0)30 b Fr(such)h(that)g(the)f(numb)l(er)g(of)i(c)l (ontiguous)f(elements)f(in)h Fn(Z)2765 3145 y Fl(\()p Fp(n)p Fl(\))2758 3197 y Fp(u)2893 3188 y Fr(do)l(es)g(not)f(exc)l(e)l (e)l(d)456 3287 y Fq(K)6 b(\030)573 3257 y Fp(n)617 3287 y Fr(.)555 3416 y Fs(Ob)n(viously)19 b(a)h Fq(\030)t Fs(-f.b.)35 b(map)20 b(satis\014es)g(condition)g FB(C2)p Fs(,)h(pro)n(vided)e(\002)p Fq(\030)27 b(<)c(\032)p Fs(,)f(since)e(if)g Fq(Z)29 b Fn(2)24 b(Z)3348 3373 y Fl(\()p Fp(n)p Fl(\))3341 3441 y Fp(f)456 3523 y Fs(then)k(\003\()p FB(1)783 3535 y Fp(Z)836 3523 y Fs(\))c Fq(>)f Fs(0.)38 b(The)28 b(p)r(oin)n(t)g(is)g (that)g(it)h(ma)n(y)e(b)r(e)h(easy)f(to)h(v)n(erify)f(that)h(a)g(map)g (is)g Fq(\030)t Fs(-f.b.)38 b(as)456 3622 y(the)28 b(next)f(lemma)h (sho)n(ws.)456 3739 y FB(Lemma)20 b(6.3.)34 b Fr(Cal)t(ling)24 b Fq(C)1309 3751 y Fp(n)1377 3739 y Fr(the)f(maximal)h(numb)l(er)e(of)i (c)l(ontiguous)e(elements)h(in)g Fn(Z)3116 3696 y Fl(\()p Fp(n)p Fl(\))3109 3749 y Fp(u)3212 3739 y Fr(,)i(holds)1526 3948 y Fq(C)1585 3960 y Fp(n)1654 3948 y Fn(\024)d Fs(2)1797 3845 y Fp(n)p Fj(\000)p Fl(1)1800 3869 y Fk(X)1806 4046 y Fp(i)p Fl(=0)1923 3948 y Fs(\()p Fq(C)2014 3960 y Fl(1)2070 3948 y Fs(+)c(2\))2227 3914 y Fp(i)2254 3948 y Fq(C)2313 3960 y Fl(1)2351 3948 y Fq(:)456 4170 y Fr(Pr)l(o)l(of.)43 b Fs(The)27 b(pro)r(of)f(is)g(b)n(y)h(induction)g(on)g Fq(n)p Fs(.)36 b(Clearly)26 b(it)h(is)g(true)f(for)g Fq(n)d Fs(=)g(1.)36 b(Let)27 b(us)g(supp)r(ose)456 4281 y(it)34 b(true)h(for)e Fq(n)p Fs(.)57 b(The)35 b(elemen)n(ts)f(of)g (the)h(partition)f Fn(Z)2186 4238 y Fl(\()p Fp(n)p Fl(+1\))2179 4293 y Fj(\003)2401 4281 y Fs(are)f(formed)h(b)n(y)g Fn(f)p Fq(T)3057 4251 y Fj(\000)p Fl(1)3145 4281 y Fq(Z)29 b Fn(\\)23 b Fq(Z)3366 4293 y Fl(1)3403 4281 y Fn(g)456 4397 y Fs(where)36 b Fq(Z)44 b Fn(2)39 b(Z)967 4354 y Fl(\()p Fp(n)p Fl(\))960 4409 y Fj(\003)1101 4397 y Fs(and)e Fq(Z)1329 4409 y Fl(1)1404 4397 y Fn(2)i(Z)1565 4354 y Fl(\(1\))1558 4409 y Fj(\003)1654 4397 y Fs(.)65 b(No)n(w)36 b(if)i Fq(Z)2083 4409 y Fl(1)2158 4397 y Fn(2)h(Z)2319 4354 y Fl(\(1\))2312 4422 y Fp(f)2408 4397 y Fs(,)g(the)f(elemen)n(ts)f (main)n(tain)f(the)456 4529 y(same)f(nature)g(\(i.e.)62 b(if)36 b Fq(Z)43 b Fn(2)37 b(Z)1486 4485 y Fl(\()p Fp(n)p Fl(\))1479 4538 y Fp(u)1619 4529 y Fs(then)f Fq(T)1877 4498 y Fj(\000)p Fl(1)1965 4529 y Fq(Z)30 b Fn(\\)24 b Fq(Z)2188 4541 y Fl(1)2262 4529 y Fn(2)37 b(Z)2421 4485 y Fl(\()p Fp(n)p Fl(+1\))2414 4538 y Fp(u)2638 4529 y Fs(and)e(if)i Fq(Z)42 b Fn(2)37 b(Z)3150 4485 y Fl(\()p Fp(n)p Fl(\))3143 4554 y Fp(f)3283 4529 y Fs(then)456 4660 y Fq(T)517 4630 y Fj(\000)p Fl(1)605 4660 y Fq(Z)c Fn(\\)28 b Fq(Z)835 4672 y Fl(1)918 4660 y Fn(2)47 b(Z)1087 4617 y Fl(\()p Fp(n)p Fl(+1\))1080 4685 y Fp(f)1268 4660 y Fs(\).)79 b(So)41 b(w)n(e)g(ha)n(v)n(e)f(in)i Fq(Z)2040 4672 y Fl(1)2118 4660 y Fs(at)g(most)f Fq(C)2510 4672 y Fp(n)2597 4660 y Fs(con)n(tiguous)f(elemen)n(ts)i(of)456 4791 y Fn(Z)523 4748 y Fl(\()p Fp(n)p Fl(+1\))516 4801 y Fp(u)704 4791 y Fs(.)67 b(The)38 b(only)g(problem)f(can)h(arise)e (when)i(a)g(blo)r(c)n(k)f(of)h(con)n(tiguous)f(elemen)n(ts)h(ends)456 4891 y(at)i(the)h(b)r(oundary)f(of)g Fq(Z)1277 4903 y Fl(1)1355 4891 y Fs(since)g(in)h(suc)n(h)f(a)h(case)e(it)i(can)f(still) h(b)r(e)g(con)n(tiguous)f(to)g(other)456 5002 y(elemen)n(ts)31 b(of)h Fn(Z)965 4959 y Fl(\()p Fp(n)p Fl(+1\))958 5012 y Fp(u)1146 5002 y Fs(.)49 b(Y)-7 b(et,)33 b(if)g(the)f(con)n(tiguous)e (elemen)n(ts)i(of)f Fq(Z)2542 5014 y Fl(1)2611 5002 y Fs(are)g(in)h Fn(Z)2922 4959 y Fl(\(1\))2915 5027 y Fp(f)3011 5002 y Fs(,)g(then)h(there)456 5104 y(can)27 b(b)r(e)i(at)f(most)f(a)h (blo)r(c)n(k)f(of)h(length)h(2)p Fq(C)1767 5116 y Fp(n)1812 5104 y Fs(.)38 b(One)27 b(m)n(ust)i(then)f(analyze)f(what)h(can)g(happ) r(en)g(if)456 5216 y Fq(Z)513 5228 y Fl(1)581 5216 y Fn(2)j(Z)734 5172 y Fl(\(1\))727 5225 y Fp(u)823 5216 y Fs(.)52 b(In)32 b(this)h(case)e(a)h(set)h(of)f(con)n(tiguous)f (elemen)n(ts)i(can)f(either)g(ha)n(v)n(e)f(only)h(partial)p eop %%Page: 23 23 23 22 bop 1339 251 a Fl(LASOT)-5 b(A-YORKE)29 b(MAPS)f(WITH)h(HOLES)817 b(23)456 450 y Fs(preimage)33 b(in)i Fq(Z)979 462 y Fl(1)1016 450 y Fs(,)i(hence)d(w)n(e)h(get)f(a)g(shorter)g(groups)f(of)i(con)n (tiguous)e(elemen)n(ts)i(or)f(all)g(the)456 550 y(group)h(can)h(ha)n(v) n(e)g(preimage.)63 b(In)37 b(this)g(last)f(case)g(the)h(w)n(orst)f (case)g(scenario)f(is)h(when)h(the)456 661 y(elemen)n(ts)26 b(con)n(tiguous)g(to)g(the)h(groups)f(\(that)h(m)n(ust)g(b)r(elong)f (to)g Fn(Z)2564 618 y Fl(\()p Fp(n)p Fl(\))2557 686 y Fp(f)2661 661 y Fs(\))h(are)f(cut)h(while)g(taking)456 763 y(preimages.)33 b(This)20 b(means)g(that)h(at)g(most)f(t)n(w)n(o)g (new)h(con)n(tiguous)e(elemen)n(ts)h(can)h(b)r(e)g(generated,)456 863 y(but)k(in)f(this)h(case)f(the)h(group)e(m)n(ust)i(end)f(at)h(the)g (b)r(oundary)e(of)i Fq(Z)2516 875 y Fl(1)2553 863 y Fs(.)36 b(Since)24 b(there)h(are)e(at)i(most)456 974 y Fq(C)515 986 y Fl(1)582 974 y Fs(con)n(tiguous)k(elemen)n(ts)h(in)h Fn(Z)1507 931 y Fl(\(1\))1500 984 y Fp(u)1626 974 y Fs(in)f(this)h(w)n (a)n(y)e(w)n(e)g(can)h(generate,)g(at)g(most,)g Fq(C)3091 986 y Fl(1)3129 974 y Fs(\()p Fq(C)3220 986 y Fp(n)3286 974 y Fs(+)20 b(2\))456 1074 y(con)n(tiguous)32 b(elemen)n(ts)i(that,)h (again)e(in)h(the)g(w)n(orst)f(case)g(scenario,)h(can)f(b)r(e)h(con)n (tiguous)f(to)456 1185 y(t)n(w)n(o)26 b(blo)r(c)n(ks)h(b)r(elonging)g (to)h(the)g(neigh)n(b)r(oring)e(elemen)n(ts)h(in)h Fn(Z)2440 1142 y Fl(\(1\))2433 1210 y Fp(f)2529 1185 y Fs(.)37 b(Accordingly)724 1411 y Fq(C)783 1423 y Fp(n)p Fl(+1)936 1411 y Fn(\024)22 b Fq(C)1082 1423 y Fl(1)1120 1411 y Fs(\()p Fq(C)1211 1423 y Fp(n)1275 1411 y Fs(+)c(2\))h(+)f(2)p Fq(C)1635 1423 y Fp(n)1703 1411 y Fs(=)k(\()p Fq(C)1881 1423 y Fl(1)1938 1411 y Fs(+)c(2\))p Fq(C)2154 1423 y Fp(n)2217 1411 y Fs(+)g(2)p Fq(C)2401 1423 y Fl(1)2462 1411 y Fn(\024)k Fs(2)2644 1308 y Fp(n)2605 1333 y Fk(X)2611 1509 y Fp(i)p Fl(=0)2724 1411 y Fs(\()p Fq(C)2815 1423 y Fl(1)2872 1411 y Fs(+)c(2\))3029 1377 y Fp(i)3056 1411 y Fq(C)3115 1423 y Fl(1)3153 1411 y Fq(;)456 1629 y Fs(where)27 b(w)n(e)g(ha)n(v)n(e)f(used)i(the)g(induction)g(h)n(yp)r(othesis.)1264 b Ff(\003)555 1792 y Fs(The)25 b(Lemma)g(sa)n(ys)e(that)i(if)g Fq(\032)p Fs(\002)1548 1762 y Fj(\000)p Fl(1)1660 1792 y Fq(>)e(C)1807 1804 y Fl(1)1857 1792 y Fs(+)13 b(2,)24 b(then)i(the)f(h)n(yp)r(othesis)f FB(C2)h Fs(is)f(v)n(eri\014ed.)35 b(The)456 1892 y(in)n(terest)d(of)g(this)h(condition)f(is)h(that)f(it)h (applies)f(to)h(general)e(non)h(Mark)n(o)n(v)e(maps)j(pro)n(vided)456 1992 y(the)c(p)r(oten)n(tial)g(is)f(su\016cien)n(tly)h(con)n(tracting)e (and)i(there)g(are)e(enough)i(full)g(branc)n(hes)f(outside)456 2091 y(the)g(hole.)770 2059 y Fl(10)1633 2289 y Fs(7.)41 b Ft(Small)31 b(Holes)555 2439 y Fs(In)c(this)g(section)g(w)n(e)g(will) g(see)f(that,)i(if)f(one)f(is)h(in)n(terested)g(only)f(in)h(v)n(ery)f (small)h(holes)f(then)456 2538 y(results)32 b(stronger)e(than)j(the)g (one)f(in)g(the)h(previous)e(sections)h(can)g(b)r(e)h(readily)f (obtained)g(b)n(y)456 2638 y(regarding)18 b(the)i(system)g(with)h (holes)e(as)h(a)f(small)h(p)r(erturbation)g(of)g(the)h(system)e (without)i(holes.)555 2738 y(The)29 b(basic)f(idea)g(is)h(to)f (consider)g(the)h(transfer)e(op)r(erator)g Fn(L)i Fs(as)f(a)g(small)g (p)r(erturbation)h(of)456 2837 y(the)c(op)r(erator)d Fn(L)984 2849 y Fl(0)1022 2837 y Fs(.)36 b(Of)25 b(course,)f(the)h (norm)f(of)g(the)h(di\013erence)g(of)f(the)h(ab)r(o)n(v)n(e)e(op)r (erators)g(equal)456 2937 y(2)g(b)r(oth)i(in)f(the)h Fq(L)1004 2907 y Fl(1)1064 2937 y Fs(and)f(BV)h(norm,)f(hence)g (standard)f(p)r(erturbation)h(theory)f(do)r(es)h(not)g(apply)456 3036 y(directly)34 b(\(but)i(see)f([C])g(for)g(an)f(indirect)h (application\),)i(y)n(et)e(they)g(are)f(close)g(as)g(op)r(erators)456 3136 y(from)27 b(BV)h(to)f Fq(L)959 3106 y Fl(1)996 3136 y Fs(.)456 3258 y FB(De\014nition)j(7.1.)40 b Fr(F)-6 b(or)30 b(e)l(ach)h(op)l(er)l(ator)g Fn(L)23 b Fs(:)g Fq(B)t(V)c Fs(\()p Fq(I)7 b(;)14 b(m)p Fs(\))24 b Fn(!)f Fq(L)2424 3228 y Fl(1)2461 3258 y Fs(\()p Fq(I)7 b(;)14 b(m)p Fs(\))30 b Fr(let)1528 3403 y Fn(jjjLjjj)24 b Fs(:=)104 b(sup)1858 3476 y Fj(k)p Fp(f)7 b Fj(k)1965 3484 y Fh(B)r(V)2060 3476 y Fj(\024)p Fl(1)2158 3403 y Fn(jL)p Fq(f)i Fn(j)2311 3415 y Fl(1)2349 3403 y Fq(:)555 3610 y Fs(Then)31 b(the)h(exact)e (statemen)n(t)h(of)g(the)g(closeness)f(of)h(the)g(t)n(w)n(o)f(op)r (erators)f(is)i(giv)n(en)f(b)n(y)h(the)456 3709 y(follo)n(wing)26 b(lemma.)37 b(Let)28 b Fn(L)1312 3721 y Fp(Y)1397 3709 y Fs(b)r(e)g(the)g(transfer)f(op)r(erator)f(asso)r(ciated)g(to)h(the)h (hole)g Fq(Y)18 b Fs(.)456 3832 y FB(Lemma)35 b(7.2.)42 b Fr(If)35 b Fn(L)1141 3844 y Fl(0)1212 3832 y Fr(and)g Fn(L)1435 3844 y Fp(Y)1527 3832 y Fr(ar)l(e)f(the)g(two)h(op)l(er)l (ators)g(de\014ne)l(d)f(in)g(\(1.1\))h(and)g(\(1.2\),)i(r)l(e-)456 3931 y(sp)l(e)l(ctively,)31 b(then)1450 4052 y Fn(jjjL)1576 4064 y Fl(0)1632 4052 y Fn(\000)18 b(L)1772 4064 y Fp(Y)1830 4052 y Fn(jjj)23 b(\024)g Fq(e)2049 4017 y Fp(P)9 b Fl(\()p Fp(g)2160 3992 y Fi(0)2193 4017 y Fl(\))2223 4052 y Fq(m)p Fs(\()p Fq(Y)19 b Fs(\))p Fq(:)456 4216 y Fr(Pr)l(o)l(of.)43 b Fs(F)-7 b(or)27 b(eac)n(h)g Fq(f)k Fn(2)24 b Fq(B)t(V)46 b Fs(holds)925 4371 y Fn(jL)1005 4383 y Fl(0)1043 4371 y Fs(\()p Fq(f)9 b Fs(\))18 b Fn(\000)g(L)1315 4383 y Fp(Y)1373 4371 y Fs(\()p Fq(f)9 b Fs(\))p Fn(j)1510 4383 y Fl(1)1570 4371 y Fs(=)p Fn(jL)1715 4383 y Fl(0)1753 4371 y Fs(\()p FB(1)1833 4383 y Fp(Y)1890 4371 y Fq(f)g Fs(\))p Fn(j)1995 4383 y Fl(1)2055 4371 y Fn(\024)23 b Fq(e)2182 4337 y Fp(P)9 b Fl(\()p Fp(g)2293 4312 y Fi(0)2326 4337 y Fl(\))2356 4371 y Fn(j)p FB(1)2427 4383 y Fp(Y)2484 4371 y Fq(f)g Fn(j)2557 4383 y Fl(1)1570 4524 y Fn(\024)p Fq(e)1674 4490 y Fp(P)g Fl(\()p Fp(g)1785 4465 y Fi(0)1818 4490 y Fl(\))1848 4524 y Fn(j)p Fq(f)g Fn(j)1944 4536 y Fj(1)2014 4524 y Fq(m)p Fs(\()p Fq(Y)19 b Fs(\))k Fn(\024)g Fq(e)2368 4490 y Fp(P)9 b Fl(\()p Fp(g)2479 4465 y Fi(0)2511 4490 y Fl(\))2541 4524 y Fn(k)p Fq(f)g Fn(k)2675 4536 y Fp(B)s(V)2785 4524 y Fq(m)p Fs(\()p Fq(Y)19 b Fs(\))456 4669 y(from)27 b(whic)n(h)g(the)h(lemma)g(follo)n (ws.)1811 b Ff(\003)555 4833 y Fs(The)28 b(ab)r(o)n(v)n(e)f(notion)h (of)g(closeness)f(is)h(the)g(one)g(emplo)n(y)n(ed)f(in)h([KL],)g(it)h (is)f(then)g(natural)g(to)456 4933 y(try)f(to)g(v)n(erify)g(the)h (conditions)f(of)g(the)h(abstract)f(p)r(erturbation)g(result)g(con)n (tained)g(in)h(suc)n(h)f(a)456 5032 y(pap)r(er.)p 456 5117 499 4 v 555 5190 a Fl(10)621 5216 y FA(Note)e(that)g(if)d(a)i(map) f(do)r(es)h(not)h(satisfy)e(immediately)f(suc)n(h)i(a)g(criteria,)f (some)g(of)g(its)g(p)r(o)n(w)n(ers)h(ma)n(y)-6 b(.)p eop %%Page: 24 24 24 23 bop 456 251 a Fl(24)355 b(CARLANGELO)23 b(LIVERANI)f(AND)g(V)1947 236 y(\023)1941 251 y(ER)n(ONIQUE)g(MA)n(UME-DESCHAMPS)555 450 y Fs(F)-7 b(or)21 b(the)h(reader)d(con)n(v)n(enience)h(let)i(us)f (summarize)g(the)h(ab)r(o)n(v)n(e)e(men)n(tion)h(result)g(sp)r (ecialized)456 550 y(to)27 b(the)h(simple)g(case)f(under)g (consideration.)456 673 y FB(Theorem)35 b(7.3)h Fs(\([KL]\))p FB(.)44 b Fr(If)34 b(ther)l(e)g(exists)g(c)l(onstants)f Fq(A;)14 b(B)35 b(>)30 b Fs(0)p Fr(,)35 b(indep)l(endent)f(of)h Fq(Y)19 b Fr(,)35 b(and)456 784 y Fq(\022)25 b Fn(2)e Fs(\(\002\()p Fq(g)770 754 y Fl(0)807 784 y Fs(\))p Fq(;)14 b(e)915 754 y Fp(P)9 b Fl(\()p Fp(g)1026 729 y Fi(0)1059 754 y Fl(\))1089 784 y Fs(\))30 b Fr(such)g(that,)g(for)h(e)l(ach)g Fq(f)g Fn(2)24 b Fr(BV,)1355 930 y Fn(kL)1454 896 y Fp(n)1454 951 y Fl(0)1499 930 y Fq(f)9 b Fn(k)1591 942 y Fc(BV)1715 930 y Fn(\024)23 b Fq(A\022)1906 896 y Fp(n)1951 930 y Fn(k)p Fq(f)9 b Fn(k)2085 942 y Fc(BV)2205 930 y Fs(+)18 b Fq(B)t Fn(j)p Fq(f)9 b Fn(j)2451 942 y Fl(1)1355 1060 y Fn(kL)1454 1025 y Fp(n)1454 1080 y(Y)1511 1060 y Fq(f)g Fn(k)1603 1072 y Fc(BV)1727 1060 y Fn(\024)23 b Fq(A)1877 1025 y Fp(n)1922 1060 y Fq(\022)1963 1025 y Fp(n)2009 1060 y Fn(k)p Fq(f)9 b Fn(k)2143 1072 y Fc(BV)2262 1060 y Fs(+)18 b Fq(B)t Fn(j)p Fq(f)9 b Fn(j)2508 1072 y Fl(1)456 1211 y Fr(then)41 b(for)h(e)l(ach)g Fq(\022)1033 1223 y Fl(1)1114 1211 y Fn(2)j Fs(\()p Fq(\022)r(;)14 b Fs(1\))41 b Fr(and)h Fq(\016)47 b Fn(2)d Fs(\(0)p Fq(;)14 b Fs(1)26 b Fn(\000)g Fq(\022)2104 1223 y Fl(1)2142 1211 y Fs(\))p Fr(,)44 b(ther)l(e)e(exists)f Fq(")2740 1223 y Fl(0)2821 1211 y Fq(>)i Fs(0)e Fr(such)g(that)g(if)456 1311 y Fn(jjjL)582 1323 y Fl(0)640 1311 y Fn(\000)21 b(L)783 1323 y Fp(Y)841 1311 y Fn(jjj)30 b Fq(<)f(")1073 1323 y Fl(0)1144 1311 y Fr(then)k(the)h(sp)l(e)l(ctrum)f(of)h Fn(L)1985 1323 y Fp(Y)2076 1311 y Fr(outside)g(the)g(disk)g Fn(f)p Fq(z)f Fn(2)d Fo(C)51 b Fn(j)30 b(j)p Fq(z)t Fn(j)f(\024)h Fq(\022)3273 1323 y Fl(1)3310 1311 y Fn(g)j Fr(is)456 1410 y Fq(\016)s Fr(-close,)d(with)h(multiplicity,)g(to)f(the)g(one)g(of)h Fn(L)1947 1422 y Fl(0)1984 1410 y Fr(.)555 1533 y Fs(Clearly)-7 b(,)33 b(from)f(the)h(Theorem)f(7.3)f(Theorem)h(C)h(readily)e(follo)n (ws.)51 b(In)33 b(fact,)h(if)f(the)g(map)456 1633 y Fq(T)k Fs(has)26 b(a)g(unique)g(in)n(v)-5 b(arian)n(t)26 b(measure)f Fq(\026)1745 1645 y Fl(0)1809 1633 y Fs(absolutely)g(con)n(tin)n(uous)g (with)i(resp)r(ect)f(to)h Fq(m)p Fs(,)f(this)456 1739 y(means)31 b(that)h Fn(L)955 1751 y Fl(0)1025 1739 y Fs(has)f Fq(e)1216 1709 y Fp(P)9 b Fl(\()p Fp(g)1327 1684 y Fi(0)1360 1709 y Fl(\))1422 1739 y Fs(as)31 b(an)h(isolated)f (eigen)n(v)-5 b(alues)30 b(and,)j(if)g(the)f(systems)f(\()p Fq(I)7 b(;)14 b(T)7 b(;)14 b(\026)3375 1751 y Fl(0)3412 1739 y Fs(\))456 1839 y(is)26 b(mixing,)h(this)g(means)f(that)h(there)g (are)e(no)i(other)f(eigen)n(v)-5 b(alues)25 b(of)i(mo)r(dulus)g(one,)g (whic)n(h)f(in)456 1939 y(turn)h(implies)h(the)g(existence)f(of)h(a)f (sp)r(ectral)g(gap.)36 b(Let)28 b Fq(\025)2280 1951 y Fl(1)2318 1939 y Fs(,)f Fn(j)p Fq(\025)2439 1951 y Fl(1)2477 1939 y Fn(j)c Fq(<)g Fs(1)k(b)r(e)h(the)g(second)f(largest)456 2070 y(eigen)n(v)-5 b(alue)27 b(then,)h(in)h(the)f(ab)r(o)n(v)n(e)f (theorem,)h(c)n(ho)r(ose)e Fq(\022)2191 2082 y Fl(1)2252 2070 y Fn(\025)e Fs(max)o Fn(f)p Fq(\022)r(;)k(\025)2677 2082 y Fl(1)2715 2070 y Fn(g)f Fs(and)h Fq(\016)f Fs(=)3108 2036 y Fp(e)3139 2011 y Fh(P)8 b Fi(\()p Fh(g)3235 1994 y Fi(0)3269 2011 y(\))3295 2036 y Fj(\000)p Fp(\022)3379 2044 y Fi(1)p 3108 2051 304 4 v 3243 2099 a Fl(2)3421 2070 y Fs(.)456 2170 y(Theorem)22 b(7.3)g(implies)h(that,)h(for)f (su\016cien)n(tly)g(small)f(holes,)i(the)f(sp)r(ectrum)g(of)g Fn(L)g Fs(outside)g(the)456 2270 y(disk)k Fn(f)p Fq(z)f Fn(2)d Fo(C)44 b Fn(j)23 b(j)p Fq(z)t Fn(j)g(\024)f Fq(\022)1181 2282 y Fl(1)1218 2270 y Fn(g)27 b Fs(consists)g(of)g(only)g(one)g (eigen)n(v)-5 b(alue)27 b Fq(\025)2466 2282 y Fl(0)2531 2270 y Fs(\(that)h(mo)n(v)n(es)e(con)n(tin)n(uously)456 2381 y(from)h Fq(e)691 2350 y Fp(P)9 b Fl(\()p Fp(g)802 2325 y Fi(0)835 2350 y Fl(\))894 2381 y Fs(as)27 b(the)i(hole)e(gets)h (larger\))f(of)h(m)n(ultiplicit)n(y)h(one)e(and)h(of)h(mo)r(dulus)f (larger)e(than)456 2480 y(1)20 b Fn(\000)h Fq(\016)s Fs(.)667 2448 y Fl(11)782 2480 y Fs(The)32 b(pro)5 b(jector)30 b(\005)i(\()p Fn(L)1502 2492 y Fp(Y)1560 2480 y Fs(\005)e(=)g(\005)p Fn(L)1866 2492 y Fp(Y)1953 2480 y Fs(=)g Fq(\025)2096 2492 y Fl(0)2134 2480 y Fs(\005\))i(asso)r(ciated)e(to)i(suc)n(h)f(an)g (eigen)n(v)-5 b(alue)456 2580 y(is)29 b(of)h(the)h(form)e(\005\()p Fq(f)9 b Fs(\))27 b(=)g Fq(h\026)p Fs(\()p Fq(f)9 b Fs(\))30 b(where)f Fn(L)1818 2592 y Fp(Y)1876 2580 y Fq(h)e Fs(=)f Fq(\025)2090 2592 y Fl(0)2128 2580 y Fq(h)p Fs(,)k(giv)n(es)f(the)h (quasi)g(in)n(v)-5 b(arian)n(t)29 b(measure)456 2679 y(and)e Fq(h\026)h Fs(is)f(the)h(in)n(v)-5 b(arian)n(t)26 b(measure.)1637 2647 y Fl(12)555 2779 y Fs(Hence,)i(to)g(conclude)f(w)n (e)g(need)h(only)f(v)n(erify)g(the)h(h)n(yp)r(otheses)f(of)g(Theorem)g (7.3.)456 2909 y FB(Lemma)35 b(7.4.)43 b Fr(F)-6 b(or)34 b(e)l(ach)h Fq(\022)f Fn(2)d Fs(\(\002\()p Fq(g)1674 2879 y Fl(0)1711 2909 y Fs(\))p Fq(;)d(e)1833 2879 y Fp(P)9 b Fl(\()p Fp(g)1944 2854 y Fi(0)1977 2879 y Fl(\))2007 2909 y Fs(\))34 b Fr(ther)l(e)h(exists)e Fq(A;)14 b(B)36 b(>)30 b Fs(0)p Fr(,)36 b(indep)l(endent)f(of)456 3008 y Fq(Y)18 b Fr(,)30 b(such)g(that,)g(for)h(e)l(ach)g Fq(f)g Fn(2)24 b Fr(BV,)1369 3155 y Fn(kL)1468 3121 y Fp(n)1468 3175 y Fl(0)1513 3155 y Fq(f)9 b Fn(k)1605 3167 y Fp(B)s(V)1738 3155 y Fn(\024)22 b Fq(A\022)1928 3121 y Fp(n)1974 3155 y Fn(k)p Fq(f)9 b Fn(k)2108 3167 y Fp(B)s(V)2236 3155 y Fs(+)18 b Fq(B)t Fn(j)p Fq(f)9 b Fn(j)2482 3167 y Fl(1)1369 3284 y Fn(kL)1468 3250 y Fp(n)1468 3305 y(Y)1525 3284 y Fq(f)g Fn(k)1617 3296 y Fp(B)s(V)1750 3284 y Fn(\024)23 b Fq(A\022)1941 3250 y Fp(n)1986 3284 y Fn(k)p Fq(f)9 b Fn(k)2120 3296 y Fp(B)s(V)2248 3284 y Fs(+)18 b Fq(B)t Fn(j)p Fq(f)9 b Fn(j)2494 3296 y Fl(1)456 3451 y Fr(Pr)l(o)l(of.)43 b Fs(The)31 b(\014rst)g(inequalit) n(y)g(is)h(nothing)f(else)g(that)h(the)f(usual)g(Lasota-Y)-7 b(ork)n(e)29 b(inequalit)n(y)-7 b(,)456 3551 y(the)28 b(second)f(is)g(pro)n(v)n(ed)f(b)n(y)h(a)h(simpli\014ed)g(v)n(ersion)e (of)h(Lemma)h(2.5.)555 3650 y(Remem)n(b)r(er)d(that)h Fn(A)1213 3662 y Fp(n)1284 3650 y Fs(is)f(the)g(set)h(of)f(\014nite)h (partitions)e(in)i(in)n(terv)-5 b(als)24 b Fq(A)f Fs(=)g Fn(f)p Fq(A)3013 3662 y Fp(i)3040 3650 y Fn(g)i Fs(suc)n(h)g(that)456 3698 y Fk(W)525 3785 y Fp(A)575 3793 y Fh(i)619 3760 y Fq(g)659 3772 y Fp(n)732 3760 y Fn(\024)i Fs(2)p Fn(k)p Fq(g)948 3772 y Fp(n)992 3760 y Fn(k)1034 3772 y Fj(1)1104 3760 y Fs(.)45 b(Giv)n(en)31 b Fq(n)c Fn(2)i Fo(N)40 b Fs(and)31 b Fq(A)d Fn(2)g(A)2071 3772 y Fp(i)2129 3760 y Fs(let)2274 3740 y Fk(e)2252 3760 y Fn(Z)2319 3730 y Fl(\()p Fp(n)p Fl(\))2447 3760 y Fs(b)r(e)j(the)f(coarsest)f (partition)h(in)456 3889 y(in)n(terv)-5 b(als)32 b(among)g(all)h(the)g (ones)g(\014ner)g(than)g(b)r(oth)h Fq(A)f Fs(and)g Fn(Z)2452 3859 y Fl(\()p Fp(n)p Fl(\))2549 3889 y Fs(.)54 b(F)-7 b(or)32 b(eac)n(h)g Fq(Z)38 b Fn(2)33 b(Z)3222 3846 y Fl(\()p Fp(n)p Fl(\))3215 3900 y Fj(\003)3352 3889 y Fs(let)473 3973 y(~)456 3994 y Fq(Z)f Fn(2)650 3973 y Fk(e)628 3994 y Fn(Z)695 3964 y Fl(\()p Fp(n)p Fl(\))822 3994 y Fs(b)r(e)e(suc)n(h)g(that)g Fq(Z)j Fn(\032)1508 3973 y Fs(~)1491 3994 y Fq(Z)6 b Fs(.)44 b(W)-7 b(e)30 b(ha)n(v)n(e)f(then)i(the)f(follo)n(wing)f(analogous)f(of)i(equation) 456 4094 y(\(2.6\):)456 4530 y(\(7.1\))1123 4178 y Fk(_)1229 4257 y FB(1)1277 4269 y Fp(T)1325 4253 y Fh(n)1366 4269 y Fp(Z)1419 4257 y Fs(\()p Fq(g)1491 4269 y Fp(n)1536 4257 y Fq(h)p Fs(\))19 b Fn(\016)f Fq(T)1756 4221 y Fj(\000)p Fp(n)1744 4281 y(Z)1875 4257 y Fn(\024)1962 4178 y Fk(_)1984 4356 y Fp(Z)2069 4257 y Fq(hg)2157 4269 y Fp(n)2247 4257 y Fs(+)46 b(2)14 b(sup)2451 4326 y Fp(Z)2552 4257 y Fn(j)p Fq(h)19 b Fn(\001)f Fq(g)2723 4269 y Fp(n)2768 4257 y Fn(j)1442 4485 y(\024)1530 4406 y Fk(_)1551 4584 y Fp(Z)1636 4485 y Fq(hg)1724 4497 y Fp(n)1815 4485 y Fs(+)g(2)c(inf)1992 4537 y Fl(~)1979 4552 y Fp(Z)2068 4485 y Fn(j)p Fq(h)k Fn(\001)h Fq(g)2239 4497 y Fp(n)2283 4485 y Fn(j)g Fs(+)f(2)2464 4406 y Fk(_)2498 4583 y Fl(~)2485 4598 y Fp(Z)2569 4485 y Fq(hg)2657 4497 y Fp(n)1442 4727 y Fn(\024)23 b Fs(9)p Fn(k)p Fq(g)1654 4739 y Fp(n)1698 4727 y Fn(k)1740 4739 y Fj(1)1823 4648 y Fk(_)1858 4825 y Fl(~)1845 4840 y Fp(Z)1929 4727 y Fq(h)c Fs(+)f(8)p Fn(k)p Fq(g)2203 4739 y Fp(n)2246 4727 y Fn(k)2288 4739 y Fj(1)2372 4727 y Fs(inf)2411 4779 y Fl(~)2398 4794 y Fp(Z)2487 4727 y Fn(j)p Fq(h)p Fn(j)p Fq(:)p 456 4938 499 4 v 555 5019 a Fl(11)621 5044 y FA(In)26 b(fact,)f(the)g(results)g(in)f([KL])g (imply)f(that)j(there)g(exist)f(constan)n(ts)h Fy(C)h(>)21 b FA(0)k(suc)n(h)g(that)h Fy(e)3041 5021 y Fd(P)8 b Fw(\()p Fd(g)3144 5000 y Fi(0)3176 5021 y Fw(\))3221 5044 y Fx(\000)16 b Fy(\025)3333 5053 y Fw(0)3390 5044 y Fx(\024)456 5127 y Fy(C)5 b(m)p FA(\()p Fy(Y)16 b FA(\),)24 b(pro)n(vided)g Fy(\016)i FA(is)d(c)n(hosen)i(small)c(enough.)555 5190 y Fl(12)621 5216 y FA(See)k([K)o(])e(for)g(the)i(pro)r(of)e(that)i Fy(\026)f FA(is)f(a)h(measure.)p eop %%Page: 25 25 25 24 bop 1339 251 a Fl(LASOT)-5 b(A-YORKE)29 b(MAPS)f(WITH)h(HOLES)817 b(25)456 454 y FB(Sublemma)28 b(7.5.)40 b Fr(F)-6 b(or)30 b(e)l(ach)1472 433 y Fs(~)1454 454 y Fq(Z)f Fn(2)1642 433 y Fs(^)1618 454 y Fn(Z)1685 424 y Fl(\()p Fp(n)p Fl(\))1782 454 y Fr(,)h Fs(#)p Fn(f)p Fq(Z)f Fn(2)23 b(Z)2179 411 y Fl(\()p Fp(n)p Fl(\))2172 466 y Fj(\003)2299 454 y Fn(j)g Fq(Z)29 b Fn(\032)2536 433 y Fs(~)2518 454 y Fq(Z)6 b Fn(g)22 b(\024)h Fq(n)18 b Fs(+)g(1)p Fr(.)456 627 y(Pr)l(o)l(of.)43 b Fs(Since,)i(b)n(y)c(de\014nition,)k Fq(T)1572 596 y Fp(i)1598 627 y Fn(j)1634 637 y Fl(~)1621 652 y Fp(Z)1675 627 y Fs(,)g Fq(i)g Fn(\024)h Fq(n)p Fs(,)e(is)d(in)n(v)n(ertible,)j(then)e Fq(T)2813 596 y Fj(\000)p Fp(i)2892 627 y Fq(Y)60 b Fs(can)41 b(ha)n(v)n(e)f(at)456 736 y(most)f(one)g(preimage)f(in)1329 715 y(~)1311 736 y Fq(Z)6 b Fs(.)73 b(Accordingly)-7 b(,)41 b Fq(Y)2010 748 y Fp(n)2082 736 y Fn(\\)2181 715 y Fs(~)2164 736 y Fq(Z)k Fs(can)39 b(consist)g(of,)k(at)c(most,)j Fq(n)e Fs(sub-)456 836 y(in)n(terv)-5 b(als,)41 b(hence)e Fq(X)1137 848 y Fp(n)1222 836 y Fs(can)f(ha)n(v)n(e,)k(at)d(most,)j Fq(n)26 b Fs(+)g(1)38 b(connected)i(comp)r(onen)n(ts)e(whic)n(h)h(are) 456 947 y(exactly)27 b Fn(f)p Fq(Z)h Fn(2)23 b(Z)1014 904 y Fl(\()p Fp(n)p Fl(\))1007 959 y Fj(\003)1134 947 y Fn(j)g Fq(Z)29 b Fn(\032)1371 926 y Fs(~)1353 947 y Fq(Z)6 b Fn(g)p Fs(.)1899 b Ff(\003)555 1136 y Fs(By)28 b(Sublemma)f(7.5)g(it)h(follo)n(ws)f(that)h(w)n(e)f(can)g(sum)h(o)n(v)n (er)e Fq(Z)i Fn(2)c(Z)2620 1093 y Fl(\()p Fp(n)p Fl(\))2613 1148 y Fj(\003)2744 1136 y Fs(and)k(obtain)648 1246 y Fk(_)754 1325 y Fn(L)811 1290 y Fp(n)857 1325 y Fq(h)23 b Fn(\024)f Fs(9\()p Fq(n)c Fs(+)g(1\))p Fn(k)p Fq(g)1396 1337 y Fp(n)1441 1325 y Fn(k)1483 1337 y Fj(1)1566 1246 y Fk(_)1672 1325 y Fq(h)h Fs(+)f(8\()p Fq(n)g Fs(+)g(1\))p Fn(k)p Fq(g)2203 1337 y Fp(n)2247 1325 y Fn(k)2289 1337 y Fj(1)2426 1325 y Fs(sup)2386 1395 y Fl(~)2373 1410 y Fp(Z)t Fj(2)2485 1395 y Fl(^)2467 1410 y Fj(Z)2520 1393 y Fi(\()p Fh(n)p Fi(\))2619 1325 y Fq(m)p Fs(\()p Fq(Z)6 b Fs(\))2819 1290 y Fj(\000)p Fl(1)2922 1212 y Fk(Z)3019 1325 y Fn(j)p Fq(h)p Fn(j)p Fq(dm:)456 1545 y Fs(Since)27 b(there)h(exists)j(\026)-46 b Fq(n)23 b Fn(2)g Fo(N)t Fs(:)43 b Fq(\022)1430 1515 y Fl(\026)-37 b Fp(n)1495 1545 y Fq(>)22 b Fs(9\()t(\026)-46 b Fq(n)18 b Fs(+)h(1\))p Fn(k)p Fq(g)1968 1557 y Fl(\026)-37 b Fp(n)2008 1545 y Fn(k)2050 1557 y Fj(1)2119 1545 y Fs(,)28 b(the)g(result)f(follo)n(ws)g(b)n(y)g(c)n(ho)r(osing)894 1695 y Fq(A)c Fs(:=)g(sup)1215 1716 y Fp(n)p Fj(\024)t Fl(\026)-37 b Fp(n)1367 1695 y Fs(9\()p Fq(n)18 b Fs(+)g(1\))p Fn(k)p Fq(g)s Fn(k)1793 1707 y Fj(1)894 1796 y Fq(B)27 b Fs(:=)c(2\(1)17 b Fn(\000)h Fq(\022)1356 1766 y Fl(\026)-37 b Fp(n)1398 1796 y Fs(\))1430 1766 y Fj(\000)p Fl(1)1533 1796 y Fs(sup)1658 1816 y Fp(n)p Fj(\024)t Fl(\026)g Fp(n)1810 1796 y Fs(8\()p Fq(n)18 b Fs(+)g(1\))p Fn(k)p Fq(g)s Fn(k)2236 1808 y Fj(1)2319 1796 y Fs(sup)2457 1808 y Fl(~)2444 1823 y Fp(Z)t Fj(2)2556 1808 y Fl(^)2538 1823 y Fj(Z)2591 1806 y Fi(\()p Fh(n)p Fi(\))2694 1796 y Fq(m)p Fs(\()p Fq(Z)6 b Fs(\))2894 1766 y Fj(\000)p Fl(1)2983 1796 y Fq(;)456 1944 y Fs(and)29 b(using)g(the)g(same)g (iteration)f(sc)n(heme)h(emplo)n(y)n(ed)g(in)g(the)h(pro)r(of)e(of)i (Lemma)e(3.7.)41 b(Notice)456 2044 y(that,)28 b(as)e(announced,)i Fq(A)g Fs(and)f Fq(B)32 b Fs(do)27 b(not)h(dep)r(end)g(on)f(the)h(hole) g Fq(Y)18 b Fs(.)761 b Ff(\003)1708 2260 y Ft(References)456 2393 y FA([BC])147 b(H.)23 b(v)l(an)h(den)g(Bedem)f(and)i(N.)e(Cherno)n (v)h Fm(Exp)l(anding)j(maps)g(of)f(an)g(interval)f(with)h(holes)f FA(preprin)n(t.)456 2476 y([Bi])178 b(G.)30 b(Birkho\013)f(:)44 b(\\)p Fm(L)l(attic)l(e)32 b(The)l(ory)p FA(".)e(25)h(3rd)f(ed.,)h (A.M.S.)e(Collo)r(q.)g(Publ.,)i(Pro)n(vidence,)h(Rho)r(de)744 2559 y(Island)24 b(\(1967\))456 2642 y([BK])143 b(V.)33 b(Baladi)g(&)h(G.)f(Keller)g Fm(Zeta)i(functions)g(and)h(tr)l(ansfer)f (op)l(er)l(ator)i(for)f(pie)l(c)l(ewise)f(monotone)744 2725 y(tr)l(ansformations.)25 b FA(Comm.)c(Math.)j(Ph)n(y)-6 b(.)23 b Fb(127)p FA(,)g(459-477,)h(\(1990\).)456 2808 y([CMT])81 b(N.)37 b(Cherno)n(v,)h(R.)f(Mark)l(arian)g(&)g(S.)h(T)-6 b(roub)r(etzk)n(o)n(y)g(.)39 b Fm(Conditional)t(ly)h(invariant)f(me)l (asur)l(es)h(for)744 2891 y(A)n(nosov)20 b(maps)h(with)e(smal)t(l)i (holes.)c FA(Ergo)r(dic)h(Theory)f(Dynam.)f(Systems)h Fb(18)p FA(,)g(5,)h(1049-1073)h(\(1998\).)456 2974 y([C])197 b(P)-6 b(.)25 b(Collet)h Fm(Er)l(go)l(dic)i(pr)l(op)l(erties)h(of)f (maps)g(of)g(the)f(interval)f FA(in)f(R.)g(Bamon,)g(J.-M.)f(Gam)n (baudo,)j(S.)744 3057 y(Mart)-8 b(\023)-27 b(\020nez,)23 b Fm(Dynamic)l(al)j(systems)p FA(,)d(Hermann)g(\(1996\).)456 3140 y([CMS1])58 b(P)-6 b(.)36 b(Collet,)k(S.)c(Mart)-8 b(\023)-27 b(\020nez)37 b(&)g(B.)f(Sc)n(hmitt.)g Fm(The)i(Pianigiani-Y) -5 b(orke)37 b(me)l(asur)l(e)j(for)e(top)l(olo)l(gic)l(al)744 3223 y(Markov)25 b(chains.)g FA(Israel)e(Journal)h(of)f(Math.)h Fb(97)p FA(,)e(61-70)j(\(1997\).)456 3306 y([CMS2])58 b(P)-6 b(.)22 b(Collet,)g(S.)g(Mart)-8 b(\023)-27 b(\020nez)23 b(&)f(B.)g(Sc)n(hmitt.)g Fm(The)i(Pianigiani-Y)-5 b(orke)24 b(me)l(asur)l(e)i(and)g(the)e(asymptotic)744 3389 y(law)i(on)g(the)g (limit)f(Cantor)h(set)f(of)h(exp)l(anding)h(systems.)c FA(Nonlinearit)n(y)h Fb(7)p FA(,)f(1437-1443)i(\(1994\).)456 3472 y([CMS3])58 b(P)-6 b(.)26 b(Collet,)i(S.)f(Mart)-8 b(\023)-27 b(\020nez)27 b(&)g(B.)g(Sc)n(hmitt.)g Fm(Quasi-stationary)i (distribution)g(and)g(Gibbs)g(me)l(asur)l(e)744 3555 y(of)c(exp)l(anding)i(systems.)c FA(Instabilities)h(and)g(non)h (equilibrium)c(structures)j(205-219)h(\(1996\))456 3638 y([CMM])67 b(P)-6 b(.)19 b(Collet,)i(S.)e(Mart)-8 b(\023)-27 b(\020nez)20 b(&)g(V.)g(Maume-Desc)n(hamps)e Fm(On)k(the)g(existenc)l (e)f(of)i(c)l(onditional)t(ly)g(invari-)744 3721 y(ant)i(pr)l(ob)l (ability)i(me)l(asur)l(es)g(in)e(dynamic)l(al)i(systems.)d FA(Nonlinearit)n(y)-6 b(,)23 b Fb(13)p FA(,)g(\(2000\),)i(1263-1274.) 456 3804 y([FKMP])34 b(P)-6 b(.A.)23 b(F)-6 b(errari,)22 b(H.)h(Kesten,)i(S.)e(Mart)-8 b(\023)-27 b(\020nez)25 b(&)e(P)-6 b(.)24 b(Picco.)g Fm(Existenc)l(e)h(of)i(quasi-stationary)f (distribu-)744 3887 y(tions.)f(A)h(r)l(enewal)g(dynamic)l(al)h(appr)l (o)l(ach.)g FA(Annals)c(of)h(Probabilit)n(y)f Fb(23)p FA(,)g(501-521)h(\(1995\).)456 3970 y([FS])163 b(P)-6 b(.)29 b(F)-6 b(errero,)31 b(B.)e(Sc)n(hmitt)h(:)44 b(\\Ruelle's)29 b(P)n(erron-F)-6 b(rob)r(enius)30 b(theorem)f(and)i(pro)t(jectiv)n(e)g (metrics".)744 4053 y(Coll.)22 b(Math.)i(So)r(c.)g(J.)f(Bolly)n(ai,)g Fb(27)g FA(\(1979\))456 4136 y([K])193 b(G.)15 b(Keller)g Fm(Markov)k(extensions,)g(zeta)g(functions,)g(and)h(F)-5 b(r)l(e)l(dholm)20 b(the)l(ory)f(for)g(pie)l(c)l(ewise)f(invertible)744 4219 y(dynamic)l(al)27 b(systems.)c FA(T)-6 b(rans.)23 b(Amer.)f(Math.)h(So)r(c.)h Fb(314)f FA(\(2\),)h(433-497)h(\(1989\).) 456 4302 y([KL])149 b(G.Keller,)28 b(C.Liv)n(erani.)f Fm(Stability)i(of)h(the)g(Sp)l(e)l(ctr)l(al)h(Gap)g(for)f(tr)l(ansfer)g (op)l(er)l(ators.)g FA(Annali)e(della)744 4385 y(Scuola)c(Normale)e(di) i(Pisa,)e(Classe)i(di)f(Scienze,)i(\(4\),)f(V)-6 b(ol.)23 b(XXVI)r(I)r(I,)h(pp.)f(141{152)j(\(1999\).)456 4468 y([L1])169 b(C.)23 b(Liv)n(erani)g Fm(De)l(c)l(ay)j(of)g(c)l(orr)l (elations.)f FA(Ann.)e(of)g(Math.)h(\(1995\),)h Fb(142)e FA(\(2\),)h(239-301)456 4551 y([L2])169 b(C.)28 b(Liv)n(erani)h Fm(De)l(c)l(ay)i(of)g(c)l(orr)l(elations)h(for)f(pie)l(c)l(ewise)g(exp) l(anding)h(maps.)60 b FA(J.)29 b(Statist.)h(Ph)n(ys.)f Fb(78)744 4634 y FA(no.)23 b(3-4,)g(1111-1129)j(\(1995\).)456 4717 y([LSV])112 b(C.)27 b(Liv)n(erani,)h(B.)f(Saussol)g(&)h(S.)f(V)-6 b(aien)n(ti)28 b Fm(Conformal)j(me)l(asur)l(e)g(and)f(de)l(c)l(ay)g(of) f(c)l(orr)l(elations)i(for)744 4800 y(c)l(overing)e(weighte)l(d)g (systems.)e FA(Ergo)r(dic)h(Theory)f(and)h(Dynamical)f(Systems,)h Fb(18)p FA(,)f(6,)h(1399{1420)744 4883 y(\(1998\).)456 4967 y([PY])147 b(G.)22 b(Pianigiani)h(&)f(J.A.)g(Y)-6 b(ork)n(e.)23 b Fm(Exp)l(anding)j(maps)g(on)g(sets)e(which)i(ar)l(e)f (almost)h(invariant:)33 b(de)l(c)l(ay)744 5050 y(and)26 b(chaos.)f FA(T)-6 b(rans.)23 b(Amer.)e(Math.)j(So)r(ciet)n(y)h Fb(252)p FA(,)d(433-497)j(\(1989\).)456 5133 y([R])196 b(M.)20 b(Ryc)n(hlik)i Fm(Bounde)l(d)j(variation)f(and)g(invariant)g (me)l(asur)l(es.)f FA(Studia)f(mathematica)f(\(1982\),)i Fb(71)p FA(,)744 5216 y(69-80.)p eop %%Page: 26 26 26 25 bop 456 251 a Fl(26)355 b(CARLANGELO)23 b(LIVERANI)f(AND)g(V)1947 236 y(\023)1941 251 y(ER)n(ONIQUE)g(MA)n(UME-DESCHAMPS)456 450 y FA([VJ])159 b(D.)19 b(V)-6 b(ere-Jones)21 b Fm(Ge)l(ometric)i(er) l(go)l(dicity)g(in)f(denumer)l(able)i(Markov)f(chains.)e FA(Quart.)g(J.)f(Math.)g Fb(13)p FA(,)744 533 y(2,)j(2,)g(7-28)h (\(1962\).)456 616 y([V])195 b(M.)28 b(Viana)i Fm(Sto)l(chastic)i (dynamics)g(of)f(deterministic)g(systems.)e FA(Brazillian)g(Math.)h (Collo)r(quium)744 699 y(1997,)24 b(IMP)-6 b(A.)456 782 y([Y])195 b(L.-S.)28 b(Y)-6 b(oung)31 b Fm(Dimension,)i(entr)l(opy)f (and)g(Lyapunov)g(exp)l(onents.)f FA(Erg.)e(TH.)g(&)h(Dyn.)f(Syst.,)j Fb(2)744 865 y FA(\(1982\),)25 b(109-124.)555 1021 y Fz(Dip)l(ar)l(timento)g(di)g(Ma)l(tema)l(tica,)f(Universit)l(a)l(')h (di)g(R)n(oma)g(\\Tor)g(Ver)o(ga)l(t)l(a)l(",)e(Via)i(della)g(Ricer)o (ca)456 1104 y(Scientifica,)g(I-00133)e(R)n(oma,)i(IT)-6 b(AL)g(Y)555 1187 y Fm(E-mail)26 b(addr)l(ess)5 b FA(:)33 b Fa(liverani@mat.uniroma2.)q(it)555 1328 y Fz(Universit)877 1322 y(\023)877 1328 y(e)26 b(de)f(Bour)o(gogne)f(B.P.)i(47870)e(21078) g(Dijon)h(Cedex)h(FRANCE)555 1411 y Fm(E-mail)g(addr)l(ess)5 b FA(:)33 b Fa(vmaume@topolog.u-bourg)q(ogn)q(e.fr)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0106050706468--