This is a multi-part message in MIME format. ---------------9910261529511 Content-Type: text/plain; name="99-407.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-407.keywords" conformal measures; multi-dimensional dynamical systems; equilibrium states; thermodynamical formalism; piecewise expanding maps; topological pressure; transfer operator; Schauder-Tychonov theorem ---------------9910261529511 Content-Type: application/postscript; name="conformal.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="conformal.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: conformal.dvi %%Pages: 20 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips -o conformal.ps conformal %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 1999.10.26:1625 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def 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(ciated)i(to)g(the)g(collection)g Fo(Z)7 b Fq(,)36 b(i.e,)g(for)448 1174 y Fp(n)25 b Fo(\025)g Fq(1:)678 1378 y Fo(Z)744 1392 y Fm(n)817 1378 y Fq(=)f Fo(f)p Fp(A)1025 1392 y Fh(0)1086 1378 y Fo(\\)c Fp(T)1233 1341 y Fi(\000)p Fh(1)1327 1378 y Fp(A)1395 1392 y Fh(1)1454 1378 y Fo(\\)g Fp(:)15 b(:)g(:)22 b Fo(\\)d Fp(T)1808 1341 y Fi(\000)p Fh(\()p Fm(n)p Fi(\000)p Fh(1\))2055 1378 y Fp(A)2123 1392 y Fm(n)p Fi(\000)p Fh(1)2286 1378 y Fo(6)p Fq(=)25 b Fo(;)g Fq(:)h Fp(A)2571 1392 y Fh(0)2626 1378 y Fp(:)15 b(:)g(:)h(A)2815 1392 y Fm(n)p Fi(\000)p Fh(1)2977 1378 y Fo(2)25 b(Z)7 b(g)p Fp(:)448 1582 y Fq(The)31 b Fp(n)p Fq(-cylinder)e Fp(A)1136 1596 y Fh(0)1196 1582 y Fo(\\)21 b Fp(T)1344 1549 y Fi(\000)p Fh(1)1438 1582 y Fp(A)1506 1596 y Fh(1)1566 1582 y Fo(\\)g Fp(:)15 b(:)g(:)22 b Fo(\\)e Fp(T)1922 1549 y Fm(n)p Fi(\000)p Fh(1)2059 1582 y Fp(A)2127 1596 y Fm(n)p Fi(\000)p Fh(1)2295 1582 y Fq(will)29 b(b)s(e)i(written)f([)p Fp(A)3005 1596 y Fh(0)3060 1582 y Fp(:)15 b(:)g(:)h(A)3249 1596 y Fm(n)p Fi(\000)p Fh(1)3387 1582 y Fq(].)448 1695 y Fp(@)5 b(Z)43 b Fq(denotes)35 b(the)h(top)s(ological)f(b)s(oundary)f (of)h Fp(Z)42 b Fq(as)36 b(a)g(subset)e(of)i Fj(R)2845 1662 y Fm(d)2891 1695 y Fq(.)56 b(W)-8 b(e)37 b(will)32 b(use)448 1808 y(the)f(follo)m(wing)e(notation:)1624 1930 y Fp(@)5 b Fo(Z)1743 1944 y Fm(n)1815 1930 y Fq(=)1957 1843 y Fl([)1911 2040 y Fm(Z)g Fi(2Z)2062 2048 y Fg(n)2119 1930 y Fp(@)g(Z)q(:)589 2188 y Fq(The)30 b Fn(top)-5 b(olo)g(gic)g(al)36 b(pr)-5 b(essur)g(e)39 b Fq(of)31 b Fp(S)f Fo(\032)25 b Fp(X)38 b Fq(\(w.r.t.)31 b(the)f(w)m(eigh)m(t)h Fp(g)s Fq(\))g(is)1205 2430 y Fp(P)1263 2444 y Fm(g)1303 2430 y Fq(\()p Fp(S;)15 b(T)e Fq(\))26 b(=)f(lim)15 b(sup)1704 2503 y Fm(n)p Fi(!1)1965 2369 y Fq(1)p 1960 2410 55 4 v 1960 2493 a Fp(n)2040 2430 y Fq(log)2220 2344 y Fl(X)2199 2530 y Fg(Z)s Ff(2Z)2330 2538 y Fg(n)2195 2575 y Fe(\026)2183 2588 y Fg(Z)s Ff(\\)p Fg(S)s Ff(6)p Fe(=)p Ff(;)2413 2430 y Fq(sup)2455 2509 y Fm(Z)2565 2430 y Fp(g)2608 2444 y Fm(n)2656 2430 y Fp(:)589 2770 y Fq(A)41 b Fn(c)-5 b(onformal)45 b(me)-5 b(asur)g(e)49 b Fq(is)40 b(a)h(probabilit)m(y)d (measure)j Fp(\027)46 b Fq(on)41 b Fp(X)48 b Fq(suc)m(h)40 b(that,)k(for)448 2883 y(some)31 b(n)m(um)m(b)s(er)e Fp(\025)c(>)g Fq(0,)1574 2972 y Fp(d\027)h Fo(\016)21 b Fp(T)p 1574 3012 250 4 v 1649 3096 a(d\027)1859 3033 y Fq(=)j Fp(\025)2018 2972 y Fq(1)p 2017 3012 47 4 v 2017 3096 a Fp(g)2104 3033 y(\027)6 b Fq(-a.e.)448 3252 y(Observ)m(e)34 b(that)g(this)e(is)g(equiv)-5 b(alen)m(t)33 b(to:)48 b Fp(\027)6 b Fq(\()p Fp(T)13 b(A)p Fq(\))31 b(=)f Fp(\025)2318 3178 y Fl(R)2361 3283 y Fm(A)2433 3252 y Fp(g)2479 3219 y Fi(\000)p Fh(1)2589 3252 y Fp(d\027)39 b Fq(for)34 b(all)e(measurable)448 3364 y Fp(A)43 b Fq(on)f(whic)m(h)g Fp(T)55 b Fq(is)41 b(injectiv)m(e.)77 b(It)43 b(is)e(also)i(equiv)-5 b(alen)m(t)42 b(to)h(requiring)d Fp(\027)48 b Fq(to)43 b(b)s(e)f(an)448 3477 y(eigenfunctional)26 b(of)i Fo(L)1238 3444 y Fi(\003)1304 3477 y Fq(for)f(a)h(p)s(ositiv)m(e)f(eigen)m(v)-5 b(alue)27 b(\(see)h(section)g(2)g(for)f(de\014nitions\).)448 3816 y Fk(Main)35 b(Theorem)e Fn(L)-5 b(et)32 b Fq(\()p Fp(X)r(;)15 b Fo(Z)7 b Fp(;)15 b(T)e Fq(\))33 b Fn(b)-5 b(e)32 b(a)h(pie)-5 b(c)g(ewise)32 b(invertible)g(system)h(with)g (weight)448 3929 y Fp(g)s Fn(.)43 b(Assume)32 b(that:)514 4117 y Fq(\(a\))47 b Fp(P)734 4131 y Fm(g)774 4117 y Fq(\()p Fp(@)5 b Fo(Z)i Fp(;)15 b(T)e Fq(\))27 b Fp(<)e(P)1257 4131 y Fm(g)1297 4117 y Fq(\()p Fp(X)r(;)15 b(T)e Fq(\))p Fn(.)509 4304 y Fq(\(b\))46 b Fp(g)36 b Fn(has)d(b)-5 b(ounde)g(d)35 b(distortion)f(under)f Fp(T)46 b Fn(in)32 b(the)h(fol)5 b(lowing)34 b(sense:)1300 4556 y Fq(lim)1261 4611 y Fm(m)p Fi(!1)1480 4556 y Fq(lim)15 b(sup)1527 4628 y Fm(n)p Fi(!1)1836 4556 y Fq(max)1774 4617 y Fm(Z)5 b Fi(2Z)1925 4626 y Fg(n)p Fe(+)p Fg(m)2112 4556 y Fq(sup)2083 4634 y Fm(x;y)r Fi(2)p Fm(Z)2294 4556 y Fq(log)2436 4495 y Fp(g)2479 4509 y Fm(n)2527 4495 y Fq(\()p Fp(x)p Fq(\))p 2436 4535 213 4 v 2438 4619 a Fp(g)2481 4633 y Fm(n)2529 4619 y Fq(\()p Fp(y)s Fq(\))2685 4556 y(=)25 b(0)p Fp(:)519 4862 y Fq(\(c\))47 b Fo(Z)39 b Fn(gener)-5 b(ates,)34 b(i.e.,)d Fq(lim)1520 4876 y Fm(n)p Fi(!1)1723 4862 y Fq(sup)1860 4883 y Fm(Z)5 b Fi(2Z)2011 4891 y Fg(n)2102 4862 y Fq(diam)o(\()p Fp(Z)i Fq(\))26 b(=)f(0)p Fn(.)1920 5162 y Fq(3)p eop %%Page: 4 4 4 3 bop 509 623 a Fq(\(d\))46 b Fp(T)13 b Fq(\()p Fp(Y)20 b Fq(\))25 b Fo(\033)g Fp(Y)53 b Fn(and)33 b Fq(inf)1399 637 y Fm(Y)1475 623 y Fp(g)c(>)c Fq(0)p Fn(.)519 806 y Fq(\(e\))47 b Fn(e)-5 b(ach)33 b Fp(Z)f Fo(2)25 b(Z)1126 820 y Fm(n)1173 806 y Fn(,)32 b Fp(n)25 b Fo(\025)g Fq(1)p Fn(,)33 b(has)g(\014nitely)g(many)h(c)-5 b(onne)g(cte)g(d)34 b(c)-5 b(omp)g(onents.)525 990 y Fq(\(f)7 b(\))46 b Fn(for)26 b(every)g(non-empty)i(op)-5 b(en)27 b(set)f Fp(U)36 b Fo(\032)25 b Fp(X)7 b Fn(,)27 b(ther)-5 b(e)27 b(is)g Fp(n)d(<)h Fo(1)h Fn(such)h(that)g Fp(T)3216 957 y Fm(n)3263 990 y Fq(\()p Fp(U)17 b Fo(\\)676 1103 y Fp(Y)j Fq(\))25 b Fo(\033)g Fp(Y)20 b Fn(.)448 1281 y(Then)25 b(ther)-5 b(e)24 b(exists)h(a)f(c)-5 b(onformal)26 b(me)-5 b(asur)g(e)25 b Fp(\027)6 b Fn(.)39 b(Mor)-5 b(e)g(over)25 b(the)f(asso)-5 b(ciate)g(d)27 b(eigenvalue)448 1394 y(is)33 b Fp(\025)25 b Fq(=)g Fp(e)762 1361 y Fm(P)807 1369 y Fg(g)843 1361 y Fh(\()p Fm(X)q(;T)10 b Fh(\))1032 1394 y Fn(.)448 1594 y Fk(Remark)35 b(1.1)585 1885 y Fo(\017)46 b Fn(The)37 b(c)-5 b(onditions)39 b(ab)-5 b(ove)38 b(ar)-5 b(e)38 b(satis\014e)-5 b(d)39 b(for)f(instanc)-5 b(e,)39 b(if)e Fp(T)50 b Fn(is)37 b(pie)-5 b(c)g(ewise)38 b Fp(C)3370 1852 y Fh(1)3409 1885 y Fn(,)676 1998 y(exp)-5 b(anding)34 b(and)f(generic,)f(in)h(the)g(sense)f(that)i(\(se)-5 b(e)33 b([2)q(,)f(Pr)-5 b(op.)34 b(5.2]\):)1748 2190 y Fp(h)1800 2205 y Fh(m)n(ult)1948 2190 y Fq(\()p Fo(Z)7 b Fp(;)15 b(T)e Fq(\))26 b(=)f(0)676 2382 y Fn(and)41 b(if)g Fp(g)j Fn(has)d(a)g(summable)h(mo)-5 b(dulus)42 b(of)f(c)-5 b(ontinuity)42 b(\(a)f(we)-5 b(aker)41 b(c)-5 b(ondition)676 2495 y(than)33 b(H\177)-46 b(older)34 b(c)-5 b(ontinuity\))33 b(and)1604 2696 y Fq(sup)14 b Fp(g)p 1604 2736 199 4 v 1617 2820 a Fq(inf)21 b Fp(g)1838 2757 y(<)k Fq(\003)1997 2771 y Fi(\000)2071 2629 y Fl(\022)2148 2696 y Fq(\003)2211 2710 y Fi(\000)p 2148 2736 123 4 v 2148 2820 a Fq(\003)2211 2834 y Fh(+)2280 2629 y Fl(\023)2347 2652 y Fm(d)p Fi(\000)p Fh(1)2493 2757 y Fp(;)676 2995 y Fq(\003)739 3009 y Fh(+)835 2995 y Fn(and)39 b Fq(\003)1080 3009 y Fi(\000)1177 2995 y Fn(b)-5 b(eing)37 b(the)i(maximum)f(and)h (the)f(minimum)h(of)f(the)g(r)-5 b(ate)39 b(of)f(ex-)676 3107 y(p)-5 b(ansion.)585 3291 y Fo(\017)46 b Fn(Condition)29 b Fq(\(d\))f Fn(c)-5 b(ould)28 b(b)-5 b(e)28 b(r)-5 b(eplac)g(e)g(d)30 b(by)e(the)g(\(very)g(slightly\))h(we)-5 b(aker)28 b(c)-5 b(ondition:)1679 3497 y Fq(inf)1663 3558 y Fm(y)r Fi(2)p Fm(Y)1883 3411 y Fl(X)1820 3613 y Fm(x)p Fi(2)p Fm(T)1958 3594 y Ff(\000)p Fe(1)2040 3613 y Fm(y)2092 3497 y Fp(g)s Fq(\()p Fp(y)s Fq(\))27 b Fp(>)e Fq(0)p Fp(:)585 3827 y Fo(\017)46 b Fn(Condition)34 b Fq(\(b\))e Fn(is)h(use)-5 b(d)33 b(only)h(in)e(Se)-5 b(ction)33 b(3)g(to)h(pr)-5 b(ove)33 b(that:)778 4023 y Fk({)876 3955 y Fl(Q)961 4050 y Fm(k)r Fi(\025)p Fh(1)1122 3978 y Fm(g)r Fh(\()p Fm(y)1220 3990 y Fg(k)1258 3978 y Fh(\))p 1119 4002 169 4 v 1119 4055 a Fm(g)r Fh(\()p Fm(x)1222 4067 y Fg(k)1261 4055 y Fh(\))1329 4023 y Fn(c)-5 b(onver)g(ges)31 b(for)h(every)e(se)-5 b(quenc)g(es)31 b Fq(\()p Fp(x)2606 4038 y Fm(k)2649 4023 y Fq(\))g Fn(and)h Fq(\()p Fp(y)2970 4038 y Fm(k)3013 4023 y Fq(\))f Fn(such)g(that)876 4136 y Fp(T)13 b(x)994 4151 y Fm(k)r Fh(+1)1158 4136 y Fq(=)31 b Fp(x)1312 4151 y Fm(k)1355 4136 y Fn(,)37 b Fp(T)13 b(y)1531 4151 y Fm(k)r Fh(+1)1695 4136 y Fq(=)31 b Fp(y)1842 4151 y Fm(k)1920 4136 y Fn(and)37 b Fp(x)2152 4151 y Fm(k)2195 4136 y Fp(;)15 b(y)2280 4151 y Fm(k)2359 4136 y Fn(ar)-5 b(e)37 b(in)f(the)g(same)h(element)g(of)876 4249 y Fo(Z)i Fn(for)33 b(every)g Fp(k)s Fn(.)778 4504 y Fk({)46 b Fn(the)33 b(me)-5 b(asur)g(ability)34 b(of)f Fp(m)p Fq(\()p Fo(\001)p Fp(;)15 b(x)p Fq(\))34 b Fn(\(se)-5 b(e)33 b(Se)-5 b(ction)33 b(3)g(for)g(the)g(de\014nition\).)676 4800 y(Elsewher)-5 b(e)33 b(only)h Fq(sup)1431 4822 y Fm(Y)1506 4800 y Fp(g)29 b(<)c Fo(1)33 b Fn(is)f(ne)-5 b(e)g(de)g(d.)1920 5162 y Fq(4)p eop %%Page: 5 5 5 4 bop 589 623 a Fq(As)35 b(it)g(w)m(as)g(p)s(oin)m(ted)f(in)f(the)i (in)m(tro)s(duction,)f(the)h(existence)h(of)f(conformal)f(mea-)448 736 y(sures)d(is)g(motiv)-5 b(ated)33 b(b)m(y)e(the)h(study)f(of)h Fn(e)-5 b(quilibrium)34 b(states)p Fq(.)46 b(Recall)32 b(that)g(these)h(are)448 848 y(the)e(in)m(v)-5 b(arian)m(t)29 b(probabilit)m(y)f(measures)j Fp(\026)1897 862 y Fm(e)1963 848 y Fq(suc)m(h)g(that:)695 1085 y Fp(h)p Fq(\()p Fp(\026)837 1099 y Fm(e)874 1085 y Fp(;)15 b(T)e Fq(\))21 b(+)1127 962 y Fl(Z)1178 1168 y Fm(X)1260 1085 y Fq(log)c Fp(g)h(d\026)1556 1099 y Fm(e)1619 1085 y Fq(=)25 b(sup)1762 1158 y Fm(\026)1867 1085 y Fp(h)p Fq(\()p Fp(\026;)15 b(T)e Fq(\))21 b(+)2262 962 y Fl(Z)2312 1168 y Fm(X)2395 1085 y Fq(log)16 b Fp(g)j(d\026)25 b Fq(=)g Fp(P)2870 1099 y Fm(g)2911 1085 y Fq(\()p Fp(X)r(;)15 b(T)e Fq(\))p Fp(:)589 1343 y Fq(Applying)28 b([2)q(],)j(w)m(e)g(get:) 448 1552 y Fk(Corollary)k(1.1)46 b Fn(Assume)25 b(that)i Fq(\()p Fp(X)r(;)15 b Fo(Z)7 b Fp(;)15 b(T)8 b(;)15 b(g)s Fq(\))28 b Fn(sati\014es)e(the)f(r)-5 b(e)g(quir)g(ements)27 b(of)e(the)h(the-)448 1665 y(or)-5 b(em.)43 b(Then)33 b(ther)-5 b(e)34 b(exists)f(exactly)g(one)g(er)-5 b(go)g(dic)34 b(e)-5 b(quilibrium)32 b(state)i Fp(\026)2953 1679 y Fm(e)2989 1665 y Fn(.)589 1778 y(Mor)-5 b(e)g(over)29 b(it)e(is)g(er)-5 b(go)g(dic)28 b(and)g(c)-5 b(oincides)28 b(with)g(the)f(unique)f(invariant)j(pr)-5 b(ob)g(ability)448 1890 y(me)g(asur)g(e)34 b(absolutely)g(c)-5 b(ontinuous)34 b(with)f(r)-5 b(esp)g(e)g(ct)34 b(to)g(the)f(c)-5 b(onformal)35 b(me)-5 b(asur)g(e)34 b Fp(\027)6 b Fn(.)589 2099 y Fq([3)q(])24 b(then)f(giv)m(es)h(an)f(estimate)h(of)f(the)h(rate)g(of)g(mixing)d (dep)s(ending)g(on)i(the)h(smo)s(oth-)448 2212 y(ness)30 b(of)h(\()p Fp(T)8 b(;)15 b(g)s Fq(\).)42 b(F)-8 b(or)31 b(instance,)448 2420 y Fk(Corollary)k(1.2)46 b Fn(If)g Fp(T)58 b Fn(is)45 b(pie)-5 b(c)g(ewise)46 b(uniformly)g(exp)-5 b(anding)47 b(and)f Fp(g)j Fn(is)c(pie)-5 b(c)g(ewise)448 2533 y(H\177)-46 b(older)41 b(c)-5 b(ontinuous,)42 b(then)e Fq(\()p Fp(T)8 b(;)15 b(\026)1646 2547 y Fm(e)1683 2533 y Fq(\))39 b Fn(has)i(exp)-5 b(onential)41 b(mixing)f(w.r.t.)g(H\177) -46 b(older)40 b(c)-5 b(on-)448 2646 y(tinuous)33 b(observables,)h (i.e.,)734 2759 y Fl(Z)785 2965 y Fm(X)867 2883 y Fp(f)c Fo(\001)20 b Fp(h)h Fo(\016)f Fp(T)1191 2845 y Fm(n)1253 2883 y Fp(d\026)1355 2897 y Fm(e)1417 2883 y Fo(!)1534 2759 y Fl(Z)1584 2965 y Fm(X)1667 2883 y Fp(f)10 b(d\026)1824 2897 y Fm(e)1880 2883 y Fo(\001)1926 2759 y Fl(Z)1976 2965 y Fm(X)2059 2883 y Fp(hd\026)2213 2897 y Fm(e)2436 2883 y Fn(exp)-5 b(onential)5 b(ly)34 b(fast)q Fp(:)448 3124 y Fn(for)g Fp(f)41 b Fn(H\177)-46 b(older)34 b(c)-5 b(ontinuous)33 b(and)h Fp(h)f Fn(b)-5 b(ounde)g(d)34 b(and)g(me)-5 b(asur)g(able.)589 3333 y Fq(\(In)30 b(non-H\177)-45 b(older)30 b(situations,)f([3)q(])i(giv)m(es)f(sub-exp)s(onen)m(tial)f (sp)s(eeds.\))448 3671 y Fk(Commen)m(ts.)35 b Fq(The)20 b(essen)m(tial)h(condition)e(in)g(our)h(Main)g(Theorem,)i(b)s(esides)d (b)s(ounded)448 3784 y(distortion,)25 b(is)g(the)g(smallness)f(of)h (the)h(top)s(ological)f(pressure)f(of)h(the)g(b)s(oundary)-8 b(.)38 b(This)448 3897 y(is)22 b(the)h(same)g(condition)e(as)i(in)e (previous)g(w)m(orks)h([2)q(,)h(3])g(and)f(w)m(e)h(b)s(eliev)m(e)e(it)i (to)g(b)s(e)f(a)h(v)m(ery)448 4010 y(signi\014can)m(t)f(one,)k(in)21 b(fact)j(the)g(k)m(ey)g(to)g(the)f(generalization)g(of)g(man)m(y)g (one-dimensional)448 4123 y(results.)589 4236 y(W)-8 b(e)33 b(w)m(ould)e(lik)m(e)g(to)i(stress)e(t)m(w)m(o)i(features)f(of)g (our)g(pro)s(of.)44 b(The)31 b(\014rst)g(one)h(is)f(that,)448 4349 y(in)f(con)m(trast)i(to)f(most)g(of)g(the)f(previous)f(w)m(orks,)i (w)m(e)g(do)g(not)f(enlarge)h(the)g Fn(dynamic)-5 b(al)448 4462 y Fq(space)35 b Fp(X)42 b Fq(to)35 b(ha)m(v)m(e)h(the)e(transfer)g (op)s(erator)h(acting)f(on)h(con)m(tin)m(uous)f(functions,)g(but)448 4575 y(rather)d(enlarge)f(the)h Fn(functional)g Fq(space)g(b)s(ey)m (ond)e(con)m(tin)m(uous)h(functions.)589 4687 y(The)h(second)h(one)f (is)f(that)i(w)m(e)g(are)g(able)f(to)h(w)m(ork)f(with)f Fn(b)-5 b(ounde)g(d)33 b Fq(functions)d(and)448 4800 y(do)k(not)f(need)h(fancier)e(spaces)i(lik)m(e)f(functions)f(of)i(b)s (ounded)d(v)-5 b(ariation,)34 b(whic)m(h)e(ha)m(v)m(e)448 4913 y(rather)f(subtle)e(prop)s(erties)g(in)g(higher)g(dimensions.)1920 5162 y(5)p eop %%Page: 6 6 6 5 bop 448 848 a Fk(Question.)89 b Fq(As)46 b(w)m(e)h(said,)j(w)m(e)d (w)m(ork)f(under)f(assumptions)g(routinely)g(made)h(in)448 961 y(the)40 b(one-dimensional)e(setting)i([8)q(],)j(except)d(for)g (\(e\))h(whic)m(h)e(is)g(sp)s(eci\014cally)e(m)m(ulti-)448 1074 y(dimensional.)589 1187 y(It)32 b(w)m(ould)f(b)s(e)g(in)m (teresting)g(to)i(kno)m(w)e(whether)g(one)h(can)g(suppress)e(this)h (assump-)448 1300 y(tion,)f(as)h(w)m(ell)e(as)i(\(d\))g(and)e(\(f)7 b(\).)448 1639 y Fk(Ac)m(kno)m(wledgemen)m(ts.)37 b Fq(The)20 b(authors)g(are)h(indebted)d(to)j Fn(pr)-5 b(o)g(gr)g(am)27 b(ESF/PR)n(OD)n(YN)448 1752 y Fq(whic)m(h)35 b(has)h(partially)e(supp)s (orted)g(the)i Fn(International)41 b(Confer)-5 b(enc)g(e)38 b(on)h(Dynamic)-5 b(al)448 1865 y(Systems,)30 b(A)n(bb)-5 b(ey)26 b(of)i(L)-5 b(a)28 b(Bussi)n(\022)-44 b(er)-5 b(e)32 b Fq(where)24 b(most)i(of)f(this)e(w)m(ork)i(w)m(as)g(carried)f (through.)448 1978 y(J.B.)39 b(thanks)f(also)h(the)f Fn(L)-5 b(ab)g(or)g(atoir)g(e)44 b(de)c(T)-7 b(op)i(olo)g(gie)42 b(de)e(Dijon)46 b Fq(and)38 b(the)g Fn(Michigan)448 2090 y(State)c(University)k Fq(Mathematics)31 b(Departmen)m(t)h(where)e(it)g (w)m(as)g(completed.)448 2372 y Fr(2)135 b(Construction)45 b(of)h(a)f(p)t(ositiv)l(e)h(eigenfunctional.)448 2575 y Fq(Let)39 b(\()p Fp(X)r(;)15 b Fo(Z)7 b Fp(;)15 b(T)8 b(;)15 b(g)s Fq(\))42 b(b)s(e)37 b(as)i(in)e(the)i(Main)f(Theorem.)65 b(A)39 b(standard,)h(k)m(ey)f(to)s(ol)f(is)g(the)448 2688 y Fn(tr)-5 b(ansfer)38 b(op)-5 b(er)g(ator)47 b Fq(asso)s(ciated)35 b(to)g(\()p Fp(T)8 b(;)15 b(g)s Fq(\).)54 b(It)34 b(is)f(the)i(map)f Fo(L)g Fq(acting)g(on)g(functions)448 2801 y(according)d(to:)1389 2922 y(\()p Fo(L)p Fp(f)10 b Fq(\)\()p Fp(x)p Fq(\))25 b(=)1883 2836 y Fl(X)1820 3038 y Fm(y)r Fi(2)p Fm(T)1955 3019 y Ff(\000)p Fe(1)2038 3038 y Fm(x)2093 2922 y Fp(g)s Fq(\()p Fp(y)s Fq(\))c Fo(\001)g Fp(f)10 b Fq(\()p Fp(y)s Fq(\))448 3169 y(whic)m(h)29 b(also)i(can)f(b)s(e)g(written:)1412 3341 y Fo(L)p Fp(f)k Fq(=)1662 3254 y Fl(X)1650 3451 y Fm(Z)5 b Fi(2Z)1805 3341 y Fq(\()p Fp(g)s(f)10 b Fq(\))21 b Fo(\016)g Fp(T)2129 3303 y Fi(\000)p Fh(1)2116 3369 y Fm(Z)2243 3341 y Fo(\001)g Fk(1)2341 3355 y Fm(T)10 b(Z)2449 3341 y Fp(:)448 3600 y Fq(Under)25 b(condition)f(\(b\))h(of)h(the)f(Main)g(Theorem,)h(sup) 2281 3622 y Fm(Y)2357 3600 y Fp(g)j(<)c Fo(1)g Fq(so)g(that)h(sup)3076 3622 y Fm(X)3159 3600 y Fo(L)p Fk(1)3274 3614 y Fm(X)3366 3600 y Fp(<)448 3713 y Fo(1)p Fq(:)48 b Fo(L)33 b Fq(acts)i(on)e(the)h (Banac)m(h)h(space)f Fo(B)i Fq(of)e(b)s(ounded)e(functions.)49 b(Finally)32 b(let)h Fo(B)3287 3680 y Fi(0)3344 3713 y Fq(b)s(e)448 3826 y(the)e(top)s(ological)f(dual)f(space)i(of)f Fo(B)j Fq(and)d Fo(L)1948 3793 y Fi(\003)2017 3826 y Fq(b)s(e)g(the)h(dual)e(op)s(erator)h(of)h Fo(L)p Fq(,)f(i.e.)1616 3997 y Fo(L)1679 3960 y Fi(\003)1718 3997 y Fp(\013)p Fq(\()p Fp(f)10 b Fq(\))26 b(=)f Fp(\013)p Fq(\()p Fo(L)p Fp(f)10 b Fq(\))448 4169 y(for)30 b(all)g Fp(\013)25 b Fo(2)56 b(B)977 4136 y Fi(0)1030 4169 y Fq(and)29 b Fp(f)35 b Fo(2)55 b(B)s Fq(.)589 4282 y(Observ)m(e)37 b(that)g(a)g(probabilit)m(y)d(measure)i(b)s(elongs)g(to)h Fo(B)2574 4249 y Fi(0)2633 4282 y Fq(and)f(that)h(b)s(eing)e(con-)448 4395 y(formal)41 b(is)g(the)g(same)h(as)g(b)s(eing)e(an)h (eigenfunctional)f(for)i Fo(L)2624 4362 y Fi(\003)2704 4395 y Fq(and)f(some)h(p)s(ositiv)m(e)448 4508 y(eigen)m(v)-5 b(alue.)41 b(Hence)31 b(the)g(\014rst)e(step)i(of)f(the)h(pro)s(of)f (of)g(the)h(Main)f(Theorem)g(is)f(the)448 4687 y Fk(Prop)s(osition)37 b(2.1)46 b Fn(Ther)-5 b(e)38 b(exists)g(a)g(p)-5 b(ositive)38 b(functional)g Fq(\003)d Fo(2)e(B)2816 4654 y Fi(0)2876 4687 y Fn(with)39 b Fq(\003\()p Fk(1)3229 4701 y Fm(X)3297 4687 y Fq(\))34 b(=)448 4800 y(\003\()p Fk(1)598 4814 y Fm(Y)660 4800 y Fq(\))25 b(=)g(1)33 b Fn(such)g(that)1741 4913 y Fo(L)1804 4876 y Fi(\003)1844 4913 y Fq(\003)25 b(=)g Fp(\025)p Fq(\003)1920 5162 y(6)p eop %%Page: 7 7 7 6 bop 448 623 a Fn(for)34 b(some)f(numb)-5 b(er)33 b Fq(0)26 b Fp(<)f(\025)g(<)g Fo(1)p Fn(.)448 835 y Fk(Remark)35 b(2.1)46 b Fn(A)n(ny)37 b(such)h Fq(\003)f Fn(has)i(ful)5 b(l)38 b(supp)-5 b(ort:)54 b Fq(\003\()p Fk(1)2388 849 y Fm(U)2447 835 y Fq(\))35 b Fp(>)f Fq(0)39 b Fn(for)f(al)5 b(l)38 b(non-empty,)448 948 y(op)-5 b(en)25 b(subsets)e Fp(U)35 b Fo(\032)25 b Fp(X)7 b Fn(.)39 b(This)24 b(is)f(an)h(imme)-5 b(diate)25 b(c)-5 b(onse)g(quenc)g(e)24 b(of)f(c)-5 b(onditions)26 b Fq(\(d\))d Fn(and)448 1061 y Fq(\(f)7 b(\))p Fn(.)42 b(Namely,)33 b(take)f Fp(U)43 b Fn(a)32 b(non-emtpy)h(op)-5 b(en)34 b(set)e(and)h Fp(n)f Fn(such)g(that)h Fp(T)2881 1028 y Fm(n)2928 1061 y Fq(\()p Fp(U)d Fo(\\)19 b Fp(Y)h Fq(\))25 b Fo(\033)g Fp(Y)448 1174 y Fn(then:)916 1394 y Fq(\003\()p Fk(1)1066 1408 y Fm(U)1126 1394 y Fq(\))83 b(=)g(\003\()p Fk(1)1548 1408 y Fm(U)7 b Fi(\\)p Fm(Y)1711 1394 y Fq(\))26 b Fo(\025)1905 1332 y Fq(1)p 1878 1373 100 4 v 1878 1456 a Fp(\025)1931 1430 y Fm(n)1988 1394 y Fo(L)2051 1356 y Fm(n)2097 1394 y Fk(1)2149 1408 y Fm(U)7 b Fi(\\)p Fm(Y)1244 1619 y Fo(\025)1435 1557 y Fq(1)p 1408 1598 V 1408 1681 a Fp(\025)1461 1655 y Fm(n)1518 1619 y Fq(\(inf)1580 1679 y Fm(Y)1679 1619 y Fo(L)1742 1581 y Fm(n)1789 1619 y Fk(1)1841 1633 y Fm(U)g Fi(\\)p Fm(Y)2003 1619 y Fq(\)\003\()p Fk(1)2188 1633 y Fm(Y)2250 1619 y Fq(\))26 b Fo(\025)2417 1557 y Fq(\(inf)2563 1571 y Fm(Y)2639 1557 y Fp(g)s Fq(\))2720 1524 y Fm(n)p 2417 1598 351 4 v 2542 1681 a Fp(\025)2595 1655 y Fm(n)2803 1619 y Fp(>)f Fq(0)p Fp(:)448 1860 y Fk(Remark)35 b(2.2)46 b Fn(A)n(ny)35 b(such)i Fq(\003)f Fn(do)-5 b(esn)c('t)37 b(give)e(mass)i(to)g(the)g(b)-5 b(oundary:)50 b Fo(L)3020 1827 y Fm(n)3067 1860 y Fk(1)3119 1875 y Fm(@)t Fi(Z)3211 1883 y Fg(n)3289 1860 y Fq(=)32 b(0)448 1972 y Fn(and)1347 2101 y Fq(\003\()p Fk(1)1497 2116 y Fm(@)t Fi(Z)1589 2124 y Fg(n)1636 2101 y Fq(\))26 b(=)1830 2039 y(1)p 1803 2080 100 4 v 1803 2163 a Fp(\025)1856 2137 y Fm(n)1912 2101 y Fq(\003\()p Fo(L)2073 2063 y Fm(n)2121 2101 y Fk(1)2173 2116 y Fm(@)t Fi(Z)2265 2124 y Fg(n)2311 2101 y Fq(\))g(=)f(0)p Fp(:)589 2334 y Fq(Since)f(w)m(e)h(are)f(lo)s(oking)f (for)h(a)h(probabilit)m(y)d(measure,)k(w)m(e)e(will)e(restrict)i(the)h (action)448 2447 y(of)31 b Fo(L)615 2414 y Fi(\003)684 2447 y Fq(to)g(the)g(follo)m(wing)e(subset)h Fo(C)35 b Fq(of)c Fo(B)1865 2414 y Fi(0)1887 2447 y Fq(:)708 2651 y Fo(C)f Fq(=)25 b Fo(f)p Fp(\013)i Fo(2)d(B)1160 2614 y Fi(0)1183 2651 y Fp(;)46 b(\013)p Fq(\()p Fk(1)1399 2665 y Fm(X)1467 2651 y Fq(\))26 b(=)f Fp(\013)p Fq(\()p Fk(1)1769 2665 y Fm(Y)1831 2651 y Fq(\))g(=)g(1)31 b(and)f Fp(\013)p Fq(\()p Fp(f)10 b Fq(\))26 b Fo(\025)f Fq(0)30 b(for)h(all)e Fp(f)34 b Fo(\025)25 b Fq(0)p Fo(g)p Fp(:)448 2856 y Fq(W)-8 b(e)42 b(shall)e(consider)f(the)i Fn(we)-5 b(ak)44 b(star)f(top)-5 b(olo)g(gy)52 b Fq(on)41 b Fo(B)2358 2823 y Fi(0)2421 2856 y Fq(and)f(its)h(subset)f Fo(C)46 b Fq(without)448 2969 y(further)29 b(notice.)589 3081 y(The)h(normalized)f(op)s(erator)i Fo(N)38 b Fq(:)26 b Fo(C)k(!)25 b(C)36 b Fq(is)29 b(de\014ned)g(b)m(y:)1199 3319 y Fo(N)13 b Fp(\013)26 b Fq(=)1626 3257 y(1)p 1476 3298 345 4 v 1476 3381 a Fo(L)1539 3355 y Fi(\003)1579 3381 y Fp(\013)p Fq(\()p Fk(1)1724 3395 y Fm(Y)1785 3381 y Fq(\))1831 3319 y Fo(L)1894 3281 y Fi(\003)1933 3319 y Fp(\013)182 b Fq(for)30 b(all)g Fp(\013)25 b Fo(2)g(C)5 b Fp(:)448 3567 y Fq(Th)m(us)35 b(w)m(e)h(are)g(lo)s(oking)e(for)h (\014xed)g(p)s(oin)m(ts)g(of)g Fo(N)47 b Fq(:)34 b Fo(C)39 b(!)34 b(C)5 b Fq(.)57 b(T)-8 b(o)36 b(\014nd)e(them,)j(w)m(e)f(are)448 3680 y(going)31 b(to)h(apply)d(the)i(Sc)m(hauder-T)m(yc)m(hono\013)h (theorem)f([6)q(,)g(V.10.5].)44 b(T)-8 b(o)32 b(b)s(egin)d(with,)448 3793 y(remark)34 b(that)h Fo(C)k Fq(is)33 b(ob)m(viously)f(a)j(con)m(v) m(ex)g(subset)f(of)g(the)g(top)s(ological)g(v)m(ector)h(space)448 3905 y Fo(B)511 3872 y Fi(0)534 3905 y Fq(,)d(whic)m(h)e(is)h(lo)s (cally)f(con)m(v)m(ex.)46 b(Also)31 b Fo(C)37 b Fq(is)31 b(non-empt)m(y)g(as)h(it)f(con)m(tains)h(for)f(instance)448 4018 y Fp(\016)488 4032 y Fm(x)567 4018 y Fq(\()p Fp(\016)642 4032 y Fm(x)686 4018 y Fq(\()p Fp(f)10 b Fq(\))32 b(=)f Fp(f)10 b Fq(\()p Fp(x)p Fq(\),)35 b Fp(x)c Fo(2)g Fp(X)42 b Fq(is)33 b(giv)m(en\).)52 b(Th)m(us)33 b(it)h(remains)f(to)i(c)m(hec) m(k)g(the)g(follo)m(wing)448 4131 y(h)m(yp)s(othesis:)556 4319 y Fn(1.)46 b Fo(N)38 b Fq(:)25 b Fo(C)31 b(!)25 b(C)38 b Fn(is)32 b(wel)5 b(l-de\014ne)-5 b(d.)556 4507 y(2.)46 b Fo(C)37 b Fn(is)c(c)-5 b(omp)g(act.)556 4694 y(3.)46 b Fo(N)38 b Fq(:)25 b Fo(C)31 b(!)25 b(C)38 b Fn(is)32 b(c)-5 b(ontinuous.)1920 5162 y Fq(7)p eop %%Page: 8 8 8 7 bop 448 623 a Fk(Condition)35 b(1:)41 b Fo(N)d Fq(:)26 b Fo(C)k(!)25 b(C)36 b Fq(is)29 b(w)m(ell-de\014ned.)589 736 y(T)-8 b(o)31 b(c)m(hec)m(k)h(this)e(condition,)f(tak)m(e)j Fp(\013)25 b Fo(2)g(C)5 b Fq(.)589 848 y(First)35 b(w)m(e)g(claim)f (that)i Fo(N)13 b Fq(\()p Fp(\013)p Fq(\))34 b(=)f(\()p Fo(L)1855 815 y Fi(\003)1895 848 y Fp(\013)p Fq(\(1)2033 862 y Fm(Y)2095 848 y Fq(\)\))2165 815 y Fi(\000)p Fh(1)2260 848 y Fo(L)2323 815 y Fi(\003)2362 848 y Fp(\013)i Fq(is)f(w)m (ell-de\014ned)g(as)h(an)g(ele-)448 961 y(men)m(t)c(of)f Fo(B)844 928 y Fi(0)867 961 y Fq(.)40 b(Indeed,)30 b Fo(L)1316 928 y Fi(\003)1355 961 y Fp(\013)c Fo(2)e(B)1587 928 y Fi(0)1640 961 y Fq(b)m(y)30 b(construction)g(and)f Fo(L)2526 928 y Fi(\003)2565 961 y Fp(\013)p Fq(\(1)2703 975 y Fm(Y)2765 961 y Fq(\))d Fp(>)f Fq(0:)41 b(using)28 b(that)448 1074 y Fp(\013)p Fq(\(1)586 1088 y Fm(Y)648 1074 y Fq(\))e(=)f(1)31 b(and)e(the)i(p)s(ositivit)m(y)e(of)h Fp(\013)1128 1274 y(\013)p Fq(\()p Fo(L)p Fk(1)1336 1288 y Fm(Y)1397 1274 y Fq(\))c Fo(\025)f Fq(\(inf)1616 1335 y Fm(Y)1715 1274 y Fo(L)p Fk(1)1830 1288 y Fm(Y)1891 1274 y Fq(\))p Fp(\013)p Fq(\()p Fk(1)2071 1288 y Fm(Y)2133 1274 y Fq(\))h(=)f(inf)2317 1335 y Fm(Y)2415 1274 y Fo(L)p Fk(1)2530 1288 y Fm(Y)2616 1274 y Fp(>)g Fq(0)448 1508 y(b)m(y)31 b(condition)e(\(d\))h(of)h(the)f(Main)g(Theorem.)589 1621 y(Second,)h Fo(N)13 b Fp(\013)31 b Fq(is)e(p)s(ositiv)m(e,)h(\()p Fo(N)13 b Fp(\013)p Fq(\)\(1)1849 1635 y Fm(Y)1911 1621 y Fq(\))26 b(=)f(1)31 b(b)m(y)f(construction)g(and)609 1877 y Fo(N)13 b Fp(\013)p Fq(\()p Fk(1)842 1891 y Fm(X)910 1877 y Fq(\))26 b(=)1207 1816 y(1)p 1077 1857 305 4 v 1077 1940 a Fp(\013)p Fq(\()p Fo(L)p Fk(1)1285 1954 y Fm(Y)1346 1940 y Fq(\))1392 1877 y Fp(\013)p Fq(\()p Fo(L)p Fk(1)1600 1891 y Fm(X)1668 1877 y Fq(\))g(=)1834 1807 y Fp(\013)p Fq(\()p Fo(L)p Fk(1)2042 1825 y Fm(X)5 b Fi(n)p Fm(Y)2203 1807 y Fq(\))20 b(+)g Fp(\013)p Fq(\()p Fo(L)p Fk(1)2557 1821 y Fm(Y)2619 1807 y Fq(\))p 1834 1857 820 4 v 2092 1940 a Fp(\013)p Fq(\()p Fo(L)p Fk(1)2300 1954 y Fm(Y)2361 1940 y Fq(\))2689 1877 y(=)2795 1816 y Fp(\013)p Fq(\()p Fo(L)p Fk(1)3003 1830 y Fm(Y)3065 1816 y Fq(\))p 2795 1857 305 4 v 2795 1940 a Fp(\013)p Fq(\()p Fo(L)p Fk(1)3003 1954 y Fm(Y)3065 1940 y Fq(\))3135 1877 y(=)25 b(1)448 2126 y(as)31 b Fo(L)p Fq(1)668 2145 y Fm(X)5 b Fi(n)p Fm(Y)853 2126 y Fq(=)25 b(0)30 b(\(only)g(p)s(oin)m (ts)f(in)g Fp(Y)51 b Fq(ha)m(v)m(e)31 b(an)g(image\).)589 2239 y(Th)m(us,)f(condition)f(1)i(is)e(satis\014ed.)589 2578 y(Recall)h(the)448 2786 y Fk(Theorem)k(2.2)46 b Fn(\(Banach-A)n(lao)-5 b(glu)28 b([6)q(,)f(V.4.2]\):)39 b(The)27 b(unit)f(b)-5 b(al)5 b(l)28 b(of)e(the)h(dual)h(of)e(any)448 2899 y(norme)-5 b(d)35 b(ve)-5 b(ctor)33 b(sp)-5 b(ac)g(e)34 b(is)f(we)-5 b(akly*)34 b(c)-5 b(omp)g(act.)589 3107 y Fq(Th)m(us,)31 b Fp(B)915 3121 y Fh(1)980 3107 y Fq(=)26 b Fo(f)p Fp(\013)i Fo(2)d(B)1357 3074 y Fi(0)1406 3107 y Fq(:)i Fo(k)p Fp(\013)p Fo(k)h(\024)d Fq(1)p Fo(g)p Fq(,)33 b(the)e(unit)e(ball)h(of)h Fo(B)2572 3074 y Fi(0)2595 3107 y Fq(,)g(is)f(compact)i(and)e(it)h(is)448 3220 y(enough)f(to)i(c)m (hec)m(k)g(that)e Fo(C)36 b Fq(is)29 b(a)i(closed)f(subset)g(of)h Fo(B)2276 3234 y Fh(1)2345 3220 y Fq(to)g(establish)448 3559 y Fk(Condition)k(2:)41 b Fo(C)35 b Fq(is)30 b(compact.)589 3672 y(T)-8 b(o)31 b(c)m(hec)m(k)h(it,)e(let)h Fp(\013)25 b Fo(2)g(C)36 b Fq(and)30 b Fp(f)k Fo(2)25 b(B)33 b Fq(with)c(sup)2248 3694 y Fm(X)2330 3672 y Fo(j)p Fp(f)10 b Fo(j)25 b(\024)g Fq(1:)1482 3872 y(0)h Fo(\024)f Fq(sup)1686 3950 y Fm(X)1801 3872 y Fo(j)p Fp(f)10 b Fo(j)20 b(\000)g Fp(f)34 b Fo(\024)25 b Fq(1)c Fo(\000)f Fp(f)448 4123 y Fq(so)30 b(that)g Fp(\013)p Fq(\(1)19 b Fo(\000)f Fp(f)10 b Fq(\))25 b Fo(\025)g Fq(0)k(and,)g(since)g Fp(\013)p Fq(\(1)19 b Fo(\000)f Fp(f)10 b Fq(\))25 b(=)g(1)18 b Fo(\000)g Fp(\013)p Fq(\()p Fp(f)10 b Fq(\),)30 b Fp(\013)p Fq(\()p Fp(f)10 b Fq(\))26 b Fo(\024)f Fq(1.)41 b(In)28 b(the)i(same)448 4236 y(w)m(a)m(y:)42 b Fo(\000)p Fp(\013)p Fq(\()p Fp(f)10 b Fq(\))25 b Fo(\024)g Fq(1.)42 b(Therefore)30 b Fp(\013)25 b Fo(2)g Fp(B)1805 4250 y Fh(1)1875 4236 y Fq(and)30 b(w)m(e)h(ha)m(v)m(e)g(sho)m(wn)f Fo(C)g(\032)25 b Fp(B)2914 4250 y Fh(1)2954 4236 y Fq(.)589 4349 y(Recall)34 b(that)g(for)g(an)m (y)g Fp(f)40 b Fo(2)31 b(B)s Fq(,)j Fp(\013)e Fo(7!)f Fp(\013)p Fq(\()p Fp(f)10 b Fq(\))34 b(is)f(con)m(tin)m(uous)h(b)m(y)f (de\014nition)f(of)i(the)448 4462 y(w)m(eak*)f(top)s(ology)-8 b(.)43 b(Hence)32 b Fo(C)5 b Fq(,)32 b(whic)m(h)e(is)g(de\014ned)g(b)m (y)h(a)g(collection)g(of)g(inequalities)e(of)448 4575 y(the)i(t)m(yp)s(e)f Fp(\013)p Fq(\()p Fp(f)10 b Fq(\))26 b Fo(\024)f Fq(const)q(,)30 b(is)g(closed)g(and)g(therefore)g(compact.) 448 4913 y Fk(Condition)35 b(3:)41 b Fo(N)d Fq(:)26 b Fo(C)k(!)25 b(C)36 b Fq(is)29 b(con)m(tin)m(uous.)1920 5162 y(8)p eop %%Page: 9 9 9 8 bop 589 623 a Fq(Recall)30 b(that)h(a)g(basis)e(for)i(the)f(w)m (eak*)i(top)s(ology)f(of)f Fo(B)2468 590 y Fi(0)2521 623 y Fq(is)g(giv)m(en)g(b)m(y)g(the)h(sets:)638 818 y Fp(V)20 b Fq(\()p Fp(\013)804 832 y Fh(0)844 818 y Fp(;)15 b(";)g(f)1011 832 y Fh(1)1051 818 y Fp(;)g(:)g(:)g(:)i(;)e(f) 1298 832 y Fm(n)1345 818 y Fq(\))26 b(:=)f Fo(f)p Fp(\013)h Fo(2)f(B)1805 780 y Fi(0)1853 818 y Fq(:)g Fo(8)p Fp(i)g Fq(=)g(1)p Fp(;)15 b(:)g(:)g(:)i(;)e(n)26 b Fo(j)p Fp(\013)p Fq(\()p Fp(f)2597 832 y Fm(i)2625 818 y Fq(\))21 b Fo(\000)f Fp(\013)2830 832 y Fh(0)2870 818 y Fq(\()p Fp(f)2950 832 y Fm(i)2978 818 y Fq(\))p Fo(j)25 b Fp(<)g(")p Fo(g)448 1013 y Fq(with)38 b Fp(\013)722 1027 y Fh(0)801 1013 y Fo(2)h(B)964 980 y Fi(0)987 1013 y Fq(,)i Fp(")f(>)f Fq(0,)j Fp(n)d Fq(=)g(1)p Fp(;)15 b Fq(2)p Fp(;)g(:)g(:)g(:)42 b Fq(and)c Fp(f)2108 1027 y Fh(1)2147 1013 y Fp(;)15 b(:)g(:)g(:)i(;)e(f)2394 1027 y Fm(n)2480 1013 y Fo(2)39 b(B)s Fq(.)66 b(Th)m(us)38 b(to)i(establish)448 1126 y(condition)26 b(3,)j(i.e.,)g(con)m(tin)m(uit)m(y)-8 b(,)28 b(w)m(e)g(ha)m(v)m(e)h(to)f(\014nd,)f(for)g(ev)m(ery)i Fp(V)20 b Fq(\()p Fo(N)13 b Fp(\013)2836 1140 y Fh(0)2876 1126 y Fp(;)i(";)g(f)3043 1140 y Fh(1)3083 1126 y Fp(;)g(:)g(:)g(:)i(;) e(f)3330 1140 y Fm(n)3377 1126 y Fq(\),)448 1239 y(a)31 b(n)m(um)m(b)s(er)e Fp(\021)g(>)c Fq(0)31 b(and)e Fp(g)1320 1253 y Fh(1)1360 1239 y Fp(;)15 b(:)g(:)g(:)i(;)e(g)1605 1253 y Fm(m)1697 1239 y Fo(2)25 b(B)33 b Fq(suc)m(h)d(that:)1028 1434 y Fo(N)13 b Fp(V)20 b Fq(\()p Fp(\013)1282 1448 y Fh(0)1322 1434 y Fp(;)15 b(\021)s(;)g(g)1493 1448 y Fh(1)1534 1434 y Fp(;)g(:)g(:)g(:)h(;)f(g)1778 1448 y Fm(m)1846 1434 y Fq(\))25 b Fo(\032)g Fp(V)c Fq(\()p Fo(N)13 b Fp(\013)2257 1448 y Fh(0)2297 1434 y Fp(;)i(";)g(f)2464 1448 y Fh(1)2504 1434 y Fp(;)g(:)g(:)g(:)h(;)f(f)2750 1448 y Fm(n)2797 1434 y Fq(\))p Fp(:)589 1629 y Fq(W)-8 b(e)32 b(claim)d(that)i(one)g(can)g(tak)m(e)h Fp(m)25 b Fq(=)g Fp(n)20 b Fq(+)f(1)31 b(and:)1091 1824 y Fp(\021)e Fq(=)c Fp(")1318 1724 y Fl(\020)1860 1780 y Fh(\(inf)1973 1791 y Fg(Y)2038 1780 y Fi(L)p Fh(1)2122 1791 y Fg(Y)2175 1780 y Fh(\))2202 1756 y Fe(2)p 1383 1804 1332 4 v 1383 1857 a Fh(inf)1468 1868 y Fg(Y)1533 1857 y Fi(L)p Fh(1)1617 1868 y Fg(Y)1671 1857 y Fh(+\(sup)1859 1874 y Fg(X)1929 1857 y Fi(L)p Fh(1)2013 1868 y Fg(Y)2066 1857 y Fh(\))12 b(max)2236 1869 y Fe(1)p Ff(\024)p Fg(i)p Ff(\024)p Fg(n)2439 1857 y Fh(sup)2545 1874 y Fg(X)2615 1857 y Fi(j)p Fm(f)2669 1867 y Fg(i)2695 1857 y Fi(j)2725 1724 y Fl(\021)1223 1980 y Fq(and)30 b Fp(g)1443 1994 y Fh(1)1508 1980 y Fq(=)25 b Fo(L)p Fp(f)1712 1994 y Fh(1)1751 1980 y Fp(;)15 b(:)g(:)g(:)h(;)f(g)1995 1994 y Fm(n)2068 1980 y Fq(=)25 b Fo(L)p Fp(f)2272 1994 y Fm(m)2338 1980 y Fp(;)15 b(g)2421 1994 y Fm(n)p Fh(+1)2584 1980 y Fq(=)25 b Fo(L)p Fq(1)2788 1994 y Fm(Y)2849 1980 y Fp(:)448 2175 y Fq(Note)32 b(that)f(for)f(all)f Fp(j)i Fq(=)25 b(1)p Fp(;)15 b(:)g(:)g(:)i(;)e(n)20 b Fq(+)g(1,)31 b Fp(g)1851 2189 y Fm(j)1914 2175 y Fo(2)24 b(B)s Fq(.)589 2288 y(T)-8 b(o)32 b(pro)m(v)m(e)h(the)e(claim)g(assume) g Fp(\013)d Fo(2)e Fp(V)20 b Fq(\()p Fp(\013)2026 2302 y Fh(0)2066 2288 y Fp(;)15 b(";)g(g)2231 2302 y Fh(1)2272 2288 y Fp(;)g(:)g(:)g(:)i(;)e(g)2517 2302 y Fm(n)p Fh(+1)2655 2288 y Fq(\).)44 b(By)32 b(de\014nition,)e(w)m(e)448 2401 y(ha)m(v)m(e)i(for)e(all)f Fp(j)i Fq(=)25 b(1)p Fp(;)15 b(:)g(:)g(:)i(;)e(n)20 b Fq(+)g(1:)1536 2596 y Fo(j)p Fp(\013)p Fq(\()p Fp(g)1697 2610 y Fm(j)1735 2596 y Fq(\))g Fo(\000)g Fp(\013)1939 2610 y Fh(0)1979 2596 y Fq(\()p Fp(g)2057 2610 y Fm(j)2094 2596 y Fq(\))p Fo(j)26 b(\024)f Fp(\021)s(:)448 2791 y Fq(Th)m(us)570 3002 y Fo(jN)13 b Fp(\013)p Fq(\()p Fp(f)821 3016 y Fm(i)850 3002 y Fq(\))21 b Fo(\000)e(N)13 b Fp(\013)1142 3016 y Fh(0)1182 3002 y Fq(\()p Fp(f)1262 3016 y Fm(i)1290 3002 y Fq(\))p Fo(j)84 b(\024)e(j)1772 2940 y Fq(1)p 1622 2981 345 4 v 1622 3064 a Fo(L)1685 3038 y Fi(\003)1725 3064 y Fp(\013)p Fq(\()p Fk(1)1870 3078 y Fm(Y)1932 3064 y Fq(\))1977 3002 y Fo(L)2040 2964 y Fi(\003)2079 3002 y Fp(\013)p Fq(\()p Fp(f)2217 3016 y Fm(i)2246 3002 y Fq(\))20 b Fo(\000)2552 2940 y Fq(1)p 2402 2981 V 2402 3064 a Fo(L)2465 3038 y Fi(\003)2504 3064 y Fp(\013)p Fq(\()p Fk(1)2649 3078 y Fm(Y)2711 3064 y Fq(\))2756 3002 y Fo(L)2819 2964 y Fi(\003)2859 3002 y Fp(\013)2917 3016 y Fh(0)2956 3002 y Fq(\()p Fp(f)3036 3016 y Fm(i)3064 3002 y Fq(\))p Fo(j)1587 3252 y Fq(+)1673 3120 y Fl(\014)1673 3175 y(\014)1673 3229 y(\014)1673 3284 y(\014)1863 3191 y Fq(1)p 1714 3231 V 1714 3314 a Fo(L)1777 3288 y Fi(\003)1816 3314 y Fp(\013)p Fq(\()p Fk(1)1961 3328 y Fm(Y)2023 3314 y Fq(\))2088 3252 y Fo(\000)2358 3191 y Fq(1)p 2189 3231 384 4 v 2189 3314 a Fo(L)2252 3288 y Fi(\003)2291 3314 y Fp(\013)2349 3328 y Fh(0)2389 3314 y Fq(\()p Fk(1)2476 3328 y Fm(Y)2537 3314 y Fq(\))2582 3120 y Fl(\014)2582 3175 y(\014)2582 3229 y(\014)2582 3284 y(\014)2628 3252 y Fo(jL)2716 3214 y Fi(\003)2755 3252 y Fp(\013)2813 3266 y Fh(0)2853 3252 y Fq(\()p Fp(f)2933 3266 y Fm(i)2961 3252 y Fq(\))p Fo(j)1434 3492 y(\024)1756 3430 y Fq(1)p 1597 3471 363 4 v 1597 3554 a(inf)1708 3568 y Fm(Y)1784 3554 y Fo(L)p Fk(1)1899 3568 y Fm(Y)1970 3492 y Fo(j)p Fp(\013)p Fq(\()p Fo(L)p Fp(f)2196 3506 y Fm(i)2224 3492 y Fq(\))h Fo(\000)f Fp(\013)2429 3506 y Fh(0)2468 3492 y Fq(\()p Fo(L)p Fp(f)2611 3506 y Fm(i)2639 3492 y Fq(\))p Fo(j)1587 3721 y Fq(+)1882 3660 y(1)p 1668 3700 473 4 v 1668 3784 a(\(inf)1814 3798 y Fm(Y)1890 3784 y Fo(L)p Fk(1)2005 3798 y Fm(Y)2066 3784 y Fq(\))2101 3757 y Fh(2)2150 3721 y Fo(j)p Fp(\013)p Fq(\()p Fo(L)p Fk(1)2383 3735 y Fm(Y)2445 3721 y Fq(\))h Fo(\000)f Fp(\013)2650 3735 y Fh(0)2689 3721 y Fq(\()p Fo(L)p Fk(1)2839 3735 y Fm(Y)2900 3721 y Fq(\))p Fo(jj)p Fp(\013)3043 3735 y Fh(0)3084 3721 y Fq(\()p Fo(L)p Fp(f)3227 3735 y Fm(i)3255 3721 y Fq(\))p Fo(j)1434 3940 y(\024)1754 3879 y Fp(\021)p 1597 3919 363 4 v 1597 4003 a Fq(inf)1708 4017 y Fm(Y)1784 4003 y Fo(L)p Fk(1)1899 4017 y Fm(Y)1990 3940 y Fq(+)2303 3879 y Fp(\021)p 2091 3919 473 4 v 2091 4003 a Fq(\(inf)2237 4017 y Fm(Y)2313 4003 y Fo(L)p Fk(1)2428 4017 y Fm(Y)2488 4003 y Fq(\))2523 3976 y Fh(2)2573 3940 y Fo(j)p Fp(\013)2656 3954 y Fh(0)2696 3940 y Fq(\()p Fo(L)p Fp(f)2839 3954 y Fm(i)2867 3940 y Fq(\))p Fo(j)p Fp(:)448 4178 y Fq(Moreo)m(v)m(er,) 1138 4291 y Fo(j)p Fp(\013)1221 4305 y Fh(0)1261 4291 y Fq(\()p Fo(L)p Fp(f)1404 4305 y Fm(i)1431 4291 y Fq(\))p Fo(j)26 b(\024)f Fq(sup)1650 4369 y Fm(X)1765 4291 y Fo(jL)p Fp(f)1898 4305 y Fm(i)1926 4291 y Fo(j)g(\024)g Fq(sup)2109 4369 y Fm(X)2224 4291 y Fo(jL)p Fk(1)2364 4305 y Fm(X)2432 4291 y Fo(j)15 b Fq(sup)2509 4369 y Fm(X)2624 4291 y Fo(j)p Fp(f)2694 4305 y Fm(i)2722 4291 y Fo(j)448 4499 y Fq(whic)m(h)29 b(giv)m(es:)640 4737 y Fo(jN)13 b Fp(\013)p Fq(\()p Fp(f)891 4751 y Fm(i)920 4737 y Fq(\))21 b Fo(\000)e(N)13 b Fp(\013)1212 4751 y Fh(0)1252 4737 y Fq(\()p Fp(f)1332 4751 y Fm(i)1360 4737 y Fq(\))p Fo(j)84 b(\024)1821 4676 y Fp(\021)p 1667 4716 356 4 v 1667 4799 a Fq(inf)1778 4813 y Fm(Y)1854 4799 y Fo(L)p Fq(1)1962 4813 y Fm(Y)2053 4737 y Fq(+)2154 4676 y Fp(\021)18 b Fq(sup)2354 4697 y Fm(x)2413 4676 y Fo(L)p Fq(1)2521 4690 y Fm(Y)2597 4676 y Fq(max)2766 4690 y Fm(i)p Fi(2)p Fm(I)2893 4676 y Fq(sup)3030 4697 y Fm(X)3112 4676 y Fo(j)p Fp(f)3182 4690 y Fm(i)3210 4676 y Fo(j)p 2154 4716 1082 4 v 2462 4799 a Fq(\(inf)2608 4813 y Fm(Y)2684 4799 y Fo(L)p Fq(1)2792 4813 y Fm(Y)2852 4799 y Fq(\))2887 4773 y Fh(2)1504 4913 y Fo(\024)82 b Fp(":)1920 5162 y Fq(9)p eop %%Page: 10 10 10 9 bop 448 623 a Fq(This)29 b(pro)m(v)m(es)i(condition)e(3.)589 736 y(W)-8 b(e)29 b(are)f(no)m(w)f(in)g(p)s(osition)e(to)j(apply)e(the) i(Sc)m(hauder-T)m(yc)m(hono\013)g(theorem:)40 b(there)448 848 y(exists)30 b(\003)c Fo(2)f(C)35 b Fq(suc)m(h)30 b(that)h Fo(N)13 b Fq(\(\003\))26 b(=)f(\003,)31 b(i.e.:)1741 1046 y Fo(L)1804 1009 y Fi(\003)1844 1046 y Fq(\003)25 b(=)g Fp(\025)p Fq(\003)448 1244 y(with)k Fp(\025)d Fq(=)f(\003\()p Fo(L)p Fk(1)1043 1258 y Fm(Y)1104 1244 y Fq(\))g Fo(2)15 b Fq(]0)p Fp(;)g Fo(1)p Fq([.)42 b(This)29 b(ends)g(the)i(pro)s(of)e (of)i(the)g(prop)s(osition)d(\(2.1\).)448 1529 y Fr(3)135 b(Computing)45 b(the)h(p)t(ositiv)l(e)g(eigen)l(v)-7 b(alue.)448 1732 y Fq(A)29 b(prerequisite)e(for)i(pro)m(ving)f(that)h (\003)g(obtained)g(ab)s(o)m(v)m(e)h(is)e(indeed)f(a)i(measure)g(is)f (the)448 1845 y(follo)m(wing)448 2051 y Fk(Prop)s(osition)37 b(3.1)46 b Fn(L)-5 b(et)42 b Fq(\003)g Fn(b)-5 b(e)43 b(an)f(eigenfunctional)h(for)g Fo(L)2585 2018 y Fi(\003)2624 2051 y Fn(.)70 b(Assume)42 b(that)i Fq(\003)e Fn(is)448 2164 y(p)-5 b(ositive,)34 b(satis\014es)g Fq(\003\(1\))27 b(=)f(1)33 b Fn(and)h Fq(\003\()p Fk(1)1853 2178 y Fm(U)1913 2164 y Fq(\))26 b Fp(>)f Fq(0)34 b Fn(for)f(every)g(non-empty)h(op)-5 b(en)34 b(subset)448 2277 y Fp(U)10 b Fn(.)589 2389 y(Then)33 b(the)g(eigenvalue)g(asso)-5 b(ciate)g(d)35 b(to)e Fq(\003)g Fn(is)f Fp(\025)25 b Fq(=)g(exp)15 b Fp(P)2530 2403 y Fm(g)2571 2389 y Fq(\()p Fp(X)r(;)g(T)e Fq(\))p Fn(.)589 2595 y Fq(The)33 b(inequalit)m(y)f Fp(\025)e Fo(\024)f Fq(exp)15 b Fp(P)1599 2609 y Fm(g)1640 2595 y Fq(\()p Fp(X)r(;)g(T)e Fq(\))34 b(follo)m(ws)e(directly)g(from)h(the)g (conformalit)m(y)448 2708 y(of)e(\003)f(and)g(the)h(de\014nition)d(of)i (the)h(pressure)e(as)614 2914 y(1)d(=)f(\003\()p Fk(1)931 2928 y Fm(X)999 2914 y Fq(\))h(=)f Fp(\025)1209 2877 y Fi(\000)p Fm(n)1311 2914 y Fq(\003\()p Fo(L)1472 2877 y Fm(n)1519 2914 y Fk(1)1571 2928 y Fm(X)1638 2914 y Fq(\))h Fo(\024)f Fp(\025)1848 2877 y Fi(\000)p Fm(n)1996 2828 y Fl(X)1965 3025 y Fm(Z)5 b Fi(2Z)2116 3033 y Fg(n)2173 2914 y Fq(\003\()p Fk(1)2323 2928 y Fm(T)2374 2909 y Fg(n)2417 2928 y Fm(Z)2474 2914 y Fq(\))15 b(sup)2566 2993 y Fm(Z)2676 2914 y Fp(g)2719 2928 y Fm(n)2792 2914 y Fo(\024)25 b Fp(\025)2941 2877 y Fi(\000)p Fm(n)3089 2828 y Fl(X)3058 3025 y Fm(Z)5 b Fi(2Z)3209 3033 y Fg(n)3266 2914 y Fq(sup)3308 2993 y Fm(Z)3418 2914 y Fp(g)3461 2928 y Fm(n)3508 2914 y Fp(:)448 3198 y Fq(The)35 b(pro)s(of)f(of)h (the)f(con)m(v)m(erse)j(inequalit)m(y)c(is)h(more)h(delicate)f(and)g (will)f(mak)m(e)i(hea)m(vy)448 3311 y(use)h(of)h(the)g(condition)e Fp(P)1347 3325 y Fm(g)1387 3311 y Fq(\()p Fp(@)5 b Fo(Z)i Fp(;)15 b(T)e Fq(\))37 b Fp(<)e(P)1890 3325 y Fm(g)1930 3311 y Fq(\()p Fp(X)r(;)15 b(T)e Fq(\).)60 b(W)-8 b(e)38 b(pro)s(ceed)e(prett)m(y)h(m)m(uc)m(h)g(lik)m(e)448 3424 y(in)i(the)h(pro)s(of)g(of)g(Prop)s(osition)e(E)i(in)f([2],)k(except)f (that)e(\003)g(not)h(b)s(eing)d(a)j(measure,)448 3537 y(w)m(e)36 b(shall)d(ha)m(v)m(e)j(to)g(c)m(hec)m(k)h(a)e(measurabilit)m (y)e(prop)s(ert)m(y)i(\(Lemma)g(\(3.2\))i(b)s(elo)m(w\))e(\\b)m(y)448 3650 y(hand".)589 3763 y(Recall)30 b(that)h(the)g Fn(me)-5 b(asur)g(e-the)g(or)g(etic)35 b(pr)-5 b(essur)g(e)39 b Fq(of)30 b Fp(\026)h Fq(is:)1292 3997 y Fp(P)1350 4011 y Fm(g)1391 3997 y Fq(\()p Fp(\026;)15 b(T)e Fq(\))25 b(=)g Fp(h)p Fq(\()p Fp(\026;)15 b(T)e Fq(\))22 b Fo(\000)2139 3874 y Fl(Z)2189 4080 y Fm(X)2272 3997 y Fq(log)16 b Fp(g)j(d\026:)448 4236 y Fq(W)-8 b(e)29 b(ha)m(v)m(e)g(the)f Fn(variational)k(principle)p Fq(,)d(i.e.,)f(the)g(top)s(ological)f (pressure)g(is)f(the)i(supre-)448 4349 y(m)m(um)h(of)h(the)g (measure-theoretic)h(pressures:)39 b(this)29 b(follo)m(ws)g(from)g (paragraph)g(3.2)i(of)448 4462 y([2)q(])g(using)e(that)i Fp(P)1068 4476 y Fm(g)1108 4462 y Fq(\()p Fp(@)5 b Fo(Z)i Fp(;)15 b(T)e Fq(\))27 b Fp(<)e(P)1591 4476 y Fm(g)1631 4462 y Fq(\()p Fp(T)13 b Fq(\))31 b(and)f(that)h Fo(Z)37 b Fq(generates.)42 b(This)29 b(is)g(easy)i(to)g(see)448 4575 y(as)36 b(for)g(ergo)s(dic)f(measures)g(with)g(pressure)f (strictly)h(greater)i(than)e Fp(P)2905 4589 y Fm(g)2945 4575 y Fq(\()p Fp(@)5 b Fo(Z)i Fp(;)15 b(T)e Fq(\),)39 b(the)448 4687 y(sym)m(b)s(olic)32 b(dynamics)g(can)h(b)s(e)g(iden)m (ti\014ed)e(with)h(\()p Fp(X)r(;)15 b(T)e Fq(\),)35 b(and)e(the)g(v)-5 b(ariational)32 b(prin-)448 4800 y(ciple)f(holds)f(for)h(the)h(sym)m(b) s(olic)e(dynamics)g(b)m(y)h(a)h(classical)f(result)g(\(see)h(for)f (instance)448 4913 y([4)q(,)g(c)m(hap.18].\))1897 5162 y(10)p eop %%Page: 11 11 11 10 bop 589 623 a Fq(T)-8 b(o)21 b(conclude)f(w)m(e)h(will)c(pro)m(v) m(e)k(that)g(the)g(measure-theoretic)g(pressures)e(are)h(b)s(ounded)448 736 y(b)m(y)28 b(log)16 b Fp(\025)p Fq(,)29 b(using)d(the)h(approac)m (h)h(of)g([2)q(])f(whic)m(h)g(com)m(bines)g(a)h(result)e(of)i (Ledrappier)d([7)q(])448 848 y(and)30 b(\\shado)m(wing")g(estimates.) 589 961 y(Fix)38 b(an)g(ergo)s(dic)g(in)m(v)-5 b(arian)m(t)37 b(probabilit)m(y)e(measure)j Fp(\026)p Fq(.)64 b(W)-8 b(e)39 b(can)f(assume)g(that:)448 1074 y Fp(P)506 1088 y Fm(g)547 1074 y Fq(\()p Fp(\026;)15 b(T)e Fq(\))28 b Fp(>)f(P)962 1088 y Fm(g)1002 1074 y Fq(\()p Fp(@)5 b Fo(Z)i Fp(;)15 b(T)e Fq(\))33 b(\(otherwise)f(there)g(is)e(nothing)h (to)h(pro)m(v)m(e,)h(as)f Fp(P)2995 1088 y Fm(g)3036 1074 y Fq(\()p Fp(@)5 b Fo(Z)i Fp(;)15 b(T)e Fq(\))28 b Fp(<)448 1187 y(P)506 1201 y Fm(g)547 1187 y Fq(\()p Fp(X)r(;)15 b(T)e Fq(\)\).)589 1413 y(Let)31 b Fp(p)25 b Fq(:)h(\()935 1390 y(~)909 1413 y Fp(X)8 b(;)1050 1390 y Fq(~)1032 1413 y Fp(T)13 b(;)22 b Fq(~)-52 b Fp(\026)p Fq(\))26 b Fo(!)f Fq(\()p Fp(X)r(;)15 b(T)8 b(;)15 b(\026)p Fq(\))31 b(b)s(e)f(the)h(natural)e(extension.)41 b(W)-8 b(e)31 b(realize)f(it)g(as)898 1608 y(~)872 1631 y Fp(X)j Fq(=)25 b Fo(f)6 b Fq(~)-51 b Fp(x)25 b Fq(=)g(\()p Fp(A;)15 b(x)p Fq(\))27 b Fo(2)e(Z)1710 1593 y Fd(Z)1760 1602 y Ff(\000)1830 1631 y Fo(\002)20 b Fp(X)33 b Fq(:)26 b Fp(x)f Fo(2)2259 1544 y Fl(\\)2243 1740 y Fm(n)p Fi(\025)p Fh(0)2391 1631 y Fp(T)2457 1593 y Fm(n)2503 1631 y Fq([)p Fp(A)2596 1645 y Fi(\000)p Fm(n)2714 1631 y Fp(:)15 b(:)g(:)h(A)2903 1645 y Fh(0)2943 1631 y Fq(])p Fo(g)448 1939 y Fq(with)29 b(the)i(top)s(ology)g(induced)d(b)m(y)i(that)h(of)g Fo(Z)2024 1906 y Fd(Z)2074 1915 y Ff(\000)2145 1939 y Fo(\002)20 b Fp(X)7 b Fq(.)589 2052 y(The)30 b(dynamics)f(is)h(giv)m(en)g(b)m(y)g (the)h(\(partially-de\014ned\))d(map:)1195 2233 y(~)1177 2256 y Fp(T)38 b Fq(:)26 b(\()p Fp(:)15 b(:)g(:)h(A)1543 2270 y Fi(\000)p Fh(1)1638 2256 y Fp(A)1706 2270 y Fh(0)1745 2256 y Fp(;)f(x)p Fq(\))26 b Fo(7!)g Fq(\()p Fp(:)15 b(:)g(:)h(A)2239 2270 y Fi(\000)p Fh(1)2334 2256 y Fp(A)2402 2270 y Fh(0)2441 2256 y Fp(B)5 b(;)15 b(T)e(x)p Fq(\))448 2460 y(if)30 b Fp(B)k Fq(is)c(the)g(elemen)m(t)h(of)g Fo(Z)37 b Fq(con)m(taining)30 b Fp(T)13 b(x)p Fq(.)589 2573 y(Let)38 b Fp(\030)j Fq(b)s(e)36 b(the)i(\014nite)e(partition)g Fp(p)1805 2540 y Fi(\000)p Fh(1)1899 2573 y Fo(Z)7 b Fq(.)60 b(Let)38 b Fp(\021)i Fq(b)s(e)d(the)g(measurable)f(partition) 448 2618 y Fl(W)524 2713 y Fm(n)p Fi(\025)p Fh(0)694 2663 y Fq(~)676 2686 y Fp(T)742 2653 y Fm(n)789 2686 y Fp(\030)t Fq(,)41 b(i.e.,)g(t)m(w)m(o)g(p)s(oin)m(ts)c(are)j(in)d (the)i(same)g(elemen)m(t)h(of)f Fp(\021)j Fq(i\013)c(they)h(ha)m(v)m(e) h(the)448 2816 y(same)31 b(past)g(under)1149 2793 y(~)1131 2816 y Fp(T)13 b Fq(.)41 b(Observ)m(e)30 b(that)h Fp(\021)s Fq(\()6 b(~)-51 b Fp(x)q Fq(\))25 b(=)g Fo(f)p Fp(A)p Fo(g)d(\002)d Fp(W)13 b Fq(\()6 b(~)-51 b Fp(x)p Fq(\))31 b(if)1084 3034 y(~)-51 b Fp(x)25 b Fq(=)g(\()p Fp(A;)15 b(x)p Fq(\))32 b(and)d Fp(W)13 b Fq(\()6 b(~)-51 b Fp(x)p Fq(\))26 b(:=)2073 2948 y Fl(\\)2057 3144 y Fm(n)p Fi(\025)p Fh(0)2205 3034 y Fp(T)2271 2997 y Fm(n)2318 3034 y Fq([)p Fp(A)2411 3048 y Fi(\000)p Fm(n)2528 3034 y Fp(:)15 b(:)g(:)h(A)2717 3048 y Fh(0)2757 3034 y Fq(])p Fp(:)589 3332 y Fq(The)26 b(assumption)f(\(e\))i(that)g(eac)m(h)h(cylinder)c(has)i(\014nitely)f (man)m(y)h(connected)h(com-)448 3445 y(p)s(onen)m(ts)j(is)g(used)f(for) h(the)h(pro)s(of)f(of)g(the)h(follo)m(wing)d(lemma:)448 3657 y Fk(Lemma)33 b(3.2)46 b Fn(If)31 b Fp(P)1150 3671 y Fm(g)1190 3657 y Fq(\()p Fp(\026;)15 b(T)e Fq(\))26 b Fp(>)f(P)1601 3671 y Fm(g)1641 3657 y Fq(\()p Fp(@)5 b Fo(Z)i Fp(;)15 b(T)e Fq(\))32 b Fn(then)g(ther)-5 b(e)31 b(exists)g(a)h(c)-5 b(ountable)31 b(me)-5 b(asur-)448 3770 y(able)36 b(p)-5 b(artition)1032 3747 y Fq(~)1012 3770 y Fp(P)48 b Fn(of)1253 3747 y Fq(~)1227 3770 y Fp(X)42 b Fn(mo)-5 b(dulo)44 b Fq(~)-52 b Fp(\026)35 b Fn(such)g(that)h Fp(W)13 b Fq(\()6 b(~)-51 b Fp(x)p Fq(\))35 b Fn(is)g(c)-5 b(onstant)37 b(and)f(e)-5 b(qual)35 b(to)h(a)448 3883 y(non-empty)e(op)-5 b(en)34 b(subset)e(on)h(e)-5 b(ach)34 b(element)f(of)2181 3860 y Fq(~)2161 3883 y Fp(P)13 b Fn(.)589 4096 y Fq(The)28 b(pro)s(of)f(is)g(based)h(on)f(the)h(notion)g (of)g(the)g Fn(shadowing)38 b Fq(of)28 b(a)h(measure)e(b)m(y)h(a)g(set) 448 4209 y(in)m(tro)s(duced)h(in)g([1)q(].)41 b(W)-8 b(e)31 b(recall)f(it)g(for)g(the)h(con)m(v)m(enience)g(of)g(the)f (reader.)589 4321 y(A)j(measure)40 b(~)-52 b Fp(\026)32 b Fq(on)1289 4298 y(~)1263 4321 y Fp(X)40 b Fq(is)32 b(shado)m(w)m(ed)h(b)m(y)f Fp(S)i Fo(\032)29 b Fp(X)40 b Fq(if)32 b(for)h(ev)m(ery)g Fp(")d(>)e Fq(0)34 b(and)39 b(~)-52 b Fp(\026)o Fq(-a.e.)454 4434 y(~)h Fp(x)26 b Fo(2)637 4411 y Fq(~)611 4434 y Fp(X)38 b Fq(there)31 b(exist)f Fp(n)25 b(>)g(")1390 4401 y Fi(\000)p Fh(1)1515 4434 y Fq(and)k Fp(s)c Fo(2)g Fp(S)35 b Fq(suc)m(h)30 b(that:)1088 4639 y Fp(d)p Fq(\()p Fp(T)1236 4601 y Fm(k)1279 4639 y Fq(\()p Fp(p)p Fq(\()1413 4616 y(~)1395 4639 y Fp(T)1461 4601 y Fi(\000)p Fm(n)1569 4639 y Fq(~)-51 b Fp(x)p Fq(\)\))p Fp(;)15 b(T)1791 4601 y Fm(k)1835 4639 y Fq(\()p Fp(s)p Fq(\)\))26 b Fp(<)f(")182 b Fo(8)p Fq(0)25 b Fo(\024)g Fp(k)j Fo(\024)d Fp(n:)448 4843 y Fq(i.e.,)31 b(the)g(orbit)e(is)g(made)i(of)f(arbitrarily)e(long)i(b)s (eginnings)e(of)i(orbits)f(starting)i(in)e Fp(S)5 b Fq(.)1897 5162 y(11)p eop %%Page: 12 12 12 11 bop 589 623 a Fq(The)36 b(main)e(consequence)j(of)f(shado)m(wing) f(is)g(that)h(it)g(implies)d(that)j Fp(P)3060 637 y Fm(g)3100 623 y Fq(\()7 b(~)-52 b Fp(\026)q(;)3249 600 y Fq(~)3231 623 y Fp(T)13 b Fq(\))34 b Fo(\024)448 736 y Fp(P)506 750 y Fm(g)547 736 y Fq(\()p Fp(S;)15 b(T)e Fq(\))31 b([2)q(,)f(Theorem)g(S].)448 848 y Fk(Pro)s(of)44 b Fq(:)i(\(of)33 b(lemma)f(3.2\))j(The)d(assumption)f(on)i(the)g(pressure)f(of)h Fp(\026)f Fq(ensures)g(that)448 961 y(it)g(is)f(not)h(shado)m(w)m(ed)g (b)m(y)g(\001)p Fo(Z)i Fq(:=)1636 893 y Fl(S)1712 988 y Fm(A)p Fi(2Z)1887 961 y Fp(@)5 b(T)13 b(A)32 b Fq(b)m(y)g(the)g(ab)s (o)m(v)m(e.)47 b(W)-8 b(e)33 b(claim)e(that)h(this)448 1084 y(implies)k(that)i(for)45 b(~)-52 b Fp(\026)p Fq(-a.e.)40 b(\()p Fp(A;)15 b(x)p Fq(\))39 b Fo(2)1770 1061 y Fq(~)1744 1084 y Fp(X)7 b Fq(,)41 b(there)d(exists)g Fp(N)48 b(<)38 b Fo(1)g Fq(suc)m(h)f(that,)k(for)d(all)448 1197 y Fp(n)25 b Fo(\025)g Fq(0,)1058 1310 y(\()p Fp(@)5 b(T)1212 1273 y Fm(n)1259 1310 y Fq([)p Fp(A)1352 1324 y Fi(\000)p Fm(N)i Fi(\000)p Fm(n)1588 1310 y Fp(:)15 b(:)g(:)h(A)1777 1324 y Fi(\000)p Fm(N)7 b Fi(\000)p Fh(1)1989 1310 y Fq(]\))21 b Fo(\\)f Fq([)p Fp(A)2244 1324 y Fi(\000)p Fm(N)2382 1310 y Fp(:)15 b(:)g(:)h(A)2571 1324 y Fh(0)2610 1310 y Fq(])26 b(=)f Fo(;)p Fp(:)448 1476 y Fq(By)30 b(assumption)d(\(c\),)k(w)m(e)f(can)f(\014nd)f Fp(M)39 b Fq(suc)m(h)29 b(that)g(diam)o(\()p Fo(Z)2545 1443 y Fm(M)2625 1476 y Fq(\))c Fp(<)g(")p Fq(.)41 b(Con)m(tradicting)448 1588 y(the)c(ab)s(o)m(v)m(e)h(claim)e(giv)m(es,)j(for)d(a.e.)44 b(~)-51 b Fp(x)36 b Fo(2)1908 1565 y Fq(~)1882 1588 y Fp(X)8 b Fq(,)38 b(an)f(in\014nite)d(sequence)j Fp(n)2913 1602 y Fh(1)2988 1588 y Fp(<)f(n)3150 1602 y Fh(2)3225 1588 y Fp(<)f(:)15 b(:)g(:)448 1701 y Fq(suc)m(h)30 b(that:)916 1902 y(\()p Fp(@)5 b(T)1070 1865 y Fm(n)1113 1877 y Fg(k)q Fe(+1)1228 1865 y Fi(\000)p Fm(n)1326 1877 y Fg(k)1368 1902 y Fq([)p Fp(A)1461 1916 y Fi(\000)p Fm(n)1559 1928 y Fg(k)q Fe(+1)1694 1902 y Fp(:)15 b(:)g(:)h(A)1883 1916 y Fi(\000)p Fm(n)1981 1928 y Fg(k)2019 1916 y Fi(\000)p Fh(1)2113 1902 y Fq(]\))21 b Fo(\\)f Fq([)p Fp(A)2368 1916 y Fi(\000)p Fm(n)2466 1928 y Fg(k)2524 1902 y Fp(:)15 b(:)g(:)h(A)2713 1916 y Fh(0)2752 1902 y Fq(])26 b Fo(6)p Fq(=)f Fo(;)p Fp(:)448 2103 y Fq(W)-8 b(e)33 b(ma)m(y)f(assume)f Fp(n)1174 2118 y Fm(k)r Fh(+1)1338 2103 y Fq(to)h(b)s(e)f(minimal)e (for)i(the)h(giv)m(en)f Fp(n)2521 2118 y Fm(k)2564 2103 y Fq(.)43 b(Then,)32 b(w)m(e)f(m)m(ust)h(ha)m(v)m(e)448 2216 y(that)h(\()p Fp(@)5 b(T)13 b(A)869 2230 y Fi(\000)p Fm(n)967 2242 y Fg(k)q Fe(+1)1087 2216 y Fq(\))22 b Fo(\\)f Fq([)p Fp(A)1319 2230 y Fi(\000)p Fm(n)1417 2242 y Fg(k)q Fe(+1)1532 2230 y Fh(+1)1642 2216 y Fp(:)15 b(:)g(:)h(A)1831 2230 y Fh(0)1871 2216 y Fq(])29 b Fo(6)p Fq(=)f Fo(;)p Fq(.)48 b(Th)m(us)31 b(the)i(itinerary)e(of)3030 2193 y(~)3012 2216 y Fp(T)3078 2183 y Fi(\000)p Fm(n)3176 2195 y Fg(k)q Fe(+1)3291 2183 y Fh(+1)3391 2216 y Fq(~)-51 b Fp(x)448 2329 y Fq(is)35 b(the)i(same)f(as)h(that)f(of)h(some)f(p)s (oin)m(t)f(of)h(\001)p Fo(Z)7 b Fq(.)58 b(Using)35 b(that)i(diam)o(\()p Fo(Z)2957 2296 y Fm(M)3037 2329 y Fq(\))e Fp(<)f(")j Fq(and)448 2442 y(shifting)30 b(b)m(y)922 2419 y(~)904 2442 y Fp(T)970 2409 y Fi(\000)p Fm(M)1104 2442 y Fq(,)j(w)m(e)f(get)h (that)40 b(~)-52 b Fp(\026)32 b Fq(is)f(shado)m(w)m(ed)h(b)m(y)g(\001)p Fo(Z)7 b Fq(,)33 b(a)f(con)m(tradiction)g(whic)m(h)448 2555 y(pro)m(v)m(es)f(the)g(claim.)589 2668 y(No)m(w)f(the)g(claim)e (implies)e(that)k Fp(T)1761 2635 y Fm(N)7 b Fh(+)p Fm(n)1926 2668 y Fq([)p Fp(A)2019 2682 y Fi(\000)p Fm(N)g Fi(\000)p Fm(n)2254 2668 y Fp(:)15 b(:)g(:)h(A)2443 2682 y Fh(0)2483 2668 y Fq(],)30 b(whic)m(h)d(b)m(y)i(construction)448 2781 y(is)35 b(a)h(subset)f(of)g Fp(T)1084 2748 y Fm(N)1151 2781 y Fq([)p Fp(A)1244 2795 y Fi(\000)p Fm(N)1382 2781 y Fp(:)15 b(:)g(:)h(A)1571 2795 y Fh(0)1611 2781 y Fq(],)37 b(is)e(in)f(fact)i(a)g(union)e(of)i(connected)g(comp)s(onen)m(ts)448 2894 y(of)29 b Fp(T)616 2861 y Fm(N)683 2894 y Fq([)p Fp(A)776 2908 y Fi(\000)p Fm(N)913 2894 y Fp(:)15 b(:)g(:)i(A)1103 2908 y Fh(0)1142 2894 y Fq(].)41 b(By)28 b(assumption)f(\(e\))i(of)g (the)f(Main)g(Theorem,)h(there)f(are)h(only)448 3007 y(\014nitely)g(man)m(y)h(suc)m(h)g(comp)s(onen)m(ts.)41 b(Hence,)32 b(there)e(exists)g Fp(M)36 b(<)25 b Fo(1)30 b Fq(suc)m(h)g(that)1384 3208 y Fp(W)13 b Fq(\()p Fp(A;)i(x)p Fq(\))26 b(=)f Fp(T)1901 3170 y Fm(M)1980 3208 y Fq([)p Fp(A)2073 3222 y Fi(\000)p Fm(M)2222 3208 y Fp(:)15 b(:)g(:)h(A)2411 3222 y Fh(0)2451 3208 y Fq(])p Fp(:)448 3409 y Fq(In)36 b(particular,)g Fp(W)13 b Fq(\()p Fo(\001)p Fq(\))37 b(is)e(a.e.)j(a)e(non-empt)m(y)-8 b(,)39 b(op)s(en)c(subset)h(of)g Fp(X)7 b Fq(,)39 b(and)c(tak)m(es)j(only)448 3522 y(coun)m(tably)i(man) m(y)g(distinct)e(v)-5 b(alues.)69 b(Finally)-8 b(,)40 b(observ)m(e)h(that)f(the)g(sets)h(where)e(the)448 3635 y(v)-5 b(alue)30 b(of)h Fp(W)42 b Fq(is)30 b(\014xed)g(are)g (measurable,)g(b)s(eing)f(of)i(the)f(form:)1137 3836 y Fo(f)p Fq(\()p Fp(B)5 b(;)15 b(x)p Fq(\))26 b Fo(2)1556 3813 y Fq(~)1530 3836 y Fp(X)33 b Fq(:)25 b Fp(B)1757 3850 y Fi(\000)p Fm(M)1906 3836 y Fp(:)15 b(:)g(:)h(B)2096 3850 y Fh(0)2161 3836 y Fq(=)25 b Fp(A)2325 3850 y Fi(\000)p Fm(M)2474 3836 y Fp(:)15 b(:)g(:)h(A)2663 3850 y Fh(0)2703 3836 y Fo(g)1195 3973 y(nf)p Fq(\()p Fp(B)5 b(;)15 b(x)p Fq(\))27 b Fo(2)1660 3951 y Fq(~)1634 3973 y Fp(X)33 b Fq(:)25 b Fo(8)p Fp(m)g(>)g(M)5 b(;)46 b(B)2277 3987 y Fi(\000)p Fm(m)2413 3973 y Fp(:)15 b(:)g(:)h(B)2603 3987 y Fh(0)2668 3973 y Fo(2)25 b(Ag)448 4175 y Fq(where)40 b Fo(A)g Fq(is)f(the)i(set)f(of)h(w)m(ords)e Fp(B)1706 4189 y Fi(\000)p Fm(m)1843 4175 y Fp(:)15 b(:)g(:)h(B)2033 4189 y Fh(0)2113 4175 y Fq(de\014ning)38 b(a)j(strictly)e(smaller)g Fp(W)13 b Fq(-set)448 4287 y(than)30 b Fp(B)729 4301 y Fi(\000)p Fm(M)878 4287 y Fp(:)15 b(:)g(:)i(B)1069 4301 y Fh(0)1108 4287 y Fq(.)2236 b Fc(2)589 4513 y Fq(W)-8 b(e)40 b(are)e(going)g(to)g(compare)46 b(~)-52 b Fp(\026)37 b Fq(with)g(the)h(function)f(\(ten)m(tativ)m(ely\))i(de\014ned)e(on)448 4626 y(measurable)30 b(subsets)f Fp(S)36 b Fq(of)30 b Fp(X)38 b Fq(b)m(y:)1425 4864 y Fp(m)p Fq(\()p Fp(S)5 b Fq(\))26 b(:=)1783 4740 y Fl(Z)1854 4930 y Fh(~)1834 4946 y Fm(X)1916 4864 y Fp(m)p Fq(\()6 b(~)-51 b Fp(x;)15 b(S)5 b Fq(\))24 b(~)-54 b Fp(\026)q Fq(\()p Fp(d)6 b Fq(~)-51 b Fp(x)q Fq(\))1897 5162 y(12)p eop %%Page: 13 13 13 12 bop 448 623 a Fq(where)1292 770 y Fp(m)p Fq(\()6 b(~)-51 b Fp(x;)15 b(S)5 b Fq(\))26 b(:=)1752 699 y(\003\()p Fk(1)1902 718 y Fm(W)10 b Fh(\()t(~)-39 b Fm(x)q Fh(\))2099 699 y Fo(\001)20 b Fp(\032)p Fq(\()6 b(~)-51 b Fp(x;)15 b Fo(\001)p Fq(\))21 b Fo(\001)g Fk(1)2497 713 y Fm(S)2548 699 y Fq(\))p 1752 749 831 4 v 1837 832 a(\003\()p Fk(1)1987 851 y Fm(W)10 b Fh(\()t(~)-39 b Fm(x)p Fh(\))2183 832 y Fo(\001)20 b Fp(\032)p Fq(\()6 b(~)-51 b Fp(x)q(;)15 b Fo(\001)p Fq(\)\))448 1067 y(and)27 b Fp(\032)p Fq(\()6 b(~)-51 b Fp(x)q(;)15 b(y)s Fq(\))25 b(:=)1027 999 y Fl(Q)1112 1094 y Fm(k)r Fi(\025)p Fh(1)1270 991 y Fm(g)r Fh(\()p Fm(T)1384 962 y Ff(\000)p Fe(1)1374 1014 y Fg(A)1420 1032 y Ff(\000)p Fg(k)1511 991 y Fi(\016)p Fm(:::)o(T)1656 962 y Ff(\000)p Fe(1)1646 1014 y Fg(A)1692 1029 y Ff(\000)p Fe(1)1779 991 y Fm(y)r Fh(\))p 1270 1046 574 4 v 1423 1111 a Fm(g)r Fh(\()1500 1094 y(~)1486 1111 y Fm(T)1537 1092 y Ff(\000)p Fg(k)1628 1111 y Fh(~)-39 b Fm(x)o Fh(\))1881 1067 y Fq(if)32 b(~)-50 b Fp(x)25 b Fo(2)2151 1044 y Fq(~)2125 1067 y Fp(X)7 b Fq(,)34 b(~)-51 b Fp(x)26 b Fq(=)f(\()p Fp(A;)15 b(x)p Fq(\))28 b(and)f Fp(y)h Fo(2)d Fp(W)13 b Fq(\()6 b(~)-51 b Fp(x)p Fq(\).)40 b(W)-8 b(e)448 1204 y(ha)m(v)m(e)32 b(set)f Fp(g)s Fq(\()p Fp(A;)15 b(x)p Fq(\))27 b(:=)e Fp(g)s Fq(\()p Fp(x)p Fq(\).)589 1317 y(W)-8 b(e)39 b(\014rst)d(c)m(hec)m(k)j(that)f(the)f(form)f Fp(m)h Fq(is)f(w)m(ell-de\014ned,)i(i.e.,)h(that)k(~)-51 b Fp(x)37 b Fo(7!)f Fp(m)p Fq(\()6 b(~)-51 b Fp(x;)15 b(S)5 b Fq(\))448 1430 y(is)32 b(measurable.)49 b(Observ)m(e)33 b(that)g(b)m(y)g(assumption)f(\(b\))h(of)g(the)h(Main)e(Theorem,)i Fp(\032)f Fq(is)448 1543 y(uniformly)28 b(con)m(tin)m(uous)i(on)g(its)g (domain)g(of)g(de\014nition.)39 b(Th)m(us)30 b(w)m(e)h(can)g(extend)f (it)g(to)448 1656 y(a)39 b(con)m(tin)m(uous)f(function)f(de\014ned)g (on)i(the)f(whole)g(of)2383 1633 y(~)2357 1656 y Fp(X)33 b Fo(\002)2587 1633 y Fq(~)2561 1656 y Fp(X)7 b Fq(.)65 b(It)38 b(is)g(then)g(routine)448 1769 y(to)g(c)m(hec)m(k)h(that,)h(if) c Fp(W)50 b Fq(is)36 b(an)m(y)i(giv)m(en)f(measurable)g(subset)g(of)g Fp(X)7 b Fq(,)40 b(the)d(follo)m(wing)f(is)448 1882 y(measurable:)1471 1995 y(~)-51 b Fp(x)25 b Fo(7!)h Fq(\003\()p Fk(1)1809 2009 y Fm(W)1910 1995 y Fo(\001)21 b Fp(\032)p Fq(\()6 b(~)-51 b Fp(x;)15 b Fo(\001)p Fq(\))21 b Fo(\001)g Fk(1)2309 2009 y Fm(S)2360 1995 y Fq(\))p Fp(:)448 2161 y Fq(No)m(w,)47 b(use)42 b(the)h(previous)e(Lemma)i(to)h(write:)64 b Fk(1)2213 2180 y Fm(W)10 b Fh(\()t(~)-39 b Fm(x)p Fh(\))2434 2161 y Fq(=)2551 2093 y Fl(P)2647 2191 y Fm(E)t Fi(2)2765 2175 y Fh(~)2750 2191 y Fm(P)2823 2161 y Fk(1)2875 2175 y Fm(E)2935 2161 y Fk(1)2987 2180 y Fm(W)10 b Fh(\()p Fm(E)t Fh(\))3178 2161 y Fq(.)78 b(The)448 2274 y(measurabilit)m(y)29 b(of)h Fp(m)p Fq(\()p Fo(\001)p Fp(;)15 b(S)5 b Fq(\))32 b(follo)m(ws.)40 b(In)29 b(particular,)g Fp(m)p Fq(\()p Fp(S)5 b Fq(\))31 b(is)e(w)m(ell-de\014ned.)589 2387 y(W)-8 b(e)33 b(follo)m(w)e(closely)h(Ledrappier)d([7)q(])j(for)f(the)h (rest)g(of)g(the)g(pro)s(of)f(of)h(Prop)s(osition)448 2500 y(3.1.)589 2613 y(Let)1454 2726 y Fp(Q)p Fq(\()6 b(~)-51 b Fp(x)p Fq(\))26 b(=)e(\003\()p Fk(1)1919 2744 y Fm(W)10 b Fh(\()t(~)-39 b Fm(x)q Fh(\))2116 2726 y Fo(\001)20 b Fp(\032)p Fq(\()6 b(~)-51 b Fp(x;)15 b Fo(\001)p Fq(\)\))448 2893 y(\(remark)29 b(that)g(the)f(function)f(to)i(whic)m(h) e(w)m(e)i(apply)e(\003)h(is)f(indeed)g(a)i(function)e(o)m(v)m(er)i Fp(X)7 b Fq(,)448 3006 y(not)636 2983 y(~)610 3006 y Fp(X)g Fq(\).)41 b(A)31 b(crucial)e(observ)-5 b(ation)30 b(is)g(that)h(the)f(previous)f(Lemma)h(implies:)1407 3210 y(0)c Fp(<)f(Q)p Fq(\()6 b(~)-51 b Fp(x)p Fq(\))25 b Fp(<)g Fo(1)30 b Fq(for)38 b(~)-52 b Fp(\026)o Fq(-a.e.)38 b(~)-51 b Fp(x)p Fq(.)448 3414 y(Indeed,)36 b Fp(\032)p Fq(\()6 b(~)-51 b Fp(x;)15 b Fo(\001)p Fq(\))35 b Fp(>)e Fq(0)j(on)f Fp(W)13 b Fq(\()6 b(~)-51 b Fp(x)p Fq(\))35 b(whic)m(h)f(is)g(a)i(non-empt)m(y)f(op)s(en)g(subset)g(and)f(whic)m(h) 448 3527 y(therefore)d(satis\014es)f(\003\()p Fk(1)1315 3546 y Fm(W)10 b Fh(\()t(~)-39 b Fm(x)p Fh(\))1491 3527 y Fq(\))25 b Fp(>)g Fq(0.)589 3640 y(Let)31 b Fo(Z)825 3607 y Fi(0)874 3640 y Fq(=)25 b Fp(T)1036 3607 y Fi(\000)p Fh(1)1130 3640 y Fo(Z)7 b Fq(.)41 b(Compute:)589 3877 y Fp(m)p Fq(\()6 b(~)-51 b Fp(x;)15 b Fo(Z)869 3840 y Fi(0)893 3877 y Fq(\()p Fp(x)p Fq(\)\))84 b(=)1372 3816 y(1)p 1298 3856 195 4 v 1298 3940 a Fp(Q)p Fq(\()6 b(~)-51 b Fp(x)p Fq(\))1523 3877 y Fo(\001)20 b Fq(\003\()p Fk(1)1718 3896 y Fm(W)10 b Fh(\()t(~)-39 b Fm(x)p Fh(\))1914 3877 y Fo(\001)21 b Fp(\032)p Fq(\()6 b(~)-51 b Fp(x;)15 b Fo(\001)p Fq(\))21 b Fo(\001)g Fk(1)2313 3896 y Fi(Z)2369 3877 y Ff(0)2391 3896 y Fh(\()p Fm(x)p Fh(\))2490 3877 y Fq(\))1134 4116 y(=)1372 4054 y(1)p 1298 4095 V 1298 4178 a Fp(Q)p Fq(\()6 b(~)-51 b Fp(x)p Fq(\))1502 4116 y Fp(\025)1555 4078 y Fi(\000)p Fh(1)1650 4116 y Fq(\003\()p Fo(L)p Fq(\()p Fk(1)1898 4134 y Fm(W)10 b Fh(\()t(~)-39 b Fm(x)p Fh(\))2094 4116 y Fo(\001)20 b Fp(\032)p Fq(\()6 b(~)-51 b Fp(x)q(;)15 b Fo(\001)p Fq(\))21 b Fo(\001)f Fk(1)2492 4134 y Fi(Z)2548 4115 y Ff(0)2571 4134 y Fh(\()p Fm(x)p Fh(\))2670 4116 y Fq(\)\))1134 4354 y(=)1372 4293 y(1)p 1298 4333 V 1298 4417 a Fp(Q)p Fq(\()6 b(~)-51 b Fp(x)p Fq(\))1502 4354 y Fp(\025)1555 4317 y Fi(\000)p Fh(1)1650 4354 y Fq(\003\()p Fk(1)1800 4373 y Fi(Z)5 b Fh(\()p Fm(T)10 b(x)p Fh(\))p Fi(\\)p Fm(T)g(W)g Fh(\()t(~)-39 b Fm(x)q Fh(\))2295 4354 y Fo(\001)21 b Fp(\032)p Fq(\()6 b(~)-51 b Fp(x;)15 b(T)2581 4316 y Fi(\000)p Fh(1)2568 4387 y Fi(Z)5 b Fh(\()p Fm(x)p Fh(\))2723 4354 y Fq(\()p Fo(\001)p Fq(\)\))22 b Fo(\001)e Fp(g)k Fo(\016)d Fp(T)3119 4316 y Fi(\000)p Fh(1)3106 4387 y Fi(Z)5 b Fh(\()p Fm(x)p Fh(\))3261 4354 y Fq(\))1134 4593 y(=)1372 4531 y(1)p 1298 4572 V 1298 4655 a Fp(Q)p Fq(\()h(~)-51 b Fp(x)p Fq(\))1502 4593 y Fp(\025)1555 4555 y Fi(\000)p Fh(1)1650 4593 y Fq(\003\()p Fk(1)1800 4623 y Fm(W)10 b Fh(\()1918 4606 y(~)1904 4623 y Fm(T)k Fh(~)-39 b Fm(x)p Fh(\))2047 4593 y Fo(\001)20 b Fp(\032)p Fq(\()2192 4570 y(~)2174 4593 y Fp(T)f Fq(~)-51 b Fp(x;)15 b Fo(\001)p Fq(\))21 b Fo(\001)g Fp(g)s Fq(\()p Fp(x)p Fq(\)\))1134 4856 y(=)83 b Fp(\025)1341 4818 y Fi(\000)p Fh(1)1445 4795 y Fp(Q)p Fq(\()1570 4772 y(~)1552 4795 y Fp(T)19 b Fq(~)-51 b Fp(x)p Fq(\))p 1445 4835 261 4 v 1478 4918 a Fp(Q)p Fq(\()6 b(~)-51 b Fp(x)p Fq(\))1716 4856 y Fp(g)s Fq(\()p Fp(x)p Fq(\))1437 b(\(1\))1897 5162 y(13)p eop %%Page: 14 14 14 13 bop 448 626 a Fq(using)37 b(that)h Fo(Z)7 b Fq(\()p Fp(T)13 b(x)p Fq(\))26 b Fo(\\)f Fp(T)13 b(W)g Fq(\()6 b(~)-51 b Fp(x)p Fq(\))38 b(=)g Fp(W)13 b Fq(\()1856 603 y(~)1839 626 y Fp(T)18 b Fq(~)-51 b Fp(x)p Fq(\))38 b(and)g Fp(\032)p Fq(\()2314 603 y(~)2296 626 y Fp(T)19 b Fq(~)-51 b Fp(x;)15 b(z)t Fq(\))39 b(=)e Fp(\032)p Fq(\()6 b(~)-51 b Fp(x)q(;)15 b(y)s Fq(\))26 b Fo(\001)3027 581 y Fm(g)r Fh(\()p Fm(y)r Fh(\))p 3026 605 131 4 v 3026 658 a Fm(g)r Fh(\()p Fm(x)p Fh(\))3205 626 y Fq(where)448 764 y Fp(y)i Fq(=)d Fp(T)683 726 y Fi(\000)p Fh(1)670 797 y Fi(Z)5 b Fh(\()p Fm(x)p Fh(\))825 764 y Fq(\()p Fp(z)t Fq(\).)589 877 y(W)-8 b(e)32 b(claim)d(that:)1023 969 y Fl(Z)1093 1158 y Fh(~)1073 1175 y Fm(X)1156 1093 y Fq(log)16 b Fp(m)p Fq(\()6 b(~)-51 b Fp(x;)15 b Fo(Z)1568 1055 y Fi(0)1592 1093 y Fq(\()p Fp(x)p Fq(\)\))g Fp(d)7 b Fq(~)-52 b Fp(\026)q Fq(\()6 b(~)-51 b Fp(x)q Fq(\))25 b(=)2111 969 y Fl(Z)2182 1158 y Fh(~)2162 1175 y Fm(X)2244 1093 y Fq(log)17 b Fp(g)i(d)7 b Fq(~)-52 b Fp(\026)20 b Fo(\000)g Fq(log)c Fp(\025:)448 1330 y Fq(Remark)36 b(that)g(if)e(log)17 b Fp(Q)35 b Fq(w)m(as)h(in)m(tegrable,)g(this)f (equalit)m(y)g(w)m(ould)f(b)s(e)h(an)g(immediate)448 1443 y(consequence)30 b(of)f(the)g(ab)s(o)m(v)m(e)h(computation.)40 b(But)29 b(w)m(e)g(only)f(kno)m(w)g(that)i(log)16 b Fp(Q)29 b Fq(is)f(a.e.)448 1556 y(\014nite,)i(so)h(w)m(e)f(use)g(the)h(subtler) e(argumen)m(t)i(of)f([7)q(].)589 1669 y(T)-8 b(o)27 b(b)s(egin)e(with,) h(remark)g(that)h(as)f Fp(m)p Fq(\()6 b(~)-51 b Fp(x;)15 b Fo(Z)2076 1636 y Fi(0)2100 1669 y Fq(\()p Fp(x)p Fq(\)\))27 b Fo(\024)e Fq(1,)2477 1595 y Fl(R)2540 1684 y Fh(~)2520 1701 y Fm(X)2603 1669 y Fq(log)16 b Fp(m)p Fq(\()6 b(~)-51 b Fp(x;)15 b Fo(Z)3015 1636 y Fi(0)3039 1669 y Fq(\()p Fp(x)p Fq(\)\))g Fp(d)7 b Fq(~)-52 b Fp(\026)q Fq(\()6 b(~)-51 b Fp(x)q Fq(\))448 1781 y(is)30 b(w)m(ell-de\014ned)e(in)h([)p Fo(\0001)p Fp(;)15 b Fq(0])32 b(and)d(the)i(Birkho\013)e(ergo)s(dic)h (theorem)h(giv)m(es:)480 2061 y(lim)452 2116 y Fm(n)p Fi(!1)665 2000 y Fq(1)p 661 2040 55 4 v 661 2124 a Fp(n)740 1948 y Fm(n)p Fi(\000)p Fh(1)741 1975 y Fl(X)742 2173 y Fm(k)r Fh(=0)888 2061 y Fq(log)17 b Fp(m)p Fq(\()1154 2038 y(~)1136 2061 y Fp(T)1202 2024 y Fm(k)1250 2061 y Fq(~)-50 b Fp(x;)15 b Fo(Z)1410 2024 y Fi(0)1433 2061 y Fq(\()p Fp(T)1534 2024 y Fm(k)1577 2061 y Fp(x)p Fq(\)\))31 b(exists)f(a.e.)42 b(and)30 b(is)2441 1938 y Fl(Z)2512 2127 y Fh(~)2492 2144 y Fm(X)2574 2061 y Fq(log)17 b Fp(m)p Fq(\()6 b(~)-51 b Fp(x;)15 b Fo(Z)2987 2024 y Fi(0)3011 2061 y Fq(\()p Fp(x)p Fq(\)\))g Fp(d)7 b Fq(~)-52 b Fp(\026)q Fq(\()6 b(~)-51 b Fp(x)p Fq(\))p Fp(:)448 2336 y Fq(Con)m(v)m(ergence)39 b(a.e.)f(implies)c(con)m(v)m(ergence)39 b(in)d(probabilit)m(y)-8 b(.)58 b(No)m(w)37 b(the)g(ab)s(o)m(v)m(e)h (com-)448 2449 y(putation)33 b(sho)m(ws)g(that)g(the)h(Birkho\013)e(a)m (v)m(erage)k(in)31 b(the)j(previous)d(equation)i(is)g(equal)448 2562 y(to)1068 2633 y(1)p 1063 2673 V 1063 2757 a Fp(n)1143 2593 y Fl(\020)1197 2694 y Fq(log)17 b Fp(Q)j Fo(\016)1505 2671 y Fq(~)1487 2694 y Fp(T)1553 2657 y Fm(n)1620 2694 y Fo(\000)g Fq(log)d Fp(Q)1916 2593 y Fl(\021)1990 2694 y Fq(+)2081 2571 y Fl(Z)2151 2760 y Fh(~)2131 2777 y Fm(X)2214 2694 y Fq(log)g Fp(g)h(d)7 b Fq(~)-52 b Fp(\026)21 b Fo(\000)f Fq(log)c Fp(\025:)448 2898 y Fq(As)34 b(log)17 b Fp(Q)34 b Fq(is)f(almost)h(ev)m(erywhere)g(\014nite,)g(this)f(a)m(v)m (erage)k(m)m(ust)d(con)m(v)m(erge)i(in)c(proba-)448 3011 y(bilit)m(y)d(to)797 2938 y Fl(R)860 3026 y Fh(~)840 3043 y Fm(X)922 3011 y Fq(log)17 b Fp(g)i(d)7 b Fq(~)-52 b Fp(\026)20 b Fo(\000)g Fq(log)c Fp(\025)p Fq(.)41 b(The)30 b(claim)f(follo)m(ws.)589 3135 y(Recalling)39 b(that)i Fp(\021)k Fq(=)1403 3067 y Fl(W)1479 3162 y Fm(k)r Fi(\025)p Fh(0)1644 3112 y Fq(~)1627 3135 y Fp(T)1693 3102 y Fi(\000)p Fm(k)1790 3135 y Fp(\030)f Fq(and)39 b(that)i Fp(\030)j Fq(generates,)g(it)c(is)f(easy)i(to)f(see)448 3265 y(that)31 b Fp(\021)i Fq(generates)e(and)e(that)i Fp(\021)i Fq(is)c(\014ner)g (than)2100 3242 y(~)2082 3265 y Fp(T)13 b(\021)s Fq(,)30 b(hence)g(the)g(en)m(trop)m(y)h(of)f Fp(\026)g Fq(can)g(b)s(e)448 3378 y(computed)h(as:)507 3583 y Fp(h)p Fq(\()p Fp(\026;)15 b(T)e Fq(\))84 b(=)f Fp(h)p Fq(\()7 b(~)-52 b Fp(\026;)1228 3560 y Fq(~)1210 3583 y Fp(T)13 b Fq(\))26 b(=)f Fp(h)p Fq(\()7 b(~)-52 b Fp(\026;)1634 3560 y Fq(~)1615 3583 y Fp(T)1681 3545 y Fi(\000)p Fh(1)1776 3583 y Fq(\))26 b(=)f Fp(H)2015 3597 y Fh(~)-41 b Fm(\026)2055 3583 y Fq(\()p Fp(\021)s Fo(j)2193 3496 y Fl(_)2178 3694 y Fm(k)r Fi(\025)p Fh(1)2340 3560 y Fq(~)2323 3583 y Fp(T)2389 3545 y Fm(k)2431 3583 y Fp(\021)s Fq(\))26 b(=)f Fp(H)2718 3597 y Fh(~)-41 b Fm(\026)2758 3583 y Fq(\()2811 3560 y(~)2793 3583 y Fp(T)2859 3545 y Fi(\000)p Fh(1)2954 3583 y Fp(\021)s Fo(j)3056 3496 y Fl(_)3042 3694 y Fm(k)r Fi(\025)p Fh(0)3204 3560 y Fq(~)3186 3583 y Fp(T)3252 3545 y Fm(k)3295 3583 y Fp(\021)s Fq(\))874 3831 y(=)83 b Fp(H)1110 3845 y Fh(~)-41 b Fm(\026)1150 3831 y Fq(\()1203 3808 y(~)1185 3831 y Fp(T)1251 3794 y Fi(\000)p Fh(1)1345 3831 y Fp(\021)s Fo(j)p Fp(\021)s Fq(\))27 b(=)e Fp(H)1706 3845 y Fh(~)-41 b Fm(\026)1746 3831 y Fq(\()1799 3808 y(~)1781 3831 y Fp(T)1847 3794 y Fi(\000)p Fh(1)1942 3831 y Fp(\030)t Fo(j)p Fp(\021)s Fq(\))874 4015 y(=)83 b Fo(\000)1114 3891 y Fl(Z)1184 4080 y Fh(~)1164 4097 y Fm(X)1325 3928 y Fl(X)1247 4141 y Fm(Z)1300 4122 y Ff(0)1321 4141 y Fi(2)1382 4124 y Fh(~)1368 4141 y Fm(T)1419 4122 y Ff(\000)p Fe(1)1502 4141 y Fm(\030)1551 4015 y Fk(1)1603 4030 y Fm(Z)1656 4011 y Ff(0)1682 4015 y Fq(\()6 b(~)-51 b Fp(x)p Fq(\))15 b(log)i Fj(E)2013 4029 y Fh(~)-41 b Fm(\026)2060 4015 y Fq(\()p Fk(1)2147 4030 y Fm(Z)2200 4011 y Ff(0)2226 4015 y Fo(j)p Fp(\021)s Fq(\))15 b Fp(d)7 b Fq(~)-52 b Fp(\026)r Fq(\()6 b(~)-51 b Fp(x)p Fq(\))p Fp(:)448 4318 y Fq(Th)m(us,)565 4551 y Fp(P)623 4565 y Fm(g)663 4551 y Fq(\()p Fp(\026;)15 b(T)e Fq(\))21 b Fo(\000)f Fq(log)d Fp(\025)83 b Fq(=)f Fp(h)p Fq(\()7 b(~)-52 b Fp(\026)q(;)1629 4528 y Fq(~)1611 4551 y Fp(T)13 b Fq(\))21 b(+)1824 4427 y Fl(Z)1894 4617 y Fh(~)1874 4633 y Fm(X)1957 4551 y Fq(log)16 b Fp(g)j(d)7 b Fq(~)-52 b Fp(\026)20 b Fo(\000)g Fq(log)d Fp(\025)1275 4798 y Fq(=)1428 4674 y Fl(Z)1499 4864 y Fh(~)1479 4880 y Fm(X)1640 4712 y Fl(X)1561 4924 y Fm(Z)1614 4906 y Ff(0)1636 4924 y Fi(2)1697 4908 y Fh(~)1683 4924 y Fm(T)1734 4906 y Ff(\000)p Fe(1)1817 4924 y Fm(\030)1866 4798 y Fk(1)1918 4813 y Fm(Z)1971 4794 y Ff(0)1997 4798 y Fq(\()6 b(~)-51 b Fp(x)p Fq(\))15 b(log)2278 4736 y Fp(m)p Fq(\()6 b(~)-51 b Fp(x)q(;)15 b(pZ)2601 4703 y Fi(0)2624 4736 y Fq(\))p 2277 4777 383 4 v 2277 4860 a Fj(E)2338 4874 y Fh(~)-41 b Fm(\026)2384 4860 y Fq(\()p Fk(1)2471 4876 y Fm(Z)2524 4857 y Ff(0)2551 4860 y Fo(j)p Fp(\021)s Fq(\))2685 4798 y Fp(d)7 b Fq(~)-52 b Fp(\026)p Fq(\()6 b(~)-51 b Fp(x)q Fq(\))1897 5162 y(14)p eop %%Page: 15 15 15 14 bop 1275 663 a Fo(\024)82 b Fq(log)1561 539 y Fl(Z)1631 728 y Fh(~)1611 745 y Fm(X)1773 576 y Fl(X)1694 789 y Fm(Z)1747 770 y Ff(0)1769 789 y Fi(2)1830 772 y Fh(~)1816 789 y Fm(T)1867 770 y Ff(\000)p Fe(1)1949 789 y Fm(\030)1998 663 y Fk(1)2050 678 y Fm(Z)2103 659 y Ff(0)2129 663 y Fq(\()6 b(~)-51 b Fp(x)q Fq(\))2263 601 y Fp(m)p Fq(\()6 b(~)-51 b Fp(x;)15 b(pZ)2585 568 y Fi(0)2608 601 y Fq(\))p 2262 642 383 4 v 2262 725 a Fj(E)2323 739 y Fh(~)-41 b Fm(\026)2369 725 y Fq(\()p Fk(1)2456 740 y Fm(Z)2509 721 y Ff(0)2536 725 y Fo(j)p Fp(\021)s Fq(\))2670 663 y Fp(d)7 b Fq(~)-52 b Fp(\026)p Fq(\()6 b(~)-51 b Fp(x)p Fq(\))1275 1031 y(=)82 b(log)1561 907 y Fl(Z)1631 1096 y Fh(~)1611 1113 y Fm(X)1694 1031 y Fj(E)1755 1045 y Fh(~)-41 b Fm(\026)1816 848 y Fl(0)1816 1012 y(@)1975 944 y(X)1896 1157 y Fm(Z)1949 1138 y Ff(0)1971 1157 y Fi(2)2032 1140 y Fh(~)2018 1157 y Fm(T)2069 1138 y Ff(\000)p Fe(1)2151 1157 y Fm(\030)2200 1031 y Fk(1)2252 1046 y Fm(Z)2305 1027 y Ff(0)2331 1031 y Fq(\()6 b(~)-51 b Fp(x)q Fq(\))2465 969 y Fp(m)p Fq(\()6 b(~)-51 b Fp(x;)15 b(pZ)2787 936 y Fi(0)2810 969 y Fq(\))p 2464 1010 V 2464 1093 a Fj(E)2525 1107 y Fh(~)-41 b Fm(\026)2571 1093 y Fq(\()p Fk(1)2658 1108 y Fm(Z)2711 1090 y Ff(0)2738 1093 y Fo(j)p Fp(\021)s Fq(\))2882 899 y Fl(\014)2882 954 y(\014)2882 1008 y(\014)2882 1063 y(\014)2938 1031 y Fp(\021)2986 848 y Fl(1)2986 1012 y(A)3096 1031 y Fp(d)7 b Fq(~)-52 b Fp(\026)p Fq(\()6 b(~)-51 b Fp(x)p Fq(\))1275 1344 y(=)82 b(log)1561 1220 y Fl(Z)1631 1410 y Fh(~)1611 1426 y Fm(X)1773 1258 y Fl(X)1694 1470 y Fm(Z)1747 1451 y Ff(0)1769 1470 y Fi(2)1830 1454 y Fh(~)1816 1470 y Fm(T)1867 1451 y Ff(\000)p Fe(1)1949 1470 y Fm(\030)1998 1344 y Fj(E)2059 1358 y Fh(~)-41 b Fm(\026)2105 1344 y Fq(\()p Fk(1)2192 1359 y Fm(Z)2245 1340 y Ff(0)2272 1344 y Fo(j)p Fp(\021)s Fq(\))2391 1282 y Fp(m)p Fq(\()6 b(~)-51 b Fp(x)q(;)15 b(pZ)2714 1249 y Fi(0)2737 1282 y Fq(\))p 2390 1323 V 2390 1406 a Fj(E)2452 1420 y Fh(~)-41 b Fm(\026)2498 1406 y Fq(\()p Fk(1)2585 1422 y Fm(Z)2638 1403 y Ff(0)2665 1406 y Fo(j)p Fp(\021)s Fq(\))2799 1344 y Fp(d)7 b Fq(~)-52 b Fp(\026)p Fq(\()6 b(~)-51 b Fp(x)p Fq(\))1275 1649 y(=)82 b(log)1561 1526 y Fl(Z)1631 1715 y Fh(~)1611 1732 y Fm(X)1729 1563 y Fl(X)1694 1761 y Fm(Z)1747 1742 y Ff(0)1769 1761 y Fi(2Z)1872 1742 y Ff(0)1909 1649 y Fp(m)p Fq(\()6 b(~)-51 b Fp(x)q(;)15 b(Z)2186 1612 y Fi(0)2209 1649 y Fq(\))g Fp(d)7 b Fq(~)-52 b Fp(\026)p Fq(\()6 b(~)-51 b Fp(x)q Fq(\))1275 1931 y(=)82 b(log)1561 1808 y Fl(Z)1631 1997 y Fh(~)1611 2014 y Fm(X)1694 1931 y Fp(m)p Fq(\()6 b(~)-51 b Fp(x;)15 b Fq(1\))g Fp(d)7 b Fq(~)-52 b Fp(\026)r Fq(\()6 b(~)-51 b Fp(x)p Fq(\))26 b(=)e(0)448 2181 y(using)30 b(the)i(conca)m(vit)m(y)h(of)f(the)g(log)h (and)d(that)j Fp(m)p Fq(\()p Fo(\001)p Fp(;)15 b(pZ)2333 2148 y Fi(0)2356 2181 y Fq(\))32 b(is)e Fp(\021)s Fq(-measurable:)43 b(a)32 b(simple)448 2294 y(computation)f(sho)m(ws)f(that)h Fp(m)p Fq(\()6 b(~)-51 b Fp(x;)15 b(S)5 b Fq(\))26 b(=)f Fp(m)p Fq(\()6 b(~)-51 b Fp(y)s(;)15 b(S)5 b Fq(\))31 b(if)e Fp(\021)s Fq(\()6 b(~)-51 b Fp(x)q Fq(\))25 b(=)g Fp(\021)s Fq(\()6 b(~)-51 b Fp(y)t Fq(\).)589 2407 y(This)29 b(concludes)h(the)g(pro)s(of)g(of)g(Prop)s(osition)f(\(2.1\).)448 2693 y Fr(4)135 b Fb(\003)45 b Fr(de\014nes)g(a)g(conformal)h(measure.) 448 2896 y Fq(W)-8 b(e)43 b(are)f(going)g(to)g(pro)m(v)m(e)g(that)g (\003)g(de\014nes)f(a)h(conformal)f(measure)g(and)g(this)g(will)448 3009 y(conclude)30 b(the)h(pro)s(of)e(of)i(our)f(Main)g(Theorem.)589 3122 y(Let)46 b Fp(C)832 3136 y Fm(c)867 3122 y Fq(\()p Fp(S)5 b Fq(\))45 b(b)s(e)g(the)g(set)g(of)h(con)m(tin)m(uous)e (functions)g(with)g(compact)i(compact)448 3235 y(supp)s(ort)29 b(con)m(tained)i(in)e Fp(S)5 b Fq(.)40 b(Of)30 b(course)h Fp(C)1902 3249 y Fm(c)1936 3235 y Fq(\()p Fp(X)7 b Fq(\))27 b(=)e Fp(C)7 b Fq(\()p Fp(X)g Fq(\),)31 b(as)f Fp(X)38 b Fq(is)30 b(compact.)589 3347 y(Since)e(\003)888 3366 y Fi(j)p Fm(C)5 b Fh(\()p Fm(X)g Fh(\))1114 3347 y Fq(is)28 b(a)h(p)s(ositiv)m(e)f(linear)f(form)h(on)h Fp(C)7 b Fq(\()p Fp(X)g Fq(\),)30 b(the)e(Riesz)h(represen)m(tation)448 3460 y(theorem)i(implies)d(that)j(there)f(exists)g Fp(\027)6 b Fq(,)30 b(a)h(p)s(ositiv)m(e)f(Borel)g(measure,)h(suc)m(h)f(that:) 1399 3662 y Fo(8)p Fp(f)k Fo(2)25 b Fp(C)7 b Fq(\()p Fp(X)g Fq(\))26 b(:)86 b(\003\()p Fp(f)10 b Fq(\))25 b(=)g Fp(\027)6 b Fq(\()p Fp(f)k Fq(\))p Fp(:)448 3864 y Fq(In)30 b(particular,)f Fp(\027)6 b Fq(\()p Fk(1)1145 3878 y Fm(X)1213 3864 y Fq(\))25 b(=)g(\003\()p Fk(1)1519 3878 y Fm(X)1587 3864 y Fq(\))h(=)f(1)31 b(so)f(that)h Fp(\027)36 b Fq(is)30 b(a)g(probabilit)m(y)e(measure.)589 3977 y(F)-8 b(or)36 b(all)d Fp(f)41 b Fo(2)31 b Fp(C)7 b Fq(\()p Fp(X)g Fq(\),)36 b(\003\()p Fo(L)p Fp(f)10 b Fq(\))32 b(=)g Fp(\025)p Fq(\003\()p Fp(f)10 b Fq(\))32 b(=)f Fp(\025\027)6 b Fq(\()p Fp(f)k Fq(\).)52 b(But)35 b Fo(L)p Fp(f)43 b Fq(is)34 b(not)g(necessarily)448 4090 y(con)m(tin)m(uous)d(at)h(p)s(oin)m(ts)d(of)i Fp(@)5 b(T)13 b Fo(Z)34 b Fq(:=)1734 4022 y Fl(S)1810 4117 y Fm(Z)5 b Fi(2Z)1985 4090 y Fp(@)g(T)13 b(Z)7 b Fq(,)31 b(so)g(it)f(is)g(not)h(clear)g(that)h Fp(\027)6 b Fq(\()p Fo(L)p Fp(f)k Fq(\))25 b(=)448 4203 y(\003\()p Fo(L)p Fp(f)10 b Fq(\).)589 4316 y(W)-8 b(e)32 b(claim)d(the)i(follo)m(wing:) 448 4502 y Fk(Lemma)i(4.1)1690 4615 y Fp(\027)6 b Fq(\()p Fp(@)f(T)13 b Fo(Z)7 b Fq(\))26 b(=)f(0)p Fp(:)589 4800 y Fq(Let)33 b(us)d(see)j(ho)m(w)e(this)g(claim)f(implies)f(the)j (conformalit)m(y)f(of)h Fp(\027)6 b Fq(.)44 b(Fix)31 b Fp(f)36 b Fo(2)27 b Fp(C)7 b Fq(\()p Fp(X)g Fq(\),)448 4913 y(non-negativ)m(e.)42 b(W)-8 b(e)32 b(pro)m(v)m(e)f(that)g Fp(\027)6 b Fq(\()p Fo(L)p Fp(f)k Fq(\))24 b(=)h(\003\()p Fo(L)p Fp(f)10 b Fq(\):)1897 5162 y(15)p eop %%Page: 16 16 16 15 bop 589 623 a Fq(Since)37 b(the)g(probabilit)m(y)e Fp(\027)42 b Fq(is)36 b(a)i(regular)e(measure)h(\(as)h(a)f(Borel)h (measure)f(on)g(a)448 736 y(compact)e(set\),)g(w)m(e)f(ha)m(v)m(e)g (the)g(follo)m(wing)e(prop)s(ert)m(y:)45 b(F)-8 b(or)34 b(eac)m(h)h Fp(")30 b(>)g Fq(0,)35 b(there)e(exists)448 848 y Fp(U)510 862 y Fm(")577 848 y Fq(op)s(en)d(neigh)m(b)s(ourho)s(o) s(d)e(of)i Fp(@)5 b(T)13 b Fo(Z)7 b Fq(,)31 b(suc)m(h)f(that)h Fp(\027)6 b Fq(\()p Fp(U)2320 862 y Fm(")2357 848 y Fq(\))25 b Fp(<)g(")p Fq(.)589 961 y(As)31 b(usual,)e(w)m(e)i(consider)e Fp(f)1520 975 y Fm(")1582 961 y Fo(2)24 b Fp(C)1732 975 y Fm(c)1767 961 y Fq(\()p Fp(X)k Fo(n)21 b Fp(@)5 b(T)13 b Fo(Z)7 b Fq(\))31 b(suc)m(h)f(that:)1497 1074 y Fl(\032)1606 1146 y Fp(f)1651 1160 y Fm(")1713 1146 y Fq(=)25 b Fp(f)70 b(in)60 b(X)28 b Fo(n)20 b Fp(U)2300 1160 y Fm(")1606 1259 y Fp(f)1651 1273 y Fm(")1713 1259 y Fo(\024)25 b Fp(f)70 b(in)60 b(U)2132 1273 y Fm(")2169 1259 y Fp(:)448 1439 y Fq(Then:)698 1626 y Fo(j)p Fp(\027)6 b Fq(\()p Fo(L)p Fp(f)k Fq(\))20 b Fo(\000)g Fp(\025\027)6 b Fq(\()p Fp(f)k Fq(\))p Fo(j)83 b Fq(=)f Fo(j)p Fp(\027)6 b Fq(\()p Fo(L)p Fp(f)1782 1640 y Fm(")1819 1626 y Fq(\))20 b(+)g Fp(\027)6 b Fq(\()p Fo(L)p Fq(\()p Fp(f)30 b Fo(\000)20 b Fp(f)2360 1640 y Fm(")2396 1626 y Fq(\)\))h Fo(\000)f Fp(\025\027)6 b Fq(\()p Fp(f)k Fq(\))p Fo(j)1410 1764 y(\024)82 b(j)p Fp(\025\027)6 b Fq(\()p Fp(f)1772 1778 y Fm(")1809 1764 y Fq(\))21 b(+)e(2)c(sup)2053 1842 y Fm(X)2168 1764 y Fo(L)p Fk(1)2283 1778 y Fm(X)2366 1764 y Fq(sup)2402 1842 y Fm(X)2518 1764 y Fp(f)39 b(\027)6 b Fq(\()p Fp(U)2750 1778 y Fm(")2787 1764 y Fq(\))21 b Fo(\000)f Fp(\025\027)6 b Fq(\()p Fp(f)k Fq(\))p Fo(j)1410 1953 y(\024)82 b(j)p Fq(2)15 b(sup)1686 2031 y Fm(X)1801 1953 y Fp(f)10 b Fq(\(sup)1928 2031 y Fm(X)2043 1953 y Fo(L)p Fk(1)2158 1967 y Fm(X)2246 1953 y Fq(+)19 b Fp(\025)p Fq(\))p Fp(\027)6 b Fq(\()p Fp(U)2572 1967 y Fm(")2610 1953 y Fq(\))p Fo(j)1410 2141 y(\024)82 b Fq(\(2)15 b(sup)1696 2220 y Fm(X)1811 2141 y Fp(f)10 b Fq(\(sup)1938 2220 y Fm(X)2053 2141 y Fo(L)p Fk(1)2168 2155 y Fm(X)2256 2141 y Fq(+)20 b Fp(\025)p Fq(\)\))p Fp(":)448 2375 y Fq(The)28 b(conformalit)m(y)g(of)h Fp(\027)34 b Fq(follo)m(ws)28 b(using)f(the)i(follo)m(wing)d(appro)m(ximations:)39 b(Let)29 b Fp(A)g Fq(b)s(e)448 2487 y(a)37 b(Borel)g(set)g(suc)m(h)g (that)g Fp(T)48 b Fq(:)36 b Fp(A)g Fo(!)g Fp(T)13 b(A)36 b Fq(is)g(in)m(v)m(ertible)f(and)h(let)h Fp(")f(>)f Fq(0.)60 b(Since)36 b Fp(\027)42 b Fq(is)448 2600 y(regular,)30 b(there)h(exist)f Fp(K)1307 2614 y Fm(")1374 2600 y Fq(compact)i(and)e Fp(O)1984 2614 y Fm(")2051 2600 y Fq(op)s(en)g(suc)m(h)g(that)h Fp(K)2750 2614 y Fm(")2812 2600 y Fo(\032)25 b Fp(A)g Fo(\032)g Fp(O)3166 2614 y Fm(\017)3230 2600 y Fq(and:)1593 2788 y Fp(\027)6 b Fq(\()p Fp(A)p Fq(\))21 b Fo(\000)f Fp(\027)6 b Fq(\()p Fp(K)2057 2802 y Fm(")2094 2788 y Fq(\))25 b Fp(<)g(")1597 2975 y(\027)6 b Fq(\()p Fp(O)1752 2989 y Fm(")1789 2975 y Fq(\))21 b Fo(\000)e Fp(\027)6 b Fq(\()p Fp(A)p Fq(\))26 b Fp(<)f(")448 3134 y Fq(Consider)k Fp(g)871 3148 y Fm(")938 3134 y Fq(con)m(tin)m(uous)h(with)f(compact)j (supp)s(ort)d(suc)m(h)h(that:)1447 3240 y Fl(8)1447 3322 y(<)1447 3485 y(:)1569 3313 y Fp(g)1612 3327 y Fm(")1675 3313 y Fq(=)25 b(1)p Fp(=g)65 b(in)60 b(K)2192 3327 y Fm(")1569 3426 y Fp(g)1612 3440 y Fm(")1675 3426 y Fq(=)25 b(0)61 b Fp(in)f(O)2095 3393 y Fm(c)2092 3449 y(")1569 3539 y Fp(g)1612 3553 y Fm(")1675 3539 y Fo(\024)25 b Fq(1)p Fp(=g)65 b(in)60 b(O)23 b Fo(n)e Fp(K)2350 3553 y Fm(")567 3798 y Fo(j)p Fp(\027)6 b Fq(\()p Fo(L)p Fq(\()p Fk(1)828 3812 y Fm(A)906 3798 y Fo(\001)21 b Fq(1)p Fp(=g)s Fq(\)\))85 b Fo(\000)e Fp(\025\027)6 b Fq(\()p Fk(1)1588 3812 y Fm(A)1665 3798 y Fo(\001)21 b Fq(1)p Fp(=g)s Fq(\))p Fo(j)27 b(\024)1243 3936 y(\024)83 b(j)p Fp(\027)6 b Fq(\()p Fo(L)p Fq(\()p Fk(1)1658 3950 y Fm(A)1735 3936 y Fo(\001)21 b Fq(1)p Fp(=g)s Fq(\)\))h Fo(\000)e Fp(\027)6 b Fq(\()p Fo(L)p Fp(g)2292 3950 y Fm(")2329 3936 y Fq(\))21 b(+)f Fp(\025\027)6 b Fq(\()p Fp(g)2658 3950 y Fm(")2715 3936 y Fo(\000)20 b Fp(\025\027)6 b Fq(\()p Fk(1)2997 3950 y Fm(A)3074 3936 y Fo(\001)21 b Fq(1)p Fp(=g)s Fq(\))p Fo(j)1243 4074 y(\024)83 b(j)p Fp(\027)6 b Fq(\()p Fo(L)p Fq(\()p Fk(1)1658 4088 y Fm(A)1735 4074 y Fo(\001)21 b Fq(1)p Fp(=g)k Fo(\000)20 b Fp(g)2073 4088 y Fm(")2110 4074 y Fq(\)\))p Fo(j)h Fq(+)f Fp(\025)p Fo(j)p Fp(\027)6 b Fq(\()p Fp(g)2524 4088 y Fm(")2581 4074 y Fo(\000)20 b Fk(1)2724 4088 y Fm(A)2802 4074 y Fo(\001)g Fq(1)p Fp(=g)s Fq(\))p Fo(j)1243 4255 y(\024)83 b Fq(2)1452 4193 y(sup)1589 4215 y Fm(X)1671 4193 y Fo(L)p Fk(1)1786 4207 y Fm(X)p 1452 4234 402 4 v 1533 4317 a Fq(inf)1644 4331 y Fm(X)1726 4317 y Fp(g)1864 4255 y(\027)6 b Fq(\()p Fp(O)2019 4269 y Fm(")2076 4255 y Fo(n)21 b Fp(K)2219 4269 y Fm(")2256 4255 y Fq(\))f(+)g(2)2551 4193 y Fp(\025)p 2457 4234 240 4 v 2457 4317 a Fq(inf)2568 4331 y Fm(X)2651 4317 y Fp(g)2707 4255 y(\027)6 b Fq(\()p Fp(O)2862 4269 y Fm(")2920 4255 y Fo(n)20 b Fp(K)3062 4269 y Fm(")3099 4255 y Fq(\))1243 4500 y Fo(\024)83 b Fq(4)p Fp(")1499 4371 y Fl(\022)1577 4438 y Fq(sup)1713 4460 y Fm(X)1796 4438 y Fo(L)p Fk(1)1911 4452 y Fm(X)1998 4438 y Fq(+)20 b Fp(\025)p 1577 4479 566 4 v 1740 4562 a Fq(inf)1850 4576 y Fm(X)1933 4562 y Fp(g)2152 4371 y Fl(\023)448 4732 y Fq(and:)901 4864 y Fp(\025)969 4740 y Fl(Z)1019 4946 y Fm(A)1102 4803 y Fq(1)p 1101 4843 47 4 v 1101 4926 a Fp(g)1188 4864 y(d\027)32 b Fq(=)25 b Fp(\025\027)6 b Fq(\()p Fk(1)1599 4878 y Fm(A)1676 4864 y Fo(\001)21 b Fq(1)p Fp(=g)s Fq(\))27 b(=)e Fp(\027)6 b Fq(\()p Fo(L)p Fq(\()p Fk(1)2252 4878 y Fm(A)2329 4864 y Fo(\001)21 b Fq(1)p Fp(=g)s Fq(\)\))27 b(=)e Fp(\027)6 b Fq(\()p Fp(T)13 b(A)p Fq(\))p Fp(:)1897 5162 y Fq(16)p eop %%Page: 17 17 17 16 bop 589 623 a Fq(W)-8 b(e)36 b(turn)c(to)j(the)f(pro)s(of)f(of)h (the)g(lemma)g(\(4.1\))h(.)52 b(W)-8 b(e)35 b(shall)d(need)h(the)i (follo)m(wing)448 736 y(t)m(w)m(o)d(lemmas:)448 948 y Fk(Lemma)h(4.2)1435 1061 y Fp(P)1493 1075 y Fm(g)1533 1061 y Fq(\()p Fp(@)5 b(T)13 b Fo(Z)7 b Fp(;)15 b(T)e Fq(\))27 b Fo(\024)e Fp(P)2082 1075 y Fm(g)2122 1061 y Fq(\()p Fp(@)5 b Fo(Z)i Fp(;)15 b(T)e Fq(\))p Fp(:)448 1274 y Fk(Lemma)33 b(4.3)46 b Fn(F)-7 b(or)34 b(al)5 b(l)33 b(Bor)-5 b(el)34 b(sets)e Fp(B)5 b Fn(:)1219 1478 y Fp(\027)h Fq(\()p Fp(B)f Fq(\))26 b Fo(\024)f Fq(inf)5 b Fo(f)p Fq(\003\()p Fp(O)s Fq(\))p Fp(;)49 b(O)36 b Fn(op)-5 b(en)q Fp(;)48 b(O)28 b Fo(\033)d Fp(B)5 b Fo(g)p Fp(:)448 1690 y Fk(Pro)s(of)41 b Fq(:)g(\(of)31 b(Lemma)f(4.2\).)43 b(By)30 b(de\014nition:)1091 1933 y Fp(P)1149 1947 y Fm(g)1189 1933 y Fq(\()p Fp(@)5 b(T)13 b Fo(Z)7 b Fp(;)15 b(T)e Fq(\))26 b(=)f(lim)15 b(sup)1727 2005 y Fm(n)p Fi(!1)1987 1871 y Fq(1)p 1983 1912 55 4 v 1983 1995 a Fp(n)2062 1933 y Fq(log)2288 1846 y Fl(X)2265 2032 y Fg(B)r Ff(2Z)2399 2040 y Fg(n)2219 2078 y Fe(\026)2205 2091 y Fg(B)s Ff(\\)p Fg(@)s(T)8 b Ff(Z)t(6)p Fe(=)p Ff(;)2527 1933 y Fq(sup)2567 2011 y Fm(B)2679 1933 y Fp(g)2722 1947 y Fm(n)2770 1933 y Fp(:)448 2308 y Fq(Let)22 b Fp(B)30 b Fo(2)25 b(Z)853 2322 y Fm(n)921 2308 y Fq(suc)m(h)20 b(that)1326 2285 y(\026)1304 2308 y Fp(B)6 b Fo(\\)r Fp(@)f(T)13 b Fo(Z)32 b(6)p Fq(=)25 b Fo(;)p Fq(,)f(there)d(exists)g Fp(Z)32 b Fo(2)25 b(Z)j Fq(suc)m(h)21 b(that)p 2972 2216 404 4 v 22 w Fp(Z)26 b Fo(\\)20 b Fp(T)3207 2270 y Fi(\000)p Fh(1)3194 2337 y Fm(Z)3301 2308 y Fp(B)7 b Fo(\\)448 2421 y Fp(@)e Fo(Z)33 b(6)p Fq(=)25 b Fo(;)p Fq(.)41 b(Moreo)m(v)m(er,)33 b(let)d Fp(x)25 b Fo(2)g Fp(B)5 b Fq(:)1079 2684 y Fp(g)1122 2698 y Fm(n)1169 2684 y Fq(\()p Fp(x)p Fq(\))26 b Fo(\024)1423 2620 y Fp(g)1466 2634 y Fm(n)p Fh(+1)1604 2620 y Fq(\()p Fp(T)1705 2581 y Fi(\000)p Fh(1)1692 2648 y Fm(Z)1799 2620 y Fq(\()p Fp(x)p Fq(\)\))p 1423 2663 534 4 v 1490 2755 a Fp(g)s Fq(\()p Fp(T)1637 2716 y Fi(\000)p Fh(1)1624 2783 y Fm(Z)1732 2755 y Fq(\()p Fp(x)p Fq(\)\))1992 2684 y Fo(\024)2162 2622 y Fq(1)p 2098 2663 173 4 v 2098 2746 a(inf)21 b Fp(g)2372 2684 y Fq(sup)2296 2778 y Fm(Z)5 b Fi(\\)p Fm(T)2447 2749 y Ff(\000)p Fe(1)2437 2801 y Fg(Z)2529 2778 y Fm(B)2601 2684 y Fp(g)2644 2698 y Fm(n)p Fh(+1)2781 2684 y Fp(:)448 2991 y Fq(Hence)33 b(sup)857 3013 y Fm(B)933 2991 y Fp(g)976 3005 y Fm(n)1051 2991 y Fo(\024)1209 2955 y Fh(1)p 1160 2970 134 4 v 1160 3023 a(inf)17 b Fm(g)1319 2991 y Fq(sup)1455 3013 y Fm(B)1511 2994 y Ff(0)1554 2991 y Fp(g)1597 3005 y Fm(n)p Fh(+1)1766 2991 y Fq(with)31 b Fp(B)2049 2958 y Fi(0)2100 2991 y Fq(=)d Fp(Z)g Fo(\\)21 b Fp(T)2437 2953 y Fi(\000)p Fh(1)2424 3020 y Fm(Z)2531 2991 y Fp(B)32 b Fo(2)c(Z)2787 3005 y Fm(n)p Fh(+1)2924 2991 y Fq(.)46 b(Recall)31 b(that)448 3104 y(inf)21 b Fp(g)29 b(>)c Fq(0)31 b(b)m(y)f(assumption)f(\(b\))h (of)h(the)f(Main)g(Theorem.)41 b(W)-8 b(e)31 b(ha)m(v)m(e:)1262 3260 y Fl(X)1240 3446 y Fg(B)r Ff(2Z)1374 3454 y Fg(n)1193 3491 y Fe(\026)1179 3504 y Fg(B)s Ff(\\)p Fg(@)s(T)8 b Ff(Z)t(6)p Fe(=)p Ff(;)1501 3347 y Fq(sup)1542 3425 y Fm(B)1654 3347 y Fp(g)1697 3361 y Fm(n)1769 3347 y Fo(\024)1938 3285 y Fq(1)p 1875 3326 173 4 v 1875 3409 a(inf)21 b Fp(g)2155 3260 y Fl(X)2082 3454 y Fg(B)2130 3440 y Ff(0)2153 3454 y(2Z)2239 3469 y Fg(n)p Fe(+1)2097 3513 y(\026)2083 3526 y Fg(B)2131 3512 y Ff(0)2154 3526 y(\\)p Fg(@)s Ff(Z)t(6)p Fe(=)p Ff(;)2384 3347 y Fq(sup)2413 3426 y Fm(B)2469 3407 y Ff(0)2536 3347 y Fp(g)2579 3361 y Fm(n)p Fh(+1)448 3708 y Fq(and)30 b(th)m(us:)584 3928 y(lim)15 b(sup)631 4000 y Fm(n)p Fi(!1)892 3866 y Fq(1)p 888 3907 55 4 v 888 3990 a Fp(n)1035 3928 y Fq(log)1328 3841 y Fl(X)1306 4027 y Fg(B)r Ff(2Z)1440 4035 y Fg(n)1259 4073 y Fe(\026)1245 4086 y Fg(B)s Ff(\\)p Fg(@)s(T)8 b Ff(Z)t(6)p Fe(=)p Ff(;)1568 3928 y Fq(sup)1608 4006 y Fm(B)1720 3928 y Fp(g)1763 3942 y Fm(n)1058 4258 y Fo(\024)135 b Fq(lim)1236 4312 y Fm(n)p Fi(!1)1449 4196 y Fq(1)p 1445 4237 V 1445 4320 a Fp(n)1524 4258 y Fq(log)1730 4196 y(1)p 1667 4237 173 4 v 1667 4320 a(inf)21 b Fp(g)1869 4258 y Fq(+)f(lim)15 b(sup)2007 4330 y Fm(n)p Fi(!1)2347 4196 y Fq(1)p 2264 4237 212 4 v 2264 4320 a Fp(n)k Fq(+)h(1)2500 4258 y(log)2715 4171 y Fl(X)2642 4365 y Fg(B)2690 4351 y Ff(0)2713 4365 y(2Z)2799 4380 y Fg(n)p Fe(+1)2657 4425 y(\026)2643 4438 y Fg(B)2691 4424 y Ff(0)2714 4438 y(\\)p Fg(@)s Ff(Z)t(6)p Fe(=)p Ff(;)2944 4258 y Fq(sup)2973 4337 y Fm(B)3029 4318 y Ff(0)3096 4258 y Fp(g)3139 4272 y Fm(n)p Fh(+1)3276 4258 y Fp(:)3369 4596 y Fc(2)1897 5162 y Fq(17)p eop %%Page: 18 18 18 17 bop 448 623 a Fk(Pro)s(of)41 b Fq(:)g(\(of)31 b(Lemma)f(4.3\))i (Since)e Fp(\027)36 b Fq(is)29 b(regular:)1225 818 y Fp(\027)6 b Fq(\()p Fp(B)f Fq(\))25 b(=)g(inf)6 b Fo(f)p Fp(\027)g Fq(\()p Fp(O)s Fq(\))p Fp(;)46 b(O)33 b Fq(op)s(en)p Fp(;)45 b(O)28 b Fo(\033)d Fp(B)5 b Fo(g)p Fp(:)448 1013 y Fq(Let)34 b(us)f(\014x)g(an)h(op)s(en)f(set)h Fp(O)i Fq(and)d(sho)m(w)h(that:)47 b Fp(\027)6 b Fq(\()p Fp(O)s Fq(\))31 b Fo(\024)g Fq(\003\()p Fp(O)s Fq(\),)k(this)d(will)f(pro)m(v) m(e)k(the)448 1126 y(lemma.)69 b(T)-8 b(ak)m(e)41 b Fp(")h(>)f Fq(0.)69 b(Using)39 b(again)h(the)g(regularit)m(y)f(of)i Fp(\027)6 b Fq(,)42 b(there)e(exists)f Fp(K)3287 1140 y Fm(")3324 1126 y Fq(,)k(a)448 1239 y(compact)32 b(subset)e(of)g Fp(O)s Fq(,)h(suc)m(h)f(that:)1580 1434 y Fp(\027)6 b Fq(\()p Fp(O)s Fq(\))26 b Fp(<)f(\027)6 b Fq(\()p Fp(K)2058 1448 y Fm(\017)2091 1434 y Fq(\))20 b(+)g Fp(":)448 1629 y Fq(Let)31 b Fp(g)654 1643 y Fm(")717 1629 y Fq(:)25 b Fp(X)33 b Fo(!)25 b Fq([0)p Fp(;)15 b Fq(1])32 b(b)s(e)e(con)m(tin)m (uous)g(and)g(suc)m(h)g(that:)1480 1748 y Fl(8)1480 1829 y(<)1480 1993 y(:)1603 1821 y Fp(g)1646 1835 y Fm(")1708 1821 y Fq(=)25 b(1)61 b Fp(in)g(K)2134 1835 y Fm(")1603 1934 y Fp(g)1646 1948 y Fm(")1708 1934 y Fq(=)25 b(0)61 b Fp(in)g(O)2129 1901 y Fm(c)1603 2047 y Fp(g)1646 2061 y Fm(")1708 2047 y Fo(\024)25 b Fq(1)61 b Fp(in)g(O)23 b Fo(n)d Fp(K)2291 2061 y Fm(")2328 2047 y Fp(:)448 2234 y Fq(On)30 b(the)g(one)h(hand,)f Fp(g)1219 2248 y Fm(")1281 2234 y Fo(\024)25 b Fk(1)1429 2248 y Fm(O)1519 2234 y Fq(so)31 b(that)1047 2429 y Fp(\027)6 b Fq(\()p Fp(g)1176 2443 y Fm(")1213 2429 y Fq(\))26 b(=)f(\003\()p Fp(g)1511 2443 y Fm(")1548 2429 y Fq(\))h Fo(\024)f Fq(\003\()p Fp(O)s Fq(\))31 b(and)45 b(sup)2140 2505 y Fm(">)p Fh(0)2285 2429 y Fp(\027)6 b Fq(\()p Fp(g)2414 2443 y Fm(")2451 2429 y Fq(\))26 b Fo(\024)e Fq(\003\()p Fp(O)s Fq(\))p Fp(:)448 2677 y Fq(On)30 b(the)g(other)h(hand,)f Fp(\027)6 b Fq(\()p Fp(g)1376 2691 y Fm(")1413 2677 y Fq(\))25 b Fo(\025)g Fp(\027)6 b Fq(\()p Fp(K)1732 2691 y Fm(")1769 2677 y Fq(\))26 b Fp(>)f(\027)6 b 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b(T)13 b Fo(Z)34 b(6)p Fq(=)26 b Fo(;g)p Fq(.)44 b(Using)31 b(the)g(conformalit)m(y)-8 b(,)32 b(w)m(e)f(get,)i(for)448 3631 y(an)m(y)e Fp(\016)e(>)c Fq(0,)31 b Fp(N)10 b Fq(\()p Fp(\016)s Fq(\))32 b(suc)m(h)e(that,)h(for)f(all)g Fp(n)24 b(>)h(N)10 b Fq(\()p Fp(\016)s Fq(\):)522 3906 y(\003)600 3805 y Fl(\020)654 3819 y([)770 3906 y Fo(A)843 3920 y Fm(n)890 3805 y Fl(\021)970 3906 y Fo(\024)1103 3844 y Fq(1)p 1075 3885 100 4 v 1075 3968 a Fp(\025)1128 3942 y Fm(n)1292 3819 y Fl(X)1271 4005 y Fg(Z)s Ff(2Z)1402 4013 y Fg(n)1223 4051 y Fe(\026)1211 4064 y Fg(Z)s Ff(\\)p Fg(@)s(T)e Ff(Z)t(6)p Fe(=)p Ff(;)1529 3906 y Fq(sup)14 b Fp(g)1724 3920 y Fm(n)1797 3906 y Fo(\024)1893 3750 y Fl( )1975 3844 y Fp(e)2017 3811 y Fm(P)2062 3819 y Fg(g)2098 3811 y Fh(\()p Fm(@)t(T)c Fi(Z)5 b Fm(;T)10 b Fh(\)+)p Fm(\016)p 1975 3885 490 4 v 2063 3973 a Fp(e)2105 3946 y Fm(P)2150 3954 y Fg(g)2187 3946 y Fh(\()p Fm(X)q(;T)g Fh(\))2474 3750 y Fl(!)2546 3773 y Fm(n)2618 3906 y Fo(\024)2714 3750 y Fl( )2796 3844 y Fp(e)2838 3811 y Fm(P)2883 3819 y Fg(g)2920 3811 y Fh(\()p Fm(@)t Fi(Z)5 b Fm(;T)10 b Fh(\)+)p Fm(\016)p 2796 3885 439 4 v 2860 3973 a Fp(e)2902 3946 y Fm(P)2947 3954 y Fg(g)2983 3946 y Fh(\()p Fm(X)q(;T)g Fh(\))3245 3750 y Fl(!)3317 3773 y Fm(n)448 4241 y Fq(b)m(y)32 b(Lemma)g(4.2.)46 b(T)-8 b(aking)32 b Fp(\016)f Fq(=)c(\()p Fp(P)1655 4255 y Fm(g)1696 4241 y Fq(\()p Fp(X)r(;)15 b(T)e Fq(\))23 b Fo(\000)e Fp(P)2122 4255 y Fm(g)2162 4241 y Fq(\()p Fp(@)5 b Fo(Z)i Fp(;)15 b(T)e Fq(\)\))p Fp(=)p Fq(2,)35 b(whic)m(h)30 b(is)h(p)s(ositiv)m(e)g(b)m(y)448 4354 y(assumption)e(\(a\))j(of)e(the)h(Main)f(Theorem,)g(w)m(e)h(get:) 1660 4572 y(lim)1632 4626 y Fm(n)p Fi(!1)1831 4572 y Fq(\003)1909 4471 y Fl(\020)1963 4485 y([)2079 4572 y Fo(A)2152 4586 y Fm(n)2199 4471 y Fl(\021)2279 4572 y Fq(=)24 b(0)902 b(\(2\))448 4800 y(If)44 b Fp(Z)54 b Fo(2)47 b(Z)843 4814 y Fm(n)919 4800 y Fo(n)29 b(A)1066 4814 y Fm(n)1113 4800 y Fq(,)47 b(then)1424 4777 y(\026)1405 4800 y Fp(Z)36 b Fo(\\)29 b Fp(@)5 b(T)13 b Fo(Z)54 b Fq(=)48 b Fo(;)p Fq(,)f(and)c(since)g(b)s(oth)g(sets)h(are)h(compact:) 448 4913 y Fp(d)p Fq(\()550 4890 y(\026)530 4913 y Fp(Z)8 b(;)15 b(@)5 b(T)13 b Fo(Z)7 b Fq(\))26 b Fp(>)f Fq(0)p Fp(:)1897 5162 y Fq(18)p eop %%Page: 19 19 19 18 bop 589 623 a Fq(As)21 b(the)g(set)h Fo(Z)1060 637 y Fm(n)1108 623 y Fo(n)q(A)1227 637 y Fm(n)1295 623 y Fq(is)e(\014nite,)i(inf)1738 641 y Fm(Z)5 b Fi(2Z)1889 649 y Fg(n)1930 641 y Fi(nA)2021 649 y Fg(n)2084 623 y Fp(d)p Fq(\()2186 600 y(\026)2166 623 y Fp(Z)i(;)15 b(@)5 b(T)13 b Fo(Z)7 b Fq(\))26 b Fp(>)f Fq(0)p Fp(:)d Fq(Th)m(us)d(the)j(follo)m(wing)448 736 y(set)31 b(is)f(a)g(neigh)m(b)s (orho)s(o)s(d)e(of)j Fp(@)5 b(T)13 b Fo(Z)7 b Fq(:)983 940 y Fp(O)1052 954 y Fm(n)1124 940 y Fq(=)25 b Fo(f)p Fp(x)h Fo(2)f Fp(X)r(;)15 b(d)p Fq(\()p Fp(x;)g(@)5 b(T)13 b Fo(Z)7 b Fq(\))28 b Fp(<)133 b Fq(inf)2071 1005 y Fm(Z)5 b Fi(2Z)2222 1013 y Fg(n)2263 1005 y Fi(nA)2354 1013 y Fg(n)2413 940 y Fp(d)p Fq(\()2514 917 y(\026)2495 940 y Fp(Z)i(;)15 b(@)5 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b(as)h(equilibrium)17 b(states.)28 b Fn(pr)-5 b(eprint)26 b(I.M.L.)p Fq(,)d Fk(98{23)g Fq(\(1998\).)494 2521 y([3])46 b(J.)40 b(Buzzi,)i(V.)f (Maume-Desc)m(hamps.)70 b(Deca)m(y)42 b(of)e(correlations)f(for)h (piecewise)635 2634 y(in)m(v)m(ertible)29 b(maps)h(in)f(higher)g (dimensions.)38 b Fn(pr)-5 b(eprint)32 b Fq(\(1999\).)494 2821 y([4])46 b(M.)32 b(Denk)m(er,)g(C.)f(Grillen)m(b)s(erger,)e(K.)h (Sigm)m(und.)41 b(Ergo)s(dic)30 b(theory)h(on)g(compact)635 2934 y(spaces.)61 b Fn(L)-5 b(e)g(ctur)g(e)39 b(Notes)g(in)g (Mathematics)p Fq(,)h Fk(527)p Fq(,)63 b(Springer-V)-8 b(erlag,)37 b(Berlin-)635 3047 y(New)31 b(Y)-8 b(ork,)31 b(1976.)494 3235 y([5])46 b(M.)f(Denk)m(er,)k(M.)c(Urba)s(\023)-48 b(nski.)80 b(On)43 b(the)i(existence)g(of)f(conformal)g(measures.)635 3348 y Fn(T)-7 b(r)i(ans.)34 b(A)n(mer.)e(Math.)h(So)-5 b(c.)p Fq(,)31 b Fk(328)h Fq(\(1991\),)h(563{587.)494 3535 y([6])46 b(N.)e(Dunford,)h(J.)e(Sc)m(h)m(w)m(artz.)80 b Fn(Line)-5 b(ar)46 b(op)-5 b(er)g(ators.)47 b(Part)e(I.)g(Gener)-5 b(al)45 b(the)-5 b(ory.)635 3648 y Fq(John)30 b(Wiley)f(&)h(Sons,)g (Inc.,)h(New)f(Y)-8 b(ork,)31 b(1988.)494 3836 y([7])46 b(F.)26 b(Ledrappier.)k(Some)c(prop)s(erties)d(of)j(absolutely)e(con)m (tin)m(uous)h(in)m(v)-5 b(arian)m(t)24 b(mea-)635 3949 y(sures)30 b(on)g(an)g(in)m(terv)-5 b(al.)40 b Fn(Er)-5 b(go)g(d.)34 b(Th.)f(Dynam.)g(Syst.)p Fq(,)e Fk(1)g Fq(\(1981\),)i (77{93.)494 4136 y([8])46 b(C.)37 b(Liv)m(erani,)g(B.)g(Saussol,)h(S.)e (V)-8 b(aien)m(ti.)60 b(Conformal)36 b(measure)h(and)f(deca)m(y)i(of) 635 4249 y(correlation)h(for)f(co)m(v)m(ering)i(w)m(eigh)m(ted)f (systems.)65 b Fn(Er)-5 b(go)g(d.)42 b(Th.)f(Dynam.)f(Syst.)p Fq(,)635 4362 y Fk(18)31 b Fq(\(1998\),)i(1399{1420.)448 4687 y(J)m(\023)-43 b(er^)e(ome)32 b(Buzzi)448 4800 y(C.N.R.S.)f(/)g (Institut)e(de)h(Math)m(\023)-43 b(ematiques)32 b(de)e(Lumin)m(y)448 4913 y(163,)i(a)m(v.)42 b(de)30 b(Lumin)m(y)f(-)h(Case)h(930)1897 5162 y(19)p eop %%Page: 20 20 20 19 bop 448 623 a Fq(13288)33 b(Marseille)c(cedex)i(09)g(-)g(F)-8 b(rance)448 736 y(e-mail:)40 b Fa(buzzi@iml.univ-mrs.fr)448 961 y Fq(F)-8 b(r)m(\023)-43 b(ed)m(\023)g(eric)33 b(P)m(accaut)448 1074 y(Univ)m(ersit)m(\023)-43 b(e)31 b(de)f(Bourgogne)448 1187 y(Lab)s(oratoire)h(de)f(T)-8 b(op)s(ologie)30 b(U.M.R.)h(5584)h (du)e(C.N.R.S.)448 1300 y(B.P)-8 b(.)32 b(47870)g(-)f(21078)i(Dijon)c (Cedex)i(-)f(F)-8 b(rance)448 1413 y(e-mail:)40 b Fa(fpaccaut@top)s (olog.u-b)s(ourgogne.fr)448 1639 y Fq(Bernard)30 b(Sc)m(hmitt)448 1752 y(Univ)m(ersit)m(\023)-43 b(e)31 b(de)f(Bourgogne)448 1865 y(Lab)s(oratoire)h(de)f(T)-8 b(op)s(ologie)30 b(U.M.R.)h(5584)h (du)e(C.N.R.S.)448 1978 y(B.P)-8 b(.)32 b(47870)g(-)f(21078)i(Dijon)c (Cedex)i(-)f(F)-8 b(rance)448 2090 y(e-mail:)40 b Fa (schmittb@satie.u-b)s(ourgogne.fr)1897 5162 y Fq(20)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9910261529511--