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Fp(\))p Fo(x)i Fp(:=)2676 3811 y Fg(P)2763 3832 y Fl(n)p Fi(\000)p Fm(1)2763 3898 y Fl(i)p Fm(=0)2907 3874 y Fo(f)33 b Fn(\016)24 b Fp(\002)3112 3843 y Fl(i)3139 3874 y Fp(\()p Fo(!)s(;)14 b(x)p Fp(\))38 b(=)456 3924 y Fg(P)543 3944 y Fl(n)p Fi(\000)p Fm(1)543 4011 y Fl(i)p Fm(=0)687 3986 y Fo(f)9 b Fp(\()p Fo(#)818 3956 y Fl(i)846 3986 y Fo(!)s Fp(\))18 b Fn(\016)g Fo(')p Fp(\()p Fo(i;)c(!)s Fp(\))p Fo(x)28 b Fp(and)g(put)650 4210 y Fo(\031)697 4222 y Fl(')745 4210 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(!)s(;)14 b(n;)g(\017)p Fp(\))23 b(:=)g(sup)1409 4068 y Fg(\()1483 4131 y(X)1476 4309 y Fl(x)p Fi(2)p Fl(G)1624 4210 y Fp(exp)13 b Fo(F)1817 4222 y Fl(n)1863 4210 y Fp(\()p Fo(!)s Fp(\))p Fo(x)24 b Fn(j)f Fo(G)g Fn(\032)g Fo(E)2336 4222 y Fl(!)2412 4210 y Fp(is)k(\()p Fo(!)s(;)14 b(n;)g(\017)p Fp(\)-separated)3146 4068 y Fg(\))3227 4210 y Fo(:)456 4438 y Fp(One)38 b(can)g(see)h(as)f (in)h(Bogensc)n(h)r(\177)-44 b(utz)37 b([Bog92)n(,)42 b(Lemma)c(5.3])g(that)h Fo(!)44 b Fn(7!)e Fo(\031)2911 4450 y Fl(')2959 4438 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(!)s(;)14 b(n;)g(\017)p Fp(\))39 b(is)456 4537 y(measurable)26 b(for)h(eac)n(h)g Fo(n)c Fn(2)g Fj(N)38 b Fp(and)27 b(an)n(y)g (measurable)f Fo(\017)d Fp(:)g(\012)g Fn(!)g Fp(\(0)p Fo(;)14 b Fp(1].)555 4637 y(F)-7 b(or)27 b Fo(f)32 b Fn(2)23 b Fj(L)905 4607 y Fm(1)905 4660 y Fl(E)967 4637 y Fp(\(\012)p Fo(;)14 b Fn(C)5 b Fp(\()p Fo(M)k Fp(\)\))28 b(and)f Fo(\017)c Fp(:)g(\012)g Fn(!)g Fp(\(0)p Fo(;)14 b Fp(1])27 b(put)1084 4826 y Fo(\031)1131 4838 y Fl(')1179 4826 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(\017)p Fp(\))23 b(:=)g(lim)14 b(sup)1565 4892 y Fl(n)p Fi(!1)1807 4770 y Fp(1)p 1803 4807 50 4 v 1803 4883 a Fo(n)1876 4713 y Fg(Z)1973 4826 y Fp(log)g Fo(\031)2141 4838 y Fl(')2190 4826 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(!)s(;)14 b(n;)g(\017)p Fp(\))p Fo(d)p Fj(P)p Fp(\()p Fo(!)s Fp(\))p Fo(:)555 5016 y Fp(F)-7 b(or)32 b(the)h(sak)n(e)f(of)h(distinction,)h(w)n(e)e (denote)h(small)f(n)n(um)n(b)r(ers)h(b)n(y)f Fo(\016)k Fp(and)d(use)f Fo(\017)h Fp(\(p)r(ossibly)456 5116 y(with)d (subscripts\))g(exclusiv)n(ely)e(for)i(small)f(random)g(v)-5 b(ariables)29 b(in)h(this)g(section.)43 b(With)31 b(this,)456 5216 y(the)d(pressure)e Fo(\031)970 5228 y Fl(')1018 5216 y Fp(\()p Fo(f)9 b Fp(\))28 b(of)f Fo(')h Fp(at)g Fo(f)36 b Fp(ma)n(y)27 b(b)r(e)h(de\014ned)g(b)n(y)f Fo(\031)2256 5228 y Fl(')2304 5216 y Fp(\()p Fo(f)9 b Fp(\))24 b(:=)e(lim)2668 5228 y Fl(\016)r Fi(#)p Fm(0)2785 5216 y Fo(\031)2832 5228 y Fl(')2880 5216 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(\016)s Fp(\).)p eop %%Page: 3 3 3 2 bop 947 226 a Fm(HA)n(USDORFF)28 b(DIMENSION)g(UNDER)h(RANDOM)g (PER)-5 b(TURBA)g(TION)458 b(3)555 425 y Fp(It)39 b(seems)f(to)g(b)r(e) h(a)f(bit)h(unnatural)f(to)g(\014rst)g(in)n(tegrate)f(and)i(then)g(go)e (to)h(the)h(limit)g(in)456 525 y(the)32 b(de\014nition)h(of)f Fo(\031)1123 537 y Fl(')1172 525 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(\017)p Fp(\).)51 b(This,)34 b(ho)n(w)n(ev)n(er,)d(turned)i(out)f(to) h(b)r(e)f(imp)r(ortan)n(t)g(in)h(pro)n(ving)456 624 y(the)h(v)-5 b(ariational)33 b(principle)h(\(cf.)h([Bog92)m(]\).)57 b(A)35 b(more)e(natural)g(approac)n(h)g(is)h(based)f(on)h(the)456 724 y(follo)n(wing.)456 871 y Ft(1.2.)41 b(Lemma.)d Fk(F)-6 b(or)30 b Fo(f)i Fn(2)23 b Fj(L)1377 840 y Fm(1)1377 893 y Fl(E)1439 871 y Fp(\(\012)p Fo(;)14 b Fn(C)5 b Fp(\()p Fo(M)k Fp(\)\))30 b Fk(and)g Fo(\017)23 b Fp(:)g(\012)g Fn(!)g Fp(\(0)p Fo(;)14 b Fp(1])29 b Fk(the)h(limit)1250 1046 y Fp(lim)14 b(sup)1290 1112 y Fl(n)p Fi(!1)1532 990 y Fp(1)p 1528 1027 50 4 v 1528 1103 a Fo(n)1602 1046 y Fp(log)g Fo(\031)1770 1058 y Fl(')1818 1046 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(!)s(;)14 b(n;)g(\017)p Fp(\))23 b(=:)f Fo(\031)2392 1012 y Fi(\003)2389 1067 y Fl(')2437 1046 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(\017)p Fp(\))456 1221 y Fk(is)30 b Fj(P)p Fk(-a.)13 b(s.)30 b(indep)l(endent)h(of)f Fo(!)s Fk(.)456 1368 y(Pr)l(o)l(of.)43 b Fp(W)-7 b(rite)39 b Fo(\031)1008 1338 y Fi(\003)1005 1388 y Fl(')1053 1368 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(!)s(;)14 b(\017)p Fp(\))43 b(:=)f(lim)14 b(sup)1784 1388 y Fl(n)p Fi(!1)1989 1335 y Fm(1)p 1985 1349 42 4 v 1985 1396 a Fl(n)2050 1368 y Fp(log)g Fo(\031)2218 1380 y Fl(')2266 1368 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(!)s(;)14 b(n;)g(\017)p Fp(\).)72 b(W)-7 b(e)39 b(will)h(sho)n(w)e(that)456 1470 y Fo(\031)506 1440 y Fi(\003)503 1491 y Fl(')551 1470 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(!)s(;)14 b(\017)p Fp(\))28 b Fn(\024)f Fo(\031)1025 1440 y Fi(\003)1022 1491 y Fl(')1070 1470 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(#!)s(;)14 b(\017)p Fp(\).)46 b(F)-7 b(rom)30 b(this)h(w)n(e)g(conclude)f(that)h Fo(\031)2581 1440 y Fi(\003)2578 1491 y Fl(')2626 1470 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(!)s(;)14 b(\017)p Fp(\))30 b(is)h Fo(#)p Fp(-in)n(v)-5 b(arian)n(t)456 1573 y(\(see)39 b(Arnold)g([Arn98)o(,)j (Lemma)d(3.1.4]\))g(whic)n(h,)j(b)n(y)d(ergo)r(dicit)n(y)-7 b(,)41 b(immediately)f(giv)n(es)e(the)456 1672 y(claim.)555 1772 y(Let)24 b Fo(G)f Fp(b)r(e)h(an)f(\()p Fo(!)s(;)14 b(n;)g(\017)p Fp(\)-separated)21 b(set.)36 b(Then)2074 1751 y(^)2055 1772 y Fo(G)24 b Fp(:=)e Fo(')p Fp(\()p Fo(!)s Fp(\))p Fo(G)i Fp(is)g(\()p Fo(#!)s(;)14 b(n)c Fn(\000)g Fp(1)p Fo(;)k(\017)p Fp(\)-separated.)555 1871 y(Using)28 b Fo(F)842 1883 y Fl(n)887 1871 y Fp(\()p Fo(!)s Fp(\))p Fo(x)c Fp(=)f Fo(f)9 b Fp(\()p Fo(!)s Fp(\))p Fo(x)18 b Fp(+)g Fo(F)1535 1883 y Fl(n)p Fi(\000)p Fm(1)1666 1871 y Fp(\()p Fo(#!)s Fp(\))p Fo(')p Fp(\()p Fo(!)s Fp(\))p Fo(x)29 b Fp(w)n(e)e(obtain)1048 1944 y Fg(X)1041 2122 y Fl(x)p Fi(2)p Fl(G)1189 2023 y Fp(exp)14 b Fo(F)1383 2035 y Fl(n)1429 2023 y Fp(\()p Fo(!)s Fp(\))p Fo(x)23 b Fp(=)1713 1944 y Fg(X)1706 2122 y Fl(x)p Fi(2)p Fl(G)1854 2023 y Fo(e)1893 1988 y Fl(f)7 b Fm(\()p Fl(!)r Fm(\))p Fl(x)2083 2023 y Fp(exp)2210 1955 y Fg(\000)2248 2023 y Fo(F)2301 2035 y Fl(n)p Fi(\000)p Fm(1)2431 2023 y Fp(\()p Fo(#!)s Fp(\))p Fo(')p Fp(\()p Fo(!)s Fp(\))p Fo(x)2819 1955 y Fg(\001)1618 2255 y Fn(\024)23 b Fo(e)1745 2221 y Fi(k)p Fl(f)7 b Fm(\()p Fl(!)r Fm(\))p Fi(k)1972 2176 y Fg(X)1965 2369 y Fl(x)p Fi(2)2062 2354 y Fm(^)2048 2369 y Fl(G)2113 2255 y Fp(exp)14 b Fo(F)2307 2267 y Fl(n)p Fi(\000)p Fm(1)2437 2255 y Fp(\()p Fo(#!)s Fp(\))p Fo(x)1618 2493 y Fn(\024)23 b Fo(e)1745 2459 y Fi(k)p Fl(f)7 b Fm(\()p Fl(!)r Fm(\))p Fi(k)1951 2493 y Fo(\031)1998 2505 y Fl(')2047 2493 y Fp(\()p Fo(f)i Fp(\)\()p Fo(#!)s(;)14 b(n)k Fn(\000)g Fp(1)p Fo(;)c(\017)p Fp(\))p Fo(:)456 2633 y Fp(Since)27 b(this)h(holds)g(true)f(for)g(all)g(\()p Fo(!)s(;)14 b(n;)g(\017)p Fp(\)-separated)26 b(sets)h Fo(G)h Fp(w)n(e)f(ha)n(v)n(e)1190 2774 y Fo(\031)1237 2786 y Fl(')1285 2774 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(!)s(;)14 b(n;)g(\017)p Fp(\))23 b Fn(\024)g Fo(e)1826 2740 y Fi(k)p Fl(f)7 b Fm(\()p Fl(!)r Fm(\))p Fi(k)2032 2774 y Fo(\031)2079 2786 y Fl(')2127 2774 y Fp(\()p Fo(f)i Fp(\)\()p Fo(#!)s(;)14 b(n)19 b Fn(\000)f Fp(1)p Fo(;)c(\017)p Fp(\))456 2909 y(whic)n(h)27 b(yields)h(the)g(desired.)p 3384 2909 4 57 v 3388 2856 50 4 v 3388 2909 V 3437 2909 4 57 v 456 3058 a(1.3)p Fk(.)37 b(R)l(emark.)42 b Fp(It)28 b(can)f(in)h(fact)g(b)r (e)g(sho)n(wn)f(that)h Fo(\031)2065 3070 y Fl(')2113 3058 y Fp(\()p Fo(f)9 b Fp(\))23 b(=)g(lim)2453 3070 y Fl(\016)r Fi(#)p Fm(0)2570 3058 y Fo(\031)2620 3028 y Fi(\003)2617 3079 y Fl(')2666 3058 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(\016)s Fp(\).)456 3208 y(The)25 b(follo)n(wing)f(result) h(describ)r(es)f(a)h(situation)g(where)f Fo(\031)2247 3220 y Fl(')2295 3208 y Fp(\()p Fo(f)9 b Fp(\))26 b(can)e(b)r(e)i (obtained)f(as)f(the)i(limit)456 3307 y(of)h Fo(\031)597 3319 y Fl(')645 3307 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(\017)825 3319 y Fl(k)866 3307 y Fp(\))28 b(for)f(certain)g(sequences)g(\()p Fo(\017)1773 3319 y Fl(k)1814 3307 y Fp(\))h(of)f(random)g(v)-5 b(ariables)26 b(with)j Fo(\017)2841 3319 y Fl(k)2904 3307 y Fn(!)23 b Fp(0.)456 3454 y Ft(1.4.)41 b(Prop)s(osition.)e Fk(Assume)34 b Fo(E)40 b Fk(is)35 b(a)g(c)l(omp)l(act)h(bund)t(le)f (with)h(\014b)l(ers)e(in)h(a)h(\014nite)e(dimen-)456 3553 y(sional)c(R)n(iemannian)h(manifold)g Fo(M)39 b Fk(satisfying)1468 3613 y Fg(Z)1564 3726 y Fp(log)15 b(diam)e Fo(E)1940 3738 y Fl(!)1989 3726 y Fo(d)p Fj(P)p Fp(\()p Fo(!)s Fp(\))22 b Fo(<)g Fn(1)p Fo(;)-1962 b Fp(\(1\))456 3898 y Fk(wher)l(e)30 b Fp(diam)g Fk(is)g(me)l(asur)l(e)l (d)f(with)i(r)l(esp)l(e)l(ct)e(to)h(the)g(R)n(iemannian)g(distanc)l(e)g (on)g Fo(M)9 b Fk(.)555 3998 y(If)30 b Fp(\()p Fo(\017)708 4010 y Fl(k)749 3998 y Fp(\))781 4010 y Fl(k)q Fi(2)p Ff(N)939 3998 y Fk(is)g(a)g(se)l(quenc)l(e)f(of)i(me)l(asur)l(able)f Fo(\017)1992 4010 y Fl(k)2056 3998 y Fp(:)23 b(\012)g Fn(!)g Fp(\(0)p Fo(;)14 b Fp(1])29 b Fk(with)532 4115 y Fp(\(i\))42 b(log)14 b Fo(\017)816 4127 y Fl(k)880 4115 y Fn(2)23 b Fj(L)1008 4084 y Fm(1)1051 4115 y Fp(\()p Fj(P)p Fp(\))29 b Fk(for)i(al)t(l)f Fo(k)c Fn(2)e Fj(N)39 b Fk(and)509 4214 y Fp(\(ii\))j(lim)777 4226 y Fl(k)q Fi(!1)964 4214 y Fo(\017)998 4226 y Fl(k)1061 4214 y Fp(=)23 b(0)29 b Fk(uniformly)i(in)f Fo(!)s Fk(,)456 4331 y(then)1540 4466 y Fp(lim)1513 4520 y Fl(k)q Fi(!1)1696 4466 y Fo(\031)1743 4478 y Fl(')1791 4466 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(\017)1971 4478 y Fl(k)2012 4466 y Fp(\))23 b(=)g Fo(\031)2202 4478 y Fl(')2250 4466 y Fp(\()p Fo(f)9 b Fp(\))p Fo(:)456 4653 y Fk(Pr)l(o)l(of.)43 b Fp(By)33 b(Nash's)g(Em)n(b)r(edding)h(Theorem)e Fo(M)43 b Fp(can)33 b(b)r(e)h(isometrically)e(em)n(b)r(edded)j(in)e Fj(R)3375 4623 y Fl(N)456 4753 y Fp(for)27 b Fo(N)37 b Fp(big)28 b(enough)g(\(cf.)h(Gromo)n(v)d(and)i(Rokhlin)g([GR70]\).)39 b(Therefore)27 b(it)h(su\016ces)g(to)g(pro)n(v)n(e)456 4853 y(the)g(statemen)n(t)f(for)g Fo(M)32 b Fp(=)23 b Fj(R)1366 4822 y Fl(N)1434 4853 y Fp(,)28 b(sa)n(y)-7 b(.)555 4952 y(First)40 b(note)g(that)h(clearly)e Fo(\017)1472 4964 y Fm(1)1553 4952 y Fo(<)44 b(\017)1696 4964 y Fm(2)1773 4952 y Fp(implies)d Fo(\031)2115 4964 y Fl(')2163 4952 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(\017)2343 4964 y Fm(1)2380 4952 y Fp(\))44 b Fn(\025)g Fo(\031)2612 4964 y Fl(')2660 4952 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(\017)2840 4964 y Fm(2)2878 4952 y Fp(\).)75 b(With)41 b Fo(\016)3272 4964 y Fl(k)3357 4952 y Fp(:=)456 5052 y(ess)14 b(sup)f Fo(\017)745 5064 y Fl(k)813 5052 y Fp(w)n(e)27 b(ha)n(v)n(e)g Fo(\031)1174 5064 y Fl(')1222 5052 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(\017)1402 5064 y Fl(k)1443 5052 y Fp(\))23 b Fn(\025)g Fo(\031)1633 5064 y Fl(')1681 5052 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(\016)1864 5064 y Fl(k)1905 5052 y Fp(\).)37 b(Since)28 b(lim)2329 5064 y Fl(k)q Fi(!1)2516 5052 y Fo(\016)2553 5064 y Fl(k)2617 5052 y Fp(=)23 b(0)k(b)n(y)g(\(ii\),)i(th)n(us,)1482 5186 y(lim)14 b(inf)1513 5240 y Fl(k)q Fi(!1)1726 5186 y Fo(\031)1773 5198 y Fl(')1821 5186 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(\017)2001 5198 y Fl(k)2042 5186 y Fp(\))24 b Fn(\025)e Fo(\031)2232 5198 y Fl(')2281 5186 y Fp(\()p Fo(f)9 b Fp(\))p Fo(:)p eop %%Page: 4 4 4 3 bop 456 238 a Fm(4)913 b(T.)29 b(BOGENSCH)1880 223 y(\177)1872 238 y(UTZ)h(AND)f(G.)g(OCHS)456 425 y Fp(W)-7 b(e)35 b(ha)n(v)n(e)g(to)g(sho)n(w)f(that)i(lim)14 b(sup)1570 445 y Fl(k)q Fi(!1)1757 425 y Fo(\031)1804 437 y Fl(')1852 425 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(\017)2032 437 y Fl(k)2073 425 y Fp(\))37 b Fn(\024)e Fo(\031)2289 437 y Fl(')2338 425 y Fp(\()p Fo(f)9 b Fp(\).)60 b(F)-7 b(or)35 b(that)h(purp)r(ose,)g (w)n(e)f(\014x)456 525 y Fo(\017)h Fp(with)i(log)14 b Fo(\017)38 b Fn(2)i Fj(L)1062 495 y Fm(1)1105 525 y Fp(\()p Fj(P)p Fp(\))d(and)f(sho)n(w)h(that)g(there)g(exists)f(a)h Fo(')p Fp(-in)n(v)-5 b(arian)n(t)36 b(measure)g Fo(\026)h Fp(and)g(a)456 624 y(coun)n(table)27 b(partition)g Fn(Z)34 b Fp(of)28 b Fo(E)k Fp(with)d Fo(H)1719 636 y Fl(\026)1763 624 y Fp(\()p Fn(Z)h(j)23 b Fo(\031)1981 594 y Fi(\000)p Fm(1)2071 624 y Fn(F)8 b Fp(\))23 b Fo(<)g Fn(1)k Fp(suc)n(h)h(that) 1401 824 y Fo(\031)1448 836 y Fl(')1496 824 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(\017)p Fp(\))24 b Fn(\024)f Fo(h)1868 836 y Fl(\026)1912 824 y Fp(\()p Fo(')p Fp(;)14 b Fn(Z)7 b Fp(\))19 b(+)2236 711 y Fg(Z)2333 824 y Fo(f)9 b(d\026:)-2043 b Fp(\(2\))456 1038 y(Here)34 b Fo(h)707 1050 y Fl(\026)751 1038 y Fp(\()p Fo(')p Fp(;)14 b Fn(Z)7 b Fp(\))35 b(=)f(lim)1222 1050 y Fl(n)p Fi(!1)1428 1005 y Fm(1)p 1424 1019 42 4 v 1424 1066 a Fl(n)1489 971 y Fg(R)1558 1038 y Fo(H)1627 1050 y Fl(\026)1667 1058 y Fe(!)1714 970 y Fg(\000)1752 975 y(W)1821 996 y Fl(n)p Fi(\000)p Fm(1)1821 1062 y Fl(i)p Fm(=0)1965 1038 y Fo(')2019 1007 y Fi(\000)p Fm(1)2108 1038 y Fp(\()p Fo(i;)14 b(!)s Fp(\))p Fn(Z)7 b Fp(\()p Fo(#)2441 1007 y Fl(i)2469 1038 y Fo(!)s Fp(\))2556 970 y Fg(\001)2594 1038 y Fo(d)p Fj(P)p Fp(\()p Fo(!)s Fp(\),)35 b(where)f Fn(Z)7 b Fp(\()p Fo(!)s Fp(\))35 b(de-)456 1137 y(notes)27 b(the)h(partition)f(of)h Fo(E)1318 1149 y Fl(!)1394 1137 y Fp(in)n(to)g(the)g Fo(!)s Fp(-sections)e(of)i(the)g (atoms)f(of)g Fn(Z)7 b Fp(.)37 b(The)28 b(suprem)n(um)g(of)456 1237 y Fo(h)504 1249 y Fl(\026)548 1237 y Fp(\()p Fo(')p Fp(;)14 b Fn(Z)7 b Fp(\))29 b(o)n(v)n(er)e(all)h(coun)n(table)g (partitions)f Fn(Z)36 b Fp(of)28 b Fo(E)34 b Fp(with)29 b Fo(H)2395 1249 y Fl(\026)2439 1237 y Fp(\()p Fn(Z)j(j)24 b Fo(\031)2660 1207 y Fi(\000)p Fm(1)2750 1237 y Fn(F)8 b Fp(\))24 b Fo(<)g Fn(1)29 b Fp(conincides)456 1336 y(with)d(the)h(\014b)r(er)f(en)n(trop)n(y)e Fo(h)1323 1348 y Fl(\026)1368 1336 y Fp(\()p Fo(')p Fp(\))j(whic)n(h)f(is)g (usually)f(de\014ned)i(using)e(partitions)h Fn(P)32 b Fp(of)26 b Fo(M)35 b Fp(only)456 1436 y(\(cf.)25 b([Bog93)n(]\).)36 b(By)25 b(the)g(v)-5 b(ariational)23 b(principle,)i(th)n(us,)h(\(2\))f (yields)f Fo(\031)2612 1448 y Fl(')2661 1436 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(\017)p Fp(\))23 b Fn(\024)g Fo(\031)3031 1448 y Fl(')3079 1436 y Fp(\()p Fo(f)9 b Fp(\),)25 b(hence)456 1536 y(the)j(claim.)555 1635 y(The)k(construction)e(of)i Fo(\026)f Fp(can)g(in)h(fact)f(b)r(e)h(carried)e(out)i(as)f(in)g(part)g (\(I)r(I\))h(of)g(the)g(pro)r(of)f(of)456 1735 y([Bog92)n(,)38 b(Theorem)e(6.1].)63 b(Just)37 b(note)f(that)h(the)g(set)f(of)h(all)f (random)g(probabilities)g(on)g Fo(M)456 1835 y Fp(with)c Fo(\026)699 1847 y Fl(!)747 1835 y Fp(\()p Fo(E)840 1847 y Fl(!)889 1835 y Fp(\))e(=)g(1)h(is)h(compact)f(in)h(the)g(narro)n(w)e (top)r(ology)g(\(cf.)j(Crauel)e([Cra95)n(]\).)50 b(So)31 b(the)456 1934 y(Krylo)n(v-Bogolioub)r(o)n(v)17 b(pro)r(cedure)22 b(needed)g(here)g(still)h(w)n(orks)d(\(cf.)j([Bog93)n(,)h(Theorem)d (1.2.8]\).)555 2034 y(F)-7 b(or)25 b(the)h(c)n(hoice)f(of)h Fn(Z)32 b Fp(w)n(e)26 b(need)f(the)i(Euclidian)e(structure)g(of)h Fo(M)31 b Fp(=)23 b Fj(R)2796 2004 y Fl(N)2865 2034 y Fp(.)36 b(W)-7 b(e)26 b(pro)r(ceed)f(as)456 2145 y(follo)n(ws.)34 b(F)-7 b(or)21 b Fo(k)26 b Fn(2)e Fj(N)32 b Fp(let)23 b Fn(W)1329 2157 y Fl(k)1393 2145 y Fp(=)f Fn(f)p Fo(W)1612 2102 y Fm(\()p Fl(k)q Fm(\))1600 2168 y Fl(i)1704 2145 y Fn(g)1746 2157 y Fl(i)p Fi(2)p Ff(N)1883 2145 y Fp(b)r(e)g(a)g(co)n (v)n(ering)e(of)i Fj(R)2517 2115 y Fl(N)2609 2145 y Fp(b)n(y)g(cub)r (es)g(with)h(diameter)484 2214 y Fm(1)p 466 2228 70 4 v 466 2275 a(2)p Fl(k)573 2247 y Fp(whic)n(h)k(only)g(in)n(tersect)g (at)g(their)h(b)r(oundaries.)36 b(F)-7 b(urther,)27 b(denote)g(b)n(y)g Fo(m)h Fp(the)g(marginal)e(of)456 2370 y Fo(\026)32 b Fp(on)g Fj(R)712 2340 y Fl(N)781 2370 y Fp(.)52 b(Then)33 b(eac)n(h)f Fo(W)1360 2327 y Fm(\()p Fl(k)q Fm(\))1348 2393 y Fl(i)1485 2370 y Fp(is)g(con)n(tained)g(in)h(a)f(cub)r(e)2355 2349 y(~)2331 2370 y Fo(W)2421 2327 y Fm(\()p Fl(k)q Fm(\))2409 2393 y Fl(i)2546 2370 y Fp(with)i Fo(m)p Fp(\()p Fo(@)2919 2349 y Fp(~)2895 2370 y Fo(W)2985 2327 y Fm(\()p Fl(k)q Fm(\))2973 2393 y Fl(i)3077 2370 y Fp(\))e(=)f(0)h(and)456 2488 y(diam)674 2467 y(~)649 2488 y Fo(W)739 2445 y Fm(\()p Fl(k)q Fm(\))727 2511 y Fl(i)859 2488 y Fo(<)962 2456 y Fm(1)p 960 2470 37 4 v 960 2517 a Fl(k)1037 2488 y Fp([since)e(w)n(e)f(cannot)h(ha)n(v)n(e)f(an)g(uncoun)n(table)h (collection)f(of)h(disjoin)n(t)g(sets)g(of)456 2595 y(p)r(ositiv)n(e)h (measure].)49 b(One)31 b(can)h(c)n(hec)n(k)f(that)h(eac)n(h)f(compact)h (set)g Fo(A)e Fn(\032)g Fj(R)2810 2565 y Fl(N)2911 2595 y Fp(in)n(tersects)h(with)456 2711 y(at)d(most)h([4)p Fo(k)16 b Fp(diam)e Fo(A)1143 2641 y Fn(p)p 1212 2641 76 4 v 70 x Fo(N)28 b Fp(+)19 b(4])1456 2681 y Fl(N)1547 2711 y Fp(mem)n(b)r(ers)28 b(of)2026 2690 y(~)1995 2711 y Fn(W)2077 2723 y Fl(k)2143 2711 y Fp(=)c Fn(f)2298 2690 y Fp(~)2274 2711 y Fo(W)2364 2668 y Fm(\()p Fl(k)q Fm(\))2352 2734 y Fl(i)2456 2711 y Fn(g)2498 2723 y Fl(i)p Fi(2)p Ff(N)2612 2711 y Fp(,)29 b(where)f([)p Fo(x)p Fp(])i(denotes)e(the)456 2810 y(in)n(teger)e(part)h(of)h Fo(x)23 b Fn(2)h Fj(R)p Fp(.)555 2922 y(W)-7 b(rite)32 b(\012)850 2934 y Fl(k)920 2922 y Fp(:=)d Fn(f)1132 2889 y Fm(1)p 1088 2903 121 4 v 1088 2950 a Fl(k)q Fm(+1)1248 2922 y Fo(<)g(\017)g Fn(\024)1510 2889 y Fm(1)p 1508 2903 37 4 v 1508 2950 a Fl(k)1555 2922 y Fn(g)p Fp(.)48 b(F)-7 b(urther,)32 b(write)f Fo(P)2278 2879 y Fm(\()p Fl(k)q Fm(\))2266 2944 y(1)2400 2922 y Fp(:=)2541 2901 y(~)2517 2922 y Fo(W)2607 2879 y Fm(\()p Fl(k)q Fm(\))2595 2944 y(1)2731 2922 y Fp(and)g Fo(P)2961 2879 y Fm(\()p Fl(k)q Fm(\))2949 2945 y Fl(i)3082 2922 y Fp(:=)3224 2901 y(~)3199 2922 y Fo(W)3289 2879 y Fm(\()p 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b Fn(\024)g Fp([)1162 3790 y Fm(4)11 b(diam)g Fl(E)1410 3798 y Fe(!)1452 3741 y Fi(p)p 1506 3741 59 3 v 1506 3790 a Fl(N)p 1162 3805 404 4 v 1260 3853 a(\017)p Fm(\()p Fl(!)r Fm(\)+1)1594 3824 y Fp(+)18 b(4])1742 3794 y Fl(N)1804 3824 y Fp(,)488 3943 y(\(iv\))661 3876 y Fg(R)730 3943 y Fp(log)d(card)d Fn(Z)7 b Fp(\()p Fo(!)s Fp(\))p Fo(d)p Fj(P)p Fp(\()p Fo(!)s Fp(\))23 b Fo(<)f Fn(1)p Fp(,)456 4071 y(where)27 b(\(iv\))h(is)g(a)g(consequence)f(of)h(\(iii\))g(and)g(\(1\).)38 b(Note)28 b(also)f(that)h Fo(H)2680 4083 y Fl(\026)2725 4071 y Fp(\()p Fn(Z)j(j)23 b Fo(\031)2944 4041 y Fi(\000)p Fm(1)3034 4071 y Fn(F)8 b Fp(\))24 b Fo(<)f Fn(1)28 b Fp(b)n(y)456 4171 y(\(iv\).)555 4270 y(With)i(these)f(prop)r(erties)f (the)h(calculations)e(in)i([Bog92)n(,)h(Theorem)e(6.1])g(pro)n(ving)g (\(2\))h(can)456 4370 y(b)r(e)f(carried)e(out)h(without)h(substan)n (tial)f(c)n(hanges.)p 3384 4370 4 57 v 3388 4317 50 4 v 3388 4370 V 3437 4370 4 57 v 456 4547 a(In)g(the)h(sequel)g(w)n(e)f (will)h(b)r(e)g(in)n(terested)f(in)h(expansiv)n(eness.)456 4708 y Ft(1.5.)41 b(De\014nition.)f Fp(A)25 b(bundle)i(RDS)f Fo(')g Fp(is)f(said)g(to)h(b)r(e)f(\(p)r(ositiv)n(e\))h Fk(exp)l(ansive)g Fp(if)h(there)e(exists)456 4808 y(a)i(\(0)p Fo(;)14 b Fp(1\)-v)-5 b(alued)27 b(random)f(v)-5 b(ariable)27 b(\001)h(suc)n(h)f(that)1065 4962 y Fo(d)p Fp(\()p Fo(')p Fp(\()p Fo(n;)14 b(!)s Fp(\))p Fo(x;)g(')p Fp(\()p Fo(n;)g(!)s Fp(\))p Fo(y)s Fp(\))25 b Fn(\024)d Fp(\001\()p Fo(#)2082 4928 y Fl(n)2128 4962 y Fo(!)s Fp(\))166 b(for)27 b(all)g Fo(n)c Fn(2)h Fj(N)456 5116 y Fp(implies)j Fo(x)d Fp(=)f Fo(y)s Fp(.)555 5216 y(W)-7 b(e)28 b(call)f(\001)h(an)g Fk(exp)l(ansive)i(c)l(onstant)p Fp(.)p eop %%Page: 5 5 5 4 bop 947 226 a Fm(HA)n(USDORFF)28 b(DIMENSION)g(UNDER)h(RANDOM)g (PER)-5 b(TURBA)g(TION)458 b(5)555 425 y Fp(In)28 b(studying)f (expansiv)n(eness)e(the)j(concept)f(of)h(a)e(generator)g(and)h(the)g (de\014nition)h(of)f(pres-)456 525 y(sure)g(using)g(op)r(en)g(co)n(v)n (erings)f(is)h(useful.)456 658 y Ft(1.6.)41 b(De\014nition.)f Fp(A)h Fk(gener)l(ator)g Fp(of)g(a)f(con)n(tin)n(uous)g(bundle)h(RDS)h Fo(')f Fp(is)f(a)h(family)g Fn(A)k Fp(=)456 758 y Fn(fA)p Fp(\()p Fo(!)s Fp(\))23 b(=)f(\()p Fo(A)887 770 y Fl(i)916 758 y Fp(\()p Fo(!)s Fp(\)\))h(:)g Fn(A)p Fp(\()p Fo(!)s Fp(\))28 b(is)g(an)f(op)r(en)h(co)n(v)n(ering)d(of)j Fo(E)2230 770 y Fl(!)2278 758 y Fp(\))p Fn(g)g Fp(with)555 875 y(1.)41 b Fn(A)p Fp(\()p Fo(!)s Fp(\))29 b(is)e(\014nite)h(for)f (all)h Fo(!)d Fn(2)f Fp(\012,)555 975 y(2.)41 b Fo(!)26 b Fn(7!)d Fo(d)p Fp(\()p Fo(x;)14 b(A)1066 987 y Fl(i)1095 975 y Fp(\()p Fo(!)s Fp(\)\))28 b(is)g(measurable)e(for)h(all)g Fo(x)d Fn(2)f Fo(M)36 b Fp(and)28 b(all)f Fo(i)c Fn(2)g Fj(N)t Fp(,)555 1074 y(3.)41 b(for)31 b(eac)n(h)g(sequence)g(\()p Fo(A)1425 1086 y Fl(n)1471 1074 y Fp(\))1503 1086 y Fl(n)p Fi(2)p Ff(N)1667 1074 y Fp(of)g(sets)g(w)n(e)g(ha)n(v)n(e)g(that)h Fo(A)2499 1086 y Fl(n)2574 1074 y Fn(2)d(A)p Fp(\()p Fo(#)2805 1044 y Fl(n)2851 1074 y Fo(!)s Fp(\))j(for)f(all)g Fo(n)f Fn(2)f Fj(N)661 1174 y Fp(implies)f(that)1123 1112 y Fg(T)1192 1132 y Fi(1)1192 1199 y Fl(n)p Fm(=0)1335 1174 y Fo(')p Fp(\()p Fo(n;)14 b(!)s Fp(\))1595 1144 y Fi(\000)p Fm(1)1718 1153 y Fp(\026)1685 1174 y Fo(A)1747 1186 y Fl(n)1820 1174 y Fp(con)n(tains)26 b(at)i(most)f(one)h(p)r(oin)n (t.)555 1308 y(Going)i(along)g(the)g(argumen)n(ts)g(of)g(W)-7 b(alters)30 b([W)-7 b(al82)o(,)32 b Fn(x)e Fp(5.6])g(for)g(eac)n(h)g Fo(!)j Fp(separately)c(one)456 1407 y(can)i(c)n(hec)n(k)g(that)h(a)g (bundle)h(RDS)f(is)g(expansiv)n(e)f(if)h(and)g(only)g(if)g(it)g(p)r (ossesses)f(a)h(generator.)456 1514 y(Moreo)n(v)n(er,)g(eac)n(h)h (generator)e(satis\014es)i(lim)1847 1526 y Fl(k)q Fi(!1)2034 1514 y Fp(diam)2227 1451 y Fg(W)2297 1472 y Fl(k)q Fi(\000)p Fm(1)2297 1539 y Fl(i)p Fm(=0)2436 1514 y Fo(')p Fp(\()p Fo(i;)14 b(!)s Fp(\))2675 1484 y Fi(\000)p Fm(1)2764 1514 y Fn(A)p Fp(\()p Fo(#)2911 1484 y Fl(i)2940 1514 y Fo(!)s Fp(\))33 b(=)g(0)g(for)g Fj(P)p Fp(-)456 1613 y(a.)13 b(a.)27 b Fo(!)k Fp(\(cf.)d(W)-7 b(alters)27 b([W)-7 b(al82)o(,)28 b(Theorem)f(5.21]\).)555 1713 y(One)d(ma)n(y)g (giv)n(e)g(a)g(de\014nition)h(of)g(the)g(pressure)e(of)h(a)h(bundle)g (RDS)g(using)f(op)r(en)h(co)n(v)n(erings.)456 1813 y(Ho)n(w)n(ev)n(er,) e(to)i(mak)n(e)g(the)g(argumen)n(ts)f(of)h(W)-7 b(alters)25 b([W)-7 b(al82)o(,)26 b Fn(x)f Fp(9.1])g(w)n(ork)e(here)i(one)g(needs)g (the)456 1912 y(follo)n(wing)h(notion.)456 2046 y Ft(1.7.)41 b(De\014nition.)f Fp(A)28 b(generator)d Fn(A)j Fp(of)g Fo(')f Fp(is)h(called)f(a)g Fk(str)l(ong)j(gener)l(ator)e Fp(if)990 2258 y(lim)963 2312 y Fl(k)q Fi(!1)1146 2258 y Fp(diam)1340 2154 y Fl(k)q Fi(\000)p Fm(1)1354 2179 y Fg(_)1347 2356 y Fl(i)p Fm(=0)1475 2258 y Fo(')p Fp(\()p Fo(i;)14 b(!)s Fp(\))1714 2223 y Fi(\000)p Fm(1)1803 2258 y Fn(A)p Fp(\()p Fo(#)1950 2223 y Fl(i)1979 2258 y Fo(!)s Fp(\))23 b(=)f(0)166 b(uniformly)27 b(in)h Fo(!)s Fp(.)456 2465 y(An)22 b(expansiv)n(e)g(RDS)h(is)f(said)g(to)g(b)r(e)h Fk(str)l(ongly)i(exp)l(ansive)f Fp(if)f(it)f(p)r(ossesses)f(a)h(strong) g(generator.)456 2598 y Ft(1.8.)41 b(Example.)e Fp(In)h(a)f(recen)n(t)g (pap)r(er)g(of)g(Gundlac)n(h)h(and)f(Kifer)g([GK98)o(])h(the)g(follo)n (wing)456 2698 y(uniformit)n(y)27 b(assumption)g(on)g(an)h(expansiv)n (e)e(RDS)i(app)r(ears:)456 2857 y(\(3\))83 b(T)-7 b(o)27 b(eac)n(h)g Fo(p)c Fn(2)g Fj(N)38 b Fp(there)27 b(is)g Fo(N)1547 2869 y Fl(p)1609 2857 y Fn(2)c Fj(N)38 b Fp(indep)r(enden)n (t)28 b(of)g Fo(!)i Fp(suc)n(h)d(that)904 2987 y Fo(d)p Fp(\()p Fo(')p Fp(\()p Fo(n;)14 b(!)s Fp(\))p Fo(x;)g(')p Fp(\()p Fo(n;)g(!)s Fp(\))p Fo(y)s Fp(\))25 b Fn(\024)d Fp(\001\()p Fo(#)1921 2953 y Fl(n)1967 2987 y Fo(!)s Fp(\))166 b(for)27 b Fo(n)c Fp(=)g(0)p Fo(;)14 b(:)g(:)g(:)27 b(;)14 b(N)2815 2999 y Fl(p)2871 2987 y Fn(\000)k Fp(1)645 3138 y(implies)28 b Fo(d)p Fp(\()p Fo(x;)14 b(y)s Fp(\))23 b Fo(<)1283 3098 y Fm(\001\()p Fl(!)r Fm(\))p 1283 3119 151 4 v 1341 3167 a Fl(p)1443 3138 y Fp(.)456 3283 y(This)35 b(implies)h(strong)f(expansiv)n(eness.)60 b(In)36 b(fact,)i(let)e(\()p Fo(x)2304 3295 y Fl(i)2332 3283 y Fp(\))g(b)r(e)g(a)g(coun)n(table)f (dense)g(subset)456 3396 y(of)e Fo(M)41 b Fp(and)33 b(put)h Fn(A)p Fp(\()p Fo(!)s Fp(\))e(:=)g Fn(f)p Fo(E)1443 3408 y Fl(!)1513 3396 y Fn(\\)22 b Fo(B)t Fp(\()p Fo(x)1736 3408 y Fl(i)1765 3396 y Fo(;)1812 3356 y Fm(\001\()p Fl(!)r Fm(\))p 1812 3377 V 1870 3425 a(2)1972 3396 y Fp(\))32 b(:)h Fo(i)e Fp(=)h(1)p Fo(;)14 b(:)g(:)g(:)27 b(;)14 b(l)r Fp(\()p Fo(!)s Fp(\))p Fn(g)32 b Fp(with)h Fo(l)r Fp(\()p Fo(!)s Fp(\))f(:=)g(inf)7 b Fn(f)p Fo(k)34 b Fp(:)456 3455 y Fg(S)525 3475 y Fl(k)525 3542 y(i)p Fm(=1)650 3517 y Fo(B)t Fp(\()p Fo(x)796 3529 y Fl(i)825 3517 y Fo(;)872 3477 y Fm(\001\()p Fl(!)r Fm(\))p 872 3498 V 930 3546 a(2)1032 3517 y Fp(\))25 b Fn(\033)f Fo(E)1239 3529 y Fl(!)1287 3517 y Fn(g)p Fp(.)39 b(One)28 b(easily)g(c)n(hec)n(ks)f(that)i Fn(A)g Fp(de\014ned)f(this)h(w)n(a)n (y)e(is)i(a)f(generator)456 3639 y(and)f(\(3\))h(yields)f(diam)1178 3576 y Fg(W)1247 3594 y Fl(N)1300 3602 y Fe(p)1335 3594 y Fi(\000)p Fm(1)1247 3662 y Fl(i)p Fm(=0)1437 3639 y Fo(')p Fp(\()p Fo(i;)14 b(!)s Fp(\))1676 3608 y Fi(\000)p Fm(1)1766 3639 y Fn(A)p Fp(\()p Fo(#)1913 3608 y Fl(i)1941 3639 y Fo(!)s Fp(\))23 b Fo(<)2149 3598 y Fm(\001\()p Fl(!)r Fm(\))p 2149 3620 V 2207 3667 a Fl(p)2332 3639 y Fo(<)2430 3606 y Fm(1)p 2430 3620 35 4 v 2430 3667 a Fl(p)2501 3639 y Fp(for)k(all)h Fo(!)e Fn(2)d Fp(\012.)456 3778 y Ft(1.9.)41 b(Theorem.)e Fk(L)l(et)25 b Fo(')h Fk(b)l(e)g(a)g(str)l(ongly)g(exp)l(ansive)g(bund)t(le)g(RDS)f(with)h (exp)l(ansive)g(c)l(onstant)456 3878 y Fp(\001)p Fk(.)40 b(F)-6 b(urther,)31 b(let)f Fo(f)j Fn(2)25 b Fj(L)1237 3848 y Fm(1)1237 3901 y Fl(E)1299 3878 y Fp(\(\012)p Fo(;)14 b Fn(C)5 b Fp(\()p Fo(M)k Fp(\)\))31 b Fk(and)f(let)h Fo(\017)f Fk(b)l(e)g(a)h Fp(\(0)p Fo(;)14 b Fp(1\))p Fk(-value)l(d)30 b(r)l(andom)h(variable)i(with)456 3982 y Fo(\017)22 b(<)610 3949 y Fm(\001)p 610 3963 55 4 v 621 4010 a(4)674 3982 y Fk(.)39 b(If)30 b Fp(lim)941 3994 y Fl(k)q Fi(!1)1128 3982 y Fo(\031)1175 3994 y Fl(')1223 3982 y Fp(\()p Fo(f)9 b Fp(\)\()1384 3949 y Fl(\017)p 1379 3963 37 4 v 1379 4010 a(k)1426 3982 y Fp(\))23 b(=)g Fo(\031)1616 3994 y Fl(')1664 3982 y Fp(\()p Fo(f)9 b Fp(\))30 b Fk(then)1155 4163 y Fo(\031)1202 4175 y Fl(')1250 4163 y Fp(\()p Fo(f)9 b Fp(\))24 b(=)51 b(lim)1475 4213 y Fl(n)p Fi(!1)1677 4107 y Fp(1)p 1673 4144 50 4 v 1673 4220 a Fo(n)1746 4163 y Fp(log)14 b Fo(\031)1914 4175 y Fl(')1962 4163 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(!)s(;)14 b(n;)g(\017)p Fp(\))170 b Fj(P)p Fk(-a.)13 b(s.)456 4332 y(In)21 b(the)g(situation)g(of)i(pr)l(op)l(osition)g(1.4,)h(thus,)f (the)f(ab)l(ove)g(c)l(onver)l(genc)l(e)g(holds)h(if)f Fp(log)14 b Fo(\017)23 b Fn(2)g Fj(L)3260 4302 y Fm(1)3304 4332 y Fp(\()p Fj(P)p Fp(\))p Fk(.)456 4466 y(Pr)l(o)l(of.)43 b Fp(The)35 b(pro)r(of)g(can)g(b)r(e)h(obtained)f(b)n(y)h(adapting)e (the)i(argumen)n(t)f(of)g(W)-7 b(alters)35 b([W)-7 b(al82)o(,)456 4565 y(Theorem)29 b(9.6])g(and)g(will)i(not)e(b)r(e)i(carried)d(out)i (here.)43 b(W)-7 b(e)31 b(just)f(remark)f(that)h(b)n(y)f(reasoning)456 4665 y(for)e Fj(P)p Fp(-a.)11 b(e.)28 b Fo(!)i Fp(separately)d(one)g (ends)g(up)h(with)567 4834 y Fo(\031)614 4846 y Fl(')663 4834 y Fp(\()p Fo(f)9 b Fp(\))23 b Fn(\024)f Fp(lim)14 b(inf)916 4884 y Fl(n)p Fi(!1)1146 4778 y Fp(1)p 1141 4815 V 1141 4891 a Fo(n)1215 4834 y Fp(log)g Fo(\031)1383 4846 y Fl(')1431 4834 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(!)s(;)14 b(n;)g(\017)p Fp(\))23 b Fn(\024)g Fp(lim)14 b(sup)1973 4900 y Fl(n)p Fi(!1)2215 4778 y Fp(1)p 2211 4815 V 2211 4891 a Fo(n)2284 4834 y Fp(log)g Fo(\031)2452 4846 y Fl(')2500 4834 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(!)s(;)14 b(n;)g(\017)p Fp(\))23 b(=)g Fo(\031)3052 4800 y Fi(\003)3049 4854 y Fl(')3097 4834 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(\017)p Fp(\))p Fo(:)456 5014 y Fp(The)21 b(assumption)g(lim)1168 5026 y Fl(k)q Fi(!1)1356 5014 y Fo(\031)1403 5026 y Fl(')1451 5014 y Fp(\()p Fo(f)9 b Fp(\)\()1612 4981 y Fl(\017)p 1607 4995 37 4 v 1607 5042 a(k)1654 5014 y Fp(\))23 b(=)g Fo(\031)1844 5026 y Fl(')1892 5014 y Fp(\()p Fo(f)9 b Fp(\))22 b(is)f(needed)h(to)f(ensure)g(lim)2839 5026 y Fl(k)q Fi(!1)3026 5014 y Fo(\031)3076 4983 y Fi(\003)3073 5034 y Fl(')3121 5014 y Fp(\()p Fo(f)9 b Fp(\)\()3282 4981 y Fl(\017)p 3278 4995 V 3278 5042 a(k)3324 5014 y Fp(\))24 b(=)456 5116 y Fo(\031)503 5128 y Fl(')551 5116 y Fp(\()p Fo(f)9 b Fp(\))37 b(\(whic)n(h)h(again)e(follo)n(ws)h (using)g(a)g(randomized)g(result)g(of)g(W)-7 b(alters)37 b([W)-7 b(al82)o(,)40 b(Theo-)456 5216 y(rem)27 b(9.2]\);)g(hence)h(b)n (y)f(monotonicit)n(y)g Fo(\031)1733 5185 y Fi(\003)1730 5236 y Fl(')1778 5216 y Fp(\()p Fo(f)9 b Fp(\)\()p Fo(\017)p Fp(\))23 b Fn(\024)g Fo(\031)2148 5228 y Fl(')2196 5216 y Fp(\()p Fo(f)9 b Fp(\).)p 3384 5216 4 57 v 3388 5163 50 4 v 3388 5216 V 3437 5216 4 57 v eop %%Page: 6 6 6 5 bop 456 238 a Fm(6)913 b(T.)29 b(BOGENSCH)1880 223 y(\177)1872 238 y(UTZ)h(AND)f(G.)g(OCHS)844 425 y Fp(2.)46 b Fq(Str)n(uctural)30 b(st)-6 b(ability)32 b(implies)g(stochastic)f(st) -6 b(ability)555 575 y Fp(Let)39 b Fo(U)51 b Fn(\032)42 b Fo(M)48 b Fp(b)r(e)39 b(an)g(op)r(en)g(subset)g(of)g(the)g(P)n(olish) f(space)g Fo(M)48 b Fp(with)2885 554 y(\026)2870 575 y Fo(U)j Fn(2)43 b(K)d Fp(and)f(let)456 674 y Fn(C)5 b Fp(\()p Fo(U;)14 b(M)9 b Fp(\))28 b(denote)i(the)g(space)f(of)h(all)f (con)n(tin)n(uous)g(mappings)g(from)g Fo(U)38 b Fp(to)30 b Fo(M)38 b Fp(endo)n(w)n(ed)29 b(with)456 774 y(the)f(compact)f(op)r (en)g(top)r(ology)-7 b(.)456 921 y Ft(2.1.)41 b(De\014nition.)f Fp(Assume)19 b Fo(F)35 b Fn(2)24 b(C)5 b Fp(\()p Fo(U;)14 b(M)9 b Fp(\))19 b(and)g(\004)24 b Fn(2)f(K)f Fp(with)e Fo(F)12 b Fp(\004)23 b(=)g(\004.)34 b(A)21 b(family)e Fn(f)p Fo(')3254 933 y Fl(\017)3286 921 y Fn(g)3328 933 y Fl(\017>)p Fm(0)456 1021 y Fp(of)27 b Fn(C)5 b Fp(\()p Fo(U;)14 b(M)9 b Fp(\)-v)-5 b(alued)27 b(random)f(v)-5 b(ariables)27 b(is)g(called)g(a)g Fk(r)l(andom)k(p)l(erturb)l(ation)f (of)g Fo(F)42 b Fk(on)30 b Fp(\004)e(if)532 1138 y(\(i\))42 b(lim)777 1150 y Fl(\017)p Fi(!)p Fm(0)922 1138 y Fo(')976 1150 y Fl(\017)1031 1138 y Fp(=)22 b Fo(F)40 b Fp(in)28 b(probabilit)n(y)-7 b(,)509 1238 y(\(ii\))42 b(there)28 b(exists)f(a)g(family)h Fn(f)p Fp(\004)1523 1250 y Fl(\017)1554 1238 y Fn(g)1596 1250 y Fl(\017>)p Fm(0)1740 1238 y Fp(of)g Fn(K)q Fp(-v)-5 b(alued)28 b(random)e(v)-5 b(ariables)27 b(suc)n(h)g(that)748 1337 y(-)41 b(for)27 b(eac)n(h)g Fo(\017)c(>)f Fp(0)28 b(w)n(e)f(ha)n(v)n(e)f(that)i Fj(P)p Fp(-a.)12 b(s.)27 b Fo(')2132 1349 y Fl(\017)2165 1337 y Fp(\()p Fo(!)s Fp(\)\004)2339 1349 y Fl(\017)2371 1337 y Fp(\()p Fo(!)s Fp(\))c(=)g(\004)2656 1349 y Fl(\017)2688 1337 y Fp(\()p Fo(#!)s Fp(\),)748 1437 y(-)41 b(lim)933 1449 y Fl(\017)p Fi(!)p Fm(0)1077 1437 y Fp(\004)1132 1449 y Fl(\017)1188 1437 y Fp(=)22 b(\004)28 b(in)g(probabilit)n(y)-7 b(.)456 1554 y Fn(f)p Fo(')552 1566 y Fl(\017)583 1554 y Fn(g)625 1566 y Fl(\017>)p Fm(0)767 1554 y Fp(is)25 b(said)h(to)f(b)r(e)h Fk(structur)l(al)t(ly)h(stable)f Fp(if)g(there)g(exists)f(a)g(family)g Fn(f)p Fo(h)2814 1566 y Fl(\017)2845 1554 y Fn(g)2887 1566 y Fl(\017>)p Fm(0)3029 1554 y Fp(of)h Fn(C)5 b Fp(\(\004)p Fo(;)14 b(M)9 b Fp(\)-)456 1654 y(v)-5 b(alued)27 b(random)g(v)-5 b(ariables)26 b(suc)n(h)h(that)486 1771 y(\(iii\))42 b(for)c(eac)n(h)f Fo(\017)i(>)h Fp(0)d(w)n(e)h(ha)n(v)n(e)e(that)i Fj(P)p Fp(-a.)12 b(s.)38 b Fo(h)2076 1783 y Fl(\017)2108 1771 y Fp(\()p Fo(!)s Fp(\))g(maps)f(\004)h(homeomorphically)e(on)n(to) 661 1870 y(\004)716 1882 y Fl(\017)748 1870 y Fp(\()p Fo(!)s Fp(\))28 b(and)1451 2005 y Fo(')1505 2017 y Fl(\017)1537 2005 y Fp(\()p Fo(!)s Fp(\))19 b Fn(\016)f Fo(h)1783 2017 y Fl(\017)1814 2005 y Fp(\()p Fo(!)s Fp(\))23 b(=)g Fo(h)2092 2017 y Fl(\017)2124 2005 y Fp(\()p Fo(#!)s Fp(\))c Fn(\016)e Fo(F)r(;)488 2139 y Fp(\(iv\))42 b(lim)777 2151 y Fl(\017)p Fi(!)p Fm(0)922 2139 y Fo(h)970 2151 y Fl(\017)1024 2139 y Fp(=)23 b(id)28 b(in)g(probabilit)n(y)-7 b(.)456 2287 y Ft(2.2.)41 b(Example.)e Fp(Let)49 b Fo(M)57 b Fp(b)r(e)49 b(a)g(\014nite)g(dimensional)f(Riemannian)h(manifold)g (and)f(let)456 2386 y Fn(C)505 2356 y Fl(k)545 2386 y Fp(\()p Fo(U;)14 b(M)9 b Fp(\))45 b(denote)g(the)g(space)f(of)h(all)g Fn(C)1816 2356 y Fl(k)1902 2386 y Fp(maps)f(from)h Fo(U)54 b Fp(to)45 b Fo(M)54 b Fp(\()p Fo(k)h(>)c Fp(1\).)90 b(F)-7 b(urther,)456 2493 y(let)46 b(Em)n(b)764 2456 y Fl(k)804 2493 y Fp(\()p Fo(U;)14 b(M)9 b Fp(\))46 b(denote)g(the)h (set)f(of)g(those)g(mem)n(b)r(ers)g(of)g Fn(C)2575 2463 y Fl(k)2615 2493 y Fp(\()p Fo(U;)14 b(M)9 b Fp(\))46 b(whic)n(h)g(are)f(dif-)456 2593 y(feomorphisms)d(on)n(to)h(their)g (images.)84 b(A)44 b(recen)n(t)f(result)g(of)g(Liu)h([Liu98)o(])g (yields)f(that)h(if)456 2699 y Fo(F)f Fn(2)33 b Fp(Em)n(b)809 2662 y Fl(k)850 2699 y Fp(\()p Fo(U;)14 b(M)9 b Fp(\))33 b(and)f(\004)i(is)e(an)h(Axiom)g(A)g(basic)f(set,)j(then)e(there)g (exists)f(a)h(neigh)n(b)r(our-)456 2806 y(ho)r(o)r(d)38 b Fn(U)8 b Fp(\()p Fo(F)k Fp(\))39 b(of)f Fo(F)50 b Fp(in)39 b(Em)n(b)1387 2768 y Fl(k)1428 2806 y Fp(\()p Fo(U;)14 b(M)9 b Fp(\))38 b(suc)n(h)g(that)h(eac)n(h)e(family)i Fn(f)p Fo(')2661 2818 y Fl(\017)2692 2806 y Fn(g)2734 2818 y Fl(\017>)p Fm(0)2889 2806 y Fp(of)f Fn(U)8 b Fp(\()p Fo(F)k Fp(\)-v)-5 b(alued)456 2905 y(random)22 b(v)-5 b(ariables)22 b(with)i(lim)1397 2917 y Fl(\017)p Fi(!)p Fm(0)1542 2905 y Fo(')1596 2917 y Fl(\017)1651 2905 y Fp(=)e Fo(F)36 b Fp(in)24 b(probabilit)n(y)e(is)i(a)f(structurally)f (stable)h(random)456 3005 y(p)r(erturbation)k(of)g Fo(F)40 b Fp(on)27 b(\004.)555 3105 y(If)39 b Fo(F)52 b Fn(2)40 b(C)898 3074 y Fl(k)938 3105 y Fp(\()p Fo(U;)14 b(M)9 b Fp(\))38 b(is)g(expanding)f(on)h(\004)g(this)g(statemen)n(t)g(can)f (b)r(e)h(seen)g(to)g(hold)g(true)456 3204 y(with)27 b Fn(U)8 b Fp(\()p Fo(F)k Fp(\))27 b(replaced)f(b)n(y)h(a)f(neigh)n(b)r (ourho)r(o)r(d)f(of)i Fo(F)39 b Fp(in)27 b Fn(C)2267 3174 y Fl(k)2307 3204 y Fp(\()p Fo(U;)14 b(M)9 b Fp(\).)36 b(Note)27 b(that)g(here)f(eac)n(h)g Fo(')3412 3216 y Fl(\017)456 3304 y Fp(determines)33 b(an)h(expanding)f(bundle)h(RDS;)h (if)f Fo(F)46 b Fp(is)34 b(conformal,)g(ho)n(w)n(ev)n(er,)g(the)g(same) f(will)456 3403 y(not)f(b)r(e)h(true)g(for)f Fo(')1093 3415 y Fl(\017)1157 3403 y Fp(in)h(general,)g(i.)14 b(e.)32 b(w)n(e)g(will)h(not)g(b)r(e)g(in)f(the)h(situation)g(of)f(Kifer)g ([Kif96)o(,)456 3503 y(Theorem)26 b(4.8].)36 b(W)-7 b(e)28 b(will)g(discuss)f(this)h(particular)f(situation)g(in)h(more)f(detail)g (in)h(section)f(4.)555 3603 y(Let)h(us)g(\014nally)f(remark)f(that)i (if)g(w)n(e)f(assume)g Fo(M)37 b Fp(to)27 b(b)r(e)h(compact)f(and)h (put)g(\004)c(=)e Fo(U)32 b Fp(=)23 b Fo(M)9 b Fp(,)456 3702 y(and)38 b(if)i(w)n(e)f(assume)f(supp)14 b Fo(')1387 3714 y Fl(\017)1419 3702 y Fj(P)41 b Fn(\032)h Fo(B)1682 3714 y Fl(\017)1713 3702 y Fp(\()p Fo(F)12 b Fp(\),)43 b(w)n(e)c(are)f(in)h(a)g(situation)g(whic)n(h)f(is)h(quite)h(w)n(ell) 456 3802 y(understo)r(o)r(d)i(\(cf.)h(Baladi)e([Bal97)o(],)46 b(Baladi,)f(Kondah,)h(and)c(Sc)n(hmitt)h([BKS96)o(],)j(Bogen-)456 3902 y(sc)n(h)r(\177)-44 b(utz)27 b([Bog96)n(]\).)555 4049 y(Put)d Fo(X)29 b Fp(=)914 4028 y(\026)900 4049 y Fo(U)j Fp(and)24 b(let)g Fn(C)1312 4019 y Fl(\013)1358 4049 y Fp(\()p Fo(X)r(;)14 b Fj(R)p Fp(\))30 b(denote)24 b(the)g(space)f(of)g(all)h(H\177)-42 b(older)23 b(con)n(tin)n(uous)f (functions)456 4149 y(on)28 b Fo(X)35 b Fp(with)29 b(H\177)-42 b(older)28 b(exp)r(onen)n(t)h Fo(\013)p Fp(.)40 b(W)-7 b(e)29 b(endo)n(w)f Fn(C)2062 4118 y Fl(\013)2109 4149 y Fp(\()p Fo(X)r(;)14 b Fj(R)p Fp(\))35 b(with)29 b(the)g(usual)f(norm) g Fn(k)19 b(\001)g(k)3285 4161 y Fl(\013)3357 4149 y Fp(:=)456 4248 y Fn(k)g(\001)g(k)601 4260 y Fi(1)690 4248 y Fp(+)h Fn(j)f(\001)h(j)883 4260 y Fl(\013)930 4248 y Fp(,)30 b(where)e Fn(k)19 b(\001)h(k)1370 4260 y Fi(1)1469 4248 y Fp(is)29 b(the)g(sup-norm)g(and)g Fn(j)19 b(\001)h(j)2341 4260 y Fl(\013)2417 4248 y Fp(is)29 b(the)h(least)f(H\177)-42 b(older)28 b(constan)n(t,)456 4361 y(i.)14 b(e.)27 b Fn(j)p Fo(f)9 b Fn(j)699 4373 y Fl(\013)769 4361 y Fp(:=)23 b(sup)1005 4381 y Fl(x;y)r Fi(2)p Fl(X)q(;x)p Fi(6)p Fm(=)p Fl(y)1370 4321 y Fi(j)p Fl(f)7 b Fm(\()p Fl(x)p Fm(\))p Fi(\000)p Fl(f)g Fm(\()p Fl(y)r Fm(\))p Fi(j)p 1370 4342 347 4 v 1433 4390 a Fl(d)p Fm(\()p Fl(x;y)r Fm(\))1614 4373 y Fe(\013)1727 4361 y Fp(.)456 4523 y Ft(2.3.)41 b(Prop)s(osition.)e Fk(L)l(et)30 b Fn(f)p Fo(')1439 4535 y Fl(\017)1471 4523 y Fn(g)1513 4535 y Fl(\017>)p Fm(0)1660 4523 y Fk(b)l(e)h(a)g(structur)l(al)t(ly)f (stable)h(r)l(andom)g(p)l(erturb)l(ation)g(of)g Fo(F)456 4622 y Fk(on)h Fp(\004)p Fk(.)45 b(F)-6 b(urther,)33 b(let)f Fn(f)p Fo(f)1233 4634 y Fl(\017)1264 4622 y Fn(g)1306 4634 y Fl(\017>)p Fm(0)1454 4622 y Fk(b)l(e)g(a)h(family)g(of)g Fn(C)2036 4592 y Fl(\013)2083 4622 y Fp(\()p Fo(X)r(;)14 b Fj(R)p Fp(\))p Fk(-va)q(lue)l(d)38 b(r)l(andom)32 b(variables)i (satis-)456 4722 y(fying)1393 4856 y Fp(lim)1387 4908 y Fl(\017)p Fi(!)p Fm(0)1528 4856 y Fn(k)p Fo(f)1611 4868 y Fl(\017)1660 4856 y Fn(\000)18 b Fo(f)9 b Fn(k)1835 4868 y Fl(\013)1905 4856 y Fp(=)22 b(0)169 b Fk(in)30 b Fj(L)2355 4826 y Fm(1)2398 4856 y Fp(\()p Fj(P)p Fp(\))456 5021 y Fk(for)g(some)g Fo(f)i Fn(2)23 b(C)1000 4991 y Fl(\013)1047 5021 y Fp(\()p Fo(X)r(;)14 b Fj(R)p Fp(\))p Fk(.)45 b(Then)1579 5156 y Fp(lim)1573 5207 y Fl(\017)p Fi(!)p Fm(0)1714 5156 y Fo(\031)1761 5168 y Fl(')1805 5176 y Fe(\017)1839 5156 y Fp(\()p Fo(f)1912 5168 y Fl(\017)1944 5156 y Fp(\))23 b(=)g Fo(\031)2134 5168 y Fl(F)2189 5156 y Fp(\()p Fo(f)9 b Fp(\))p Fo(:)p eop %%Page: 7 7 7 6 bop 947 226 a Fm(HA)n(USDORFF)28 b(DIMENSION)g(UNDER)h(RANDOM)g (PER)-5 b(TURBA)g(TION)458 b(7)456 425 y Fk(Pr)l(o)l(of.)43 b Fp(First)19 b(note)h(that)g(since)g Fo(')1509 437 y Fl(\017)1561 425 y Fp(and)f Fo(F)32 b Fp(are)19 b(top)r(ologically)f (conjugate)h(\(de\014nition)h(2.1\(iii\)\))456 525 y(for)34 b(eac)n(h)f Fo(\017)i(>)f Fp(0)g(w)n(e)h(ha)n(v)n(e)e Fo(\031)1402 537 y Fl(')1446 545 y Fe(\017)1480 525 y Fp(\()p Fo(f)1553 537 y Fl(\017)1585 525 y Fp(\))i(=)f Fo(\031)1798 537 y Fl(F)1853 525 y Fp(\()p Fo(f)1926 537 y Fl(\017)1981 525 y Fn(\016)23 b Fo(h)2094 537 y Fl(\017)2126 525 y Fp(\))34 b(\(cf.)i(Bogensc)n(h)r(\177)-44 b(utz)33 b([Bog93)n(,)j(Prop)r(osi-)456 624 y(tion)27 b(3.1.3]\).)36 b(F)-7 b(urther,)28 b(w)n(e)f(ha)n(v)n(e)1098 763 y Fn(k)p Fo(f)1181 775 y Fl(\017)1230 763 y Fn(\016)18 b Fo(h)1338 775 y Fl(\017)1388 763 y Fn(\000)g Fo(f)9 b Fn(k)1563 775 y Fi(1)1656 763 y Fn(\024)22 b(k)p Fo(f)1826 775 y Fl(\017)1876 763 y Fn(\016)c Fo(h)1984 775 y Fl(\017)2034 763 y Fn(\000)g Fo(f)2158 775 y Fl(\017)2189 763 y Fn(k)2231 775 y Fi(1)2319 763 y Fp(+)h Fn(k)p Fo(f)2486 775 y Fl(\017)2535 763 y Fn(\000)f Fo(f)9 b Fn(k)2710 775 y Fi(1)2779 763 y Fo(:)456 902 y Fp(The)26 b(second)f(summand)h(of)g(the)g(righ)n(t)f (hand)h(side)g(tends)g(to)g(zero)f(as)g Fo(\017)e Fn(!)g Fp(0)i(b)n(y)h(assumption.)456 1002 y(F)-7 b(or)27 b(the)h(\014rst)f (summand)h(w)n(e)f(ha)n(v)n(e)1054 1141 y Fn(k)p Fo(f)1137 1153 y Fl(\017)1187 1141 y Fn(\016)18 b Fo(h)1295 1153 y Fl(\017)1345 1141 y Fn(\000)g Fo(f)1469 1153 y Fl(\017)1500 1141 y Fn(k)1542 1153 y Fi(1)1635 1141 y Fp(=)31 b(sup)1723 1211 y Fl(x)p Fi(2)p Fl(X)1864 1141 y Fn(fj)p Fo(f)1970 1153 y Fl(\017)2001 1141 y Fp(\()p Fo(!)s(;)14 b(h)2173 1153 y Fl(\017)2205 1141 y Fp(\()p Fo(!)s Fp(\))p Fo(x)p Fp(\))19 b Fn(\000)f Fo(f)2546 1153 y Fl(\017)2578 1141 y Fp(\()p Fo(!)s(;)c(x)p Fp(\))p Fn(jg)1635 1319 y(\024)23 b(j)p Fo(f)1787 1331 y Fl(\017)1818 1319 y Fp(\()p Fo(!)s Fp(\))p Fn(j)1960 1331 y Fl(\013)2026 1319 y Fn(\001)c Fo(d)p Fp(\()p Fo(h)2191 1331 y Fl(\017)2223 1319 y Fp(\()p Fo(!)s Fp(\))p Fo(;)14 b Fp(id\))2480 1285 y Fl(\013)2528 1319 y Fo(:)456 1463 y Fp(Note)56 b(that)g Fo(!)73 b Fn(7!)e Fo(d)p Fp(\()p Fo(h)1295 1475 y Fl(\017)1327 1463 y Fp(\()p Fo(!)s Fp(\))p Fo(;)14 b Fp(id\))57 b(is)f(b)r(ounded)g (uniformly)g(in)g Fo(\017)g Fp(b)n(y)g(diam)14 b Fo(X)7 b Fp(,)62 b(hence)456 1562 y(lim)571 1574 y Fl(\017)p Fi(!)p Fm(0)716 1562 y Fo(d)p Fp(\()p Fo(h)839 1574 y Fl(\017)871 1562 y Fp(\()p Fn(\001)p Fp(\))p Fo(;)14 b Fp(id)q(\))1097 1532 y Fl(\013)1198 1562 y Fp(=)54 b(0)45 b(in)i Fj(L)1569 1532 y Fm(1)1612 1562 y Fp(\()p Fj(P)p Fp(\))f(b)n(y)f(de\014nition)i(2.1\(iv\).)92 b(With)47 b(this,)k(and)46 b(since)456 1662 y(lim)571 1674 y Fl(\017)p Fi(!)p Fm(0)716 1662 y Fn(j)p Fo(f)780 1674 y Fl(\017)811 1662 y Fp(\()p Fn(\001)p Fp(\))p Fn(j)921 1674 y Fl(\013)992 1662 y Fp(=)23 b Fn(j)p Fo(f)9 b Fn(j)1176 1674 y Fl(\013)1251 1662 y Fp(in)28 b Fj(L)1397 1632 y Fm(1)1440 1662 y Fp(\()p Fj(P)p Fp(\))f(b)n(y)h(assumption,)f(w)n(e)g(conclude)1295 1801 y(lim)1289 1852 y Fl(\017)p Fi(!)p Fm(0)1430 1801 y Fn(k)p Fo(f)1513 1813 y Fl(\017)1562 1801 y Fn(\016)18 b Fo(h)1670 1813 y Fl(\017)1720 1801 y Fn(\000)g Fo(f)9 b Fn(k)1895 1813 y Fi(1)1988 1801 y Fp(=)23 b(0)165 b(in)28 b Fj(L)2429 1771 y Fm(1)2473 1801 y Fp(\()p Fj(P)p Fp(\).)456 1966 y(F)-7 b(rom)27 b(this)h(the)g(claim)f(follo)n(ws)g(b)n(y)g(the)h (con)n(tin)n(uit)n(y)f(of)h(the)g(pressure.)p 3384 1966 4 57 v 3388 1913 50 4 v 3388 1966 V 3437 1966 4 57 v 456 2122 a(In)23 b(view)f(of)h(Bogensc)n(h)r(\177)-44 b(utz)21 b([Bog96)n(,)j(Prop)r(osition)d(2.2])h(this)h(has)g(the)g (follo)n(wing)e(consequence.)456 2274 y Ft(2.4.)41 b(Corollary)-8 b(.)41 b Fk(In)25 b(the)h(situation)g(of)h(pr)l(op)l(osition)h(2.3,)g (assume)e(that)g(for)h(e)l(ach)f Fo(\017)d(>)g Fp(0)i Fk(the)456 2373 y(entr)l(opy)30 b(map)g Fo(\026)23 b Fn(7!)g Fo(h)1163 2385 y Fl(\026)1208 2373 y Fp(\()p Fo(')1294 2385 y Fl(\017)1326 2373 y Fp(\))30 b Fk(is)g(upp)l(er)g (semi-c)l(ontinuous.)38 b(Then)31 b(the)f(fol)t(lowing)i(holds)f(true:) 555 2473 y(Assume)24 b Fn(f)p Fo(\026)952 2485 y Fl(\017)983 2473 y Fn(g)1025 2485 y Fl(\017>)p Fm(0)1166 2473 y Fk(is)h(a)g(family) i(of)e(r)l(andom)h(pr)l(ob)l(abilities)h(on)e Fo(M)33 b Fk(such)25 b(that)f(e)l(ach)i Fo(\026)3215 2485 y Fl(\017)3271 2473 y Fk(is)f(an)456 2573 y(e)l(quilibrium)31 b(state)f(for)h Fo(f)1260 2585 y Fl(\017)1322 2573 y Fk(with)g(r)l(esp)l(e)l(ct)f(to)g Fo(')1928 2585 y Fl(\017)1961 2573 y Fk(,)h(i.)15 b(e.)31 b Fo(\031)2223 2585 y Fl(')2267 2593 y Fe(\017)2300 2573 y Fp(\()p Fo(f)2373 2585 y Fl(\017)2405 2573 y Fp(\))25 b(=)e Fo(h)2598 2585 y Fl(\026)2638 2593 y Fe(\017)2672 2573 y Fp(\()p Fo(')2758 2585 y Fl(\017)2791 2573 y Fp(\))c(+)2925 2506 y Fg(R)2995 2573 y Fo(f)3036 2585 y Fl(\017)3067 2573 y Fo(d\026)3160 2585 y Fl(\017)3192 2573 y Fk(.)41 b(Then)456 2672 y(the)32 b(mar)l(ginal)h(on)f Fo(M)41 b Fk(of)33 b(e)l(ach)g(ac)l(cumulation)g(p)l(oint)g(of)g Fn(f)p Fo(\026)2390 2684 y Fl(\017)2421 2672 y Fn(g)f Fk(is)g(an)g(e)l(quilibrium)i(state)d(for)456 2772 y Fo(f)41 b Fk(with)34 b(r)l(esp)l(e)l(ct)f(to)g Fo(F)12 b Fk(.)50 b(In)32 b(p)l(articular,)k(thus,)e(if)g Fo(f)42 b Fk(p)l(ossesses)34 b(a)g(unique)f(e)l(quilibrium)h(state)456 2872 y Fo(\026)506 2884 y Fl(f)578 2872 y Fk(then)c Fp(lim)878 2884 y Fl(\017)p Fi(!)p Fm(0)1023 2872 y Fo(\026)1073 2884 y Fl(\017)1128 2872 y Fp(=)23 b Fj(P)17 b Fn(\012)h Fo(\026)1418 2884 y Fl(f)1461 2872 y Fk(.)456 3024 y Fp(2.5)p Fk(.)37 b(R)l(emark.)42 b Fp(Consider)27 b(the)h(situation)f (describ)r(ed)g(in)h(example)f(2.2.)36 b(Using)27 b(the)h(metho)r(ds) 456 3123 y(of)d(Liu)g([Liu98)o(])h(one)f(can)g(deriv)n(e)f(that,)i(b)n (y)f(making)g Fn(U)8 b Fp(\()p Fo(F)k Fp(\))26 b(smaller)e(if)i (necessary)-7 b(,)24 b Fo(')3126 3135 y Fl(\017)3183 3123 y Fp(has)h(the)456 3223 y(expansiv)n(eness)f(prop)r(ert)n(y)i(and) g(hence)g Fo(\026)e Fn(7!)f Fo(h)1935 3235 y Fl(\026)1979 3223 y Fp(\()p Fo(')2065 3235 y Fl(\017)2097 3223 y Fp(\))k(is)g(upp)r (er)f(semi-con)n(tin)n(uous.)35 b(Applying)456 3322 y(the)30 b(ab)r(o)n(v)n(e)f(results)h(to)h Fo(f)1252 3334 y Fl(\017)1311 3322 y Fn(\021)c Fp(0,)k(th)n(us,)g(w)n(e)f(obtain)g(sto)r(c)n(hastic)g (stabilit)n(y)g(of)g(the)h(top)r(ological)456 3422 y(en)n(trop)n(y)26 b(and)h(the)h(measure)f(of)g(maximal)g(en)n(trop)n(y)-7 b(.)555 3522 y(One)27 b(ma)n(y)g(also)f(deriv)n(e)g(\(again)g(using)h ([Liu98)o(]\))h(sto)r(c)n(hastic)e(stabilit)n(y)h(of)g(the)h (generalized)456 3625 y(SRB)33 b(measure)e Fo(\026)1033 3637 y Fm(S)p Fl(RB)1210 3625 y Fp(of)i Fo(F)12 b Fp(.)53 b(This)33 b(is)f(de\014ned)i(as)e(the)h(equilibrium)g(state)g(for)f Fo(f)3118 3594 y Fm(\()p Fl(u)p Fm(\))3213 3625 y Fp(\()p Fo(x)p Fp(\))h(:=)456 3724 y Fn(\000)14 b Fp(log)f Fn(j)h Fp(det)g Fo(D)890 3736 y Fl(x)932 3724 y Fo(F)e Fn(j)1020 3739 y Fl(E)1072 3723 y Fe(u)1111 3739 y Fm(\()p Fl(x)p Fm(\))1205 3724 y Fn(j)p Fp(,)28 b(where)f Fo(E)1585 3694 y Fl(u)1628 3724 y Fp(\()p Fo(x)p Fp(\))e Fn(\032)e Fo(T)1901 3736 y Fl(x)1942 3724 y Fo(M)37 b Fp(denotes)27 b(the)h(unstable)g(subspace)f(of)h Fo(F)40 b Fp(at)456 3824 y Fo(x)p Fp(.)35 b(Sto)r(c)n(hastic)23 b(stabilit)n(y)g(here)f (means)h(lim)1813 3836 y Fl(\017)p Fi(!)p Fm(0)1958 3824 y Fo(\026)2008 3836 y Fl(\017)2063 3824 y Fp(=)f Fj(P)9 b Fn(\012)g Fo(\026)2335 3836 y Fm(S)p Fl(RB)2501 3824 y Fp(where)23 b Fo(\026)2787 3836 y Fl(\017)2842 3824 y Fp(is)g(an)f(equilibrium)456 3935 y(state)27 b(for)g Fo(f)836 3892 y Fm(\()p Fl(u)p Fm(\))827 3945 y Fl(\017)931 3935 y Fp(\()p Fo(!)s(;)14 b(x)p Fp(\))24 b(:=)e Fn(\000)14 b Fp(log)g Fn(j)g Fp(det)g Fo(D)1703 3947 y Fl(x)1744 3935 y Fo(')1798 3947 y Fl(\017)1830 3935 y Fp(\()p Fo(!)s Fp(\))p Fn(j)1972 3950 y Fl(E)2024 3933 y Fe(u)2021 3967 y(\017)2064 3950 y Fm(\()p Fl(!)r(;x)p Fm(\))2221 3935 y Fn(j)p Fp(.)850 4117 y(3.)46 b Fq(A)31 b(random)f(version)i(of)f(the) h(Bo)n(wen-R)n(uelle)g(f)n(ormula)555 4267 y Fp(The)27 b(classical)e(Bo)n(w)n(en{Ruelle)h(form)n(ula)g(connects)g(the)h (Hausdor\013)f(dimension)h(of)g(a)f(con-)456 4366 y(formal)21 b(rep)r(eller)h(of)g(a)g(deterministic)h Fn(C)1700 4336 y Fm(1+)p Fl(\013)1854 4366 y Fp(map)f(with)h(top)r(ological)e (pressure.)33 b(In)23 b(the)g(sequel)456 4466 y(w)n(e)28 b(will)h(generalize)e(this)h(result)h(to)f(a)g(class)g(of)g(random)g (in)n(v)-5 b(arian)n(t)28 b(compact)g(sets)g(of)g(a)h(ran-)456 4565 y(dom)g(dynamical)h(system)f(whic)n(h)h(w)n(e)f(will)i(call)e (\\almost)g(conformal)f(rep)r(ellers".)43 b(A)30 b(map)g(is)456 4665 y(called)g(conformal,)h(if)g(the)g(expansion)f(of)h(its)g (linearization)e(at)i(eac)n(h)f(p)r(oin)n(t)h(is)g(the)g(same)f(in)456 4765 y(ev)n(ery)k(direction.)61 b(W)-7 b(e)36 b(will)g(w)n(eak)n(en)e (this)i(assumption)g(b)n(y)f(requiring)f(certain)i(lo)n(w)n(er)e(and) 456 4864 y(upp)r(er)23 b(b)r(ounds)h(of)g(expansion)f(in)h(di\013eren)n (t)f(directions.)35 b(Rep)r(eller)24 b(in)g(our)f(sense)g(means)g(that) 456 4964 y(the)h(distance)f(of)h(nearb)n(y)e(p)r(oin)n(ts)i (\\increases)d(in)j(a)n(v)n(erage".)32 b(F)-7 b(ormally)23 b(this)h(lo)r(oks)e(as)h(follo)n(ws.)456 5116 y Ft(3.1.)41 b(De\014nition.)f Fp(Let)25 b Fo(')h Fp(b)r(e)g(a)f(con)n(tin)n(uous)f (bundle)i(RDS)g(on)f(a)g(compact)g(bundle)g Fo(E)31 b Fp(with)456 5216 y(\014b)r(ers)c(in)h(some)f(complete)g(separable)f (metric)i(space)f(\()p Fo(M)t(;)14 b(d)p Fp(\).)p eop %%Page: 8 8 8 7 bop 456 238 a Fm(8)913 b(T.)29 b(BOGENSCH)1880 223 y(\177)1872 238 y(UTZ)h(AND)f(G.)g(OCHS)532 425 y Fp(\(i\))42 b Fo(')21 b Fp(is)f(said)g(to)g(b)r(e)g(\()p Fo(\025;)14 b(\021)s Fp(\))24 b Fk(almost)g(c)l(onformal)e Fp(if)e(there)g(exist)g (measurable)f(functions)i Fo(\025;)14 b(\021)26 b Fp(:)661 527 y Fo(E)i Fn(!)c Fj(R)911 539 y Fm(+)972 527 y Fp(,)h(and)f Fo(K)29 b Fp(:)23 b(\012)g Fn(!)g Fj(R)1567 539 y Fm(+)1628 527 y Fp(,)i(with)g(log)1970 490 y Fm(+)2039 527 y Fo(K)j Fn(2)23 b Fj(L)2266 497 y Fm(1)2309 527 y Fp(\()p Fj(P)p Fp(\),)i(log)14 b Fo(\025;)g Fp(log)g Fo(\021)26 b Fn(2)e Fj(L)2995 497 y Fm(1)2995 550 y Fl(E)3057 527 y Fp(\(\012)p Fo(;)14 b Fn(C)5 b Fp(\()p Fo(M)k Fp(\)\),)661 627 y(and)20 b Fo(\025)p Fp(\()p Fo(!)s(;)14 b Fn(\001)p Fp(\))p Fo(;)g(\021)s Fp(\()p Fo(!)s(;)g Fn(\001)p Fp(\))22 b(H\177)-42 b(older)19 b(con)n(tin)n(uous)h(in)g Fo(x)h Fp(with)g(uniform)f(exp)r(onen)n(t)g Fo(\013)p Fp(,)j(0)f Fo(<)h(\013)g Fn(\024)g Fp(1)661 727 y(suc)n(h)28 b(that)708 888 y Fo(\025)p Fp(\()p Fo(!)s(;)14 b(x)p Fp(\))19 b Fn(\000)f Fo(K)6 b Fp(\()p Fo(!)s Fp(\))p Fo(d)p Fp(\()p Fo(x;)14 b(y)s Fp(\))1492 854 y Fl(\013)1563 888 y Fn(\024)1661 832 y Fo(d)p Fp(\()p Fo(')p Fp(\()p Fo(!)s Fp(\))p Fo(x;)g(')p Fp(\()p Fo(!)s Fp(\))p Fo(y)s Fp(\))p 1661 869 583 4 v 1834 945 a Fo(d)p Fp(\()p Fo(x;)g(y)s Fp(\))2277 888 y Fn(\024)22 b Fo(\021)s Fp(\()p Fo(!)s(;)14 b(x)p Fp(\))19 b(+)f Fo(K)6 b Fp(\()p Fo(!)s Fp(\))p Fo(d)p Fp(\()p Fo(x;)14 b(y)s Fp(\))3144 854 y Fl(\013)661 1067 y Fp(for)27 b(ev)n(ery)g Fo(!)e Fn(2)f Fp(\012)j(and)h Fo(x)23 b Fn(6)p Fp(=)g Fo(y)j Fn(2)d Fo(E)1777 1079 y Fl(!)1826 1067 y Fp(.)509 1166 y(\(ii\))42 b Fo(')30 b Fp(is)f(said)g(to)g(b)r(e)h Fk(exp)l(anding)g Fp(on)f Fo(E)34 b Fp(\(or:)40 b Fo(E)35 b Fp(is)29 b(called)g(an)g Fk(almost)i(c)l(onformal)i(r)l(ep)l(el)t(ler)661 1266 y Fp(for)27 b Fo(')p Fp(\))i(if,)f(in)g(addition,)f(for)g(almost)g(ev)n (ery)g Fo(!)e Fn(2)f Fp(\012)j(there)h(exists)f Fo(n)c Fp(=)g Fo(n)p Fp(\()p Fo(!)s Fp(\))g 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b(DIMENSION)g(UNDER)h(RANDOM)g (PER)-5 b(TURBA)g(TION)458 b(9)555 425 y Fp(T)-7 b(o)27 b(pro)n(v)n(e)f(\(ii\),)j(tak)n(e)e Fo(x)c Fn(2)h Fo(E)1453 437 y Fl(!)1529 425 y Fp(and)j Fo(y)f Fn(2)d Fo(B)1898 437 y Fl(!)1946 425 y Fp(\()p Fo(n)c Fp(+)f(1)p Fo(;)c(x;)g(\017)p Fp(\),)27 b(i.)14 b(e.)1115 578 y Fo(d)p Fp(\()p Fo(')p Fp(\()p Fo(k)s(;)g(!)s Fp(\))p Fo(x;)g(')p Fp(\()p Fo(k)s(;)g(!)s Fp(\))p Fo(y)s Fp(\))24 b Fn(\024)f Fo(\017)p Fp(\()p Fo(#)2089 544 y Fl(k)2130 578 y Fo(!)s Fp(\))k(for)g Fo(k)f Fp(=)d(0)p Fo(;)14 b(:::;)g(n:)456 728 y Fp(By)28 b(\(5\),)g(then,)g(w)n(e)f(ha)n(v)n(e)539 902 y(1)22 b Fn(\025)h Fo(d)p Fp(\()p Fo(')p Fp(\()p Fo(n;)14 b(!)s Fp(\))p Fo(x;)g(')p Fp(\()p Fo(n;)g(!)s Fp(\))p Fo(y)s Fp(\))999 1119 y Fn(\025)1086 977 y Fg( )1152 1015 y Fl(n)p Fi(\000)p Fm(1)1162 1040 y Fg(Y)1159 1218 y Fl(i)p Fm(=)p Fl(k)1292 1119 y Fo(\015)5 b Fp(\()p Fo(#)1421 1084 y Fl(i)1448 1119 y Fo(!)s(;)14 b(')p Fp(\()p Fo(i;)g(!)s Fp(\))p Fo(x)p Fp(\))1858 977 y Fg(!)1938 1119 y Fo(d)p Fp(\()p Fo(')p Fp(\()p Fo(k)s(;)g(!)s Fp(\))p Fo(x;)g(')p Fp(\()p Fo(k)s(;)g(!)s Fp(\))p Fo(y)s Fp(\))168 b(for)27 b(0)22 b Fn(\024)h Fo(k)j Fn(\024)c Fo(n)p Fp(.)456 1370 y(With)28 b(\000)722 1382 y Fl(n)767 1370 y Fp(\()p Fo(!)s Fp(\))23 b(:=)g Fo(e)1059 1340 y Fi(\000)1121 1313 y Fe(n\014)p 1121 1327 72 3 v 1142 1360 a Fd(2)1206 1370 y Fp(\003)1264 1382 y Fl(n)1309 1370 y Fp(\()p Fo(!)s Fp(\))g(=)g(min)1677 1382 y Fl(x)p Fi(2)p Fl(E)1809 1390 y Fe(!)1869 1308 y Fg(Q)1947 1329 y Fl(n)p Fi(\000)p Fm(1)1947 1395 y Fl(k)q Fm(=0)2091 1370 y Fo(\015)5 b Fp(\()p Fo(#)2220 1340 y Fl(k)2261 1370 y Fo(!)s(;)14 b(')p Fp(\()p Fo(k)s(;)g(!)s Fp(\))p Fo(x)p Fp(\))28 b(this)g(rewrites)f(as)976 1563 y Fo(d)p Fp(\()p Fo(')p Fp(\()p Fo(k)s(;)14 b(!)s Fp(\))p Fo(x;)g(')p Fp(\()p Fo(k)s(;)g(!)s Fp(\))p Fo(y)s Fp(\))24 b Fn(\024)2021 1507 y Fp(1)p 1845 1544 395 4 v 1845 1620 a(\000)1897 1632 y Fl(n)p Fi(\000)p Fl(k)2030 1620 y Fp(\()p Fo(#)2111 1596 y Fl(k)2153 1620 y Fo(!)s Fp(\))2416 1563 y(for)j(0)22 b Fn(\024)h Fo(k)j Fn(\024)c Fo(n)p Fp(.)-2468 b(\(6\))456 1772 y(Using)27 b(the)h(fact)g(that)g(log)14 b Fo(a)k Fn(\000)g Fp(log)c Fo(b)23 b Fn(\024)1792 1739 y Fl(a)p Fi(\000)p Fl(b)p 1719 1753 263 4 v 1719 1800 a Fm(min)o Fi(f)p Fl(a;b)p Fi(g)2020 1772 y Fp(if)28 b Fo(a;)14 b(b)23 b(>)f Fp(0,)27 b(de\014nition)h(3.1\(i\))g(th)n(us)f(yields)552 2016 y(log)15 b Fo(\025)p Fp(\()p Fo(#)803 1982 y Fl(k)844 2016 y Fo(!)s(;)f(')p Fp(\()p Fo(k)s(;)g(!)s Fp(\))p Fo(x)p Fp(\))19 b Fn(\000)f Fp(log)1504 1960 y Fo(d)p Fp(\()p Fo(')p Fp(\()p Fo(k)23 b Fp(+)18 b(1)p Fo(;)c(!)s Fp(\))p Fo(x;)g(')p Fp(\()p Fo(k)21 b Fp(+)d(1)p Fo(;)c(!)s Fp(\))p Fo(y)s Fp(\))p 1504 1997 1035 4 v 1647 2073 a Fo(d)p Fp(\()p Fo(')p Fp(\()p Fo(k)s(;)g(!)s Fp(\))p Fo(x;)g(')p Fp(\()p Fo(k)s(;)g(!)s Fp(\))p Fo(y)s Fp(\))1498 2254 y Fn(\024)1602 2197 y Fo(K)6 b Fp(\()p Fo(#)1760 2167 y Fl(k)1801 2197 y Fo(!)s Fp(\))p 1596 2234 299 4 v 1596 2310 a(\000)1648 2322 y Fm(1)1685 2310 y Fp(\()p Fo(#)1766 2286 y 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Fp(\()p Fo(!)s Fp(\))e Fn(\024)i Fo(R)q Fn(g)g Fo(>)g Fp(0.)63 b(F)-7 b(or)456 1219 y(a.e.)36 b Fo(!)k Fp(there)d(exists)g(a)f (sequence)h(\()p Fo(n)p Fp(\()p Fo(j)5 b Fp(\)\))1819 1231 y Fl(j)s Fi(2)p Ff(N)1979 1219 y Fp(of)37 b(in)n(tegers)f(with)h (lim)2714 1231 y Fl(j)s Fi(!1)2895 1219 y Fo(n)p Fp(\()p Fo(j)5 b Fp(\))39 b(=)f Fn(1)g Fp(and)456 1325 y Fo(R)q Fp(\()p Fo(#)601 1295 y Fl(n)p Fm(\()p Fl(j)s Fm(\))729 1325 y Fo(!)s Fp(\))f Fn(\024)g Fo(R)f Fp(for)g(all)g Fo(j)42 b Fn(2)c Fj(N)t Fp(.)68 b(F)-7 b(or)35 b(ev)n(ery)g Fo(j)41 b Fp(c)n(ho)r(ose)35 b(a)h(maximal)f(\()p Fo(!)s(;)14 b(n)p Fp(\()p Fo(j)5 b Fp(\))25 b(+)e(1)p Fo(;)14 b(\017=)p Fp(2\){)456 1425 y(separated)26 b(set)i Fo(G)1025 1437 y Fl(j)1083 1425 y Fn(\032)22 b Fo(E)1231 1437 y Fl(!)1307 1425 y Fp(so)27 b(that)1462 1504 y Fg([)1426 1682 y Fl(x)p Fi(2)p Fl(G)1561 1690 y Fe(j)1604 1583 y Fo(B)1667 1595 y Fl(!)1716 1583 y Fp(\()p Fo(n)p Fp(\()p Fo(j)5 b Fp(\))19 b(+)f(1)p Fo(;)c(x;)g(\017)p Fp(\))22 b(=)h Fo(E)2403 1595 y Fl(!)2451 1583 y Fo(:)456 1841 y Fp(F)-7 b(urthermore,)24 b(diam)13 b Fo(B)1214 1853 y Fl(!)1262 1841 y Fp(\()p Fo(n)p Fp(\()p Fo(j)5 b Fp(\))12 b(+)g(1)p Fo(;)i(x;)g(\017)p Fp(\))23 b Fn(\024)1983 1808 y Fm(2)p Fl(e)2047 1783 y Fe(R)p 1886 1822 306 4 v 1886 1869 a Fl(\025)1925 1880 y Fe(n)p Fd(\()p Fe(j)r Fd(\))2038 1869 y Fm(\()p Fl(!)r(;x)p Fm(\))2224 1841 y Fn(\024)2393 1808 y Fm(2)p Fl(e)2457 1783 y Fe(R)p 2322 1822 254 4 v 2322 1869 a Fm(\003)2367 1880 y Fe(n)p Fd(\()p Fe(j)r Fd(\))2480 1869 y Fm(\()p Fl(!)r Fm(\))2610 1841 y Fp(b)n(y)h(lemma)g(3.3\(ii\).)36 b(Th)n(us)456 1957 y(for)27 b Fo(t)c(>)f(t)753 1969 y Fm(2)818 1957 y Fp(w)n(e)27 b(ha)n(v)n(e)505 2108 y Fg(X)482 2286 y Fl(x)p Fi(2)p Fl(G)617 2294 y Fe(j)661 2187 y Fp(\(diam)14 b Fo(B)950 2199 y Fl(!)998 2187 y Fp(\()p Fo(n)p Fp(\()p Fo(j)5 b Fp(\))19 b(+)f(1)p Fo(;)c(x;)g(\017)p Fp(\)\))1546 2145 y Fl(t)1598 2187 y Fn(\024)23 b Fp(2)1728 2153 y Fl(t)1757 2187 y Fo(e)1796 2153 y Fl(Rt)1911 2108 y Fg(X)1889 2286 y Fl(x)p Fi(2)p Fl(G)2024 2294 y Fe(j)2068 2187 y Fp(exp)2209 2020 y Fg(0)2209 2170 y(@)2281 2187 y Fn(\000)p Fo(t)2390 2076 y Fl(n)p Fm(\()p Fl(j)s Fm(\))p Fi(\000)p Fm(1)2434 2108 y Fg(X)2434 2287 y Fl(k)q Fm(=0)2612 2187 y Fp(log)14 b Fo(\025)p Fp(\()p Fo(#)2862 2153 y Fl(k)2904 2187 y Fo(!)s(;)g(')p Fp(\()p Fo(k)s(;)g(!)s Fp(\))p Fo(x)p Fp(\))3331 2020 y Fg(1)3331 2170 y(A)1598 2423 y Fn(\024)23 b Fp(2)1728 2389 y Fl(t)1757 2423 y Fo(e)1796 2389 y Fl(Rt)1875 2423 y Fo(\031)1922 2435 y Fl(')1970 2423 y Fp(\()p Fn(\000)p Fo(t)14 b Fp(log)g Fo(\025)p Fp(\)\()p Fo(!)s(;)g(n)p Fp(\()p Fo(j)5 b Fp(\))19 b(+)f(1)p Fo(;)c(\017=)p Fp(2\))456 2562 y(for)27 b(su\016cien)n(tly)g (large)f Fo(j)5 b Fp(,)28 b(b)r(ecause)686 2736 y(lim)662 2788 y Fl(j)s Fi(!1)976 2680 y Fp(1)p 849 2717 297 4 v 849 2793 a Fo(n)p Fp(\()p Fo(j)5 b Fp(\))18 b(+)g(1)1169 2736 y(log)c Fo(\031)1337 2748 y Fl(')1385 2736 y Fp(\()p Fn(\000)p Fo(t)g Fp(log)g Fo(\025)p Fp(\)\()p Fo(!)s(;)g(n)p Fp(\()p Fo(j)5 b Fp(\))19 b(+)f(1)p Fo(;)c(\017=)p Fp(2\))22 b(=)g Fo(\031)2494 2702 y Fi(\003)2491 2756 y Fl(')2539 2736 y Fp(\()p Fn(\000)p Fo(t)14 b Fp(log)g Fo(\025)p Fp(\)\()p Fo(\017=)p Fp(2\))23 b Fo(<)g Fp(0)p Fo(:)456 2923 y Fp(Since)h(sup)794 2944 y Fl(x)p Fi(2)p Fl(G)929 2952 y Fe(j)977 2923 y Fp(diam)14 b Fo(B)1234 2935 y Fl(!)1282 2923 y Fp(\()p Fo(n)p Fp(\()p Fo(j)5 b Fp(\))12 b(+)g(1)p Fo(;)i(x;)g(\017)p Fp(\))26 b(tends)e(to)h(0)f(as)g Fo(j)k Fn(!)23 b(1)i Fp(the)g Fo(t)p Fp({dimensional)e(Haus-)456 3031 y(dor\013)33 b(measure)g(of)i Fo(E)1151 3043 y Fl(!)1233 3031 y Fp(is)f(\014nite)h(for)e(ev)n(ery)g Fo(t)h(>)g(t)2094 3043 y Fm(2)2131 3031 y Fp(.)56 b(Hence)35 b(w)n(e)e(ha)n(v)n(e)g Fo(H)7 b(D)r Fp(\()p Fo(E)3030 3043 y Fl(!)3079 3031 y Fp(\))34 b Fn(\024)g Fo(t)3274 3043 y Fm(2)3345 3031 y Fp(for)456 3130 y Fj(P)p Fp({a.e.)25 b Fo(!)s Fp(.)p 3384 3130 4 57 v 3388 3078 50 4 v 3388 3130 V 3437 3130 4 57 v 456 3286 a(W)-7 b(e)25 b(can)h(impro)n(v)n(e)e(statemen)n(t)h (\(ii\))h(as)f(so)r(on)g(as)g(w)n(e)g(are)g(in)g(the)h(situation)g(of)f (theorem)g(1.9,)g(in)456 3386 y(that)j(then)g(w)n(e)f(ma)n(y)g(lo)r(ok) g(at)g(the)h(zero)f(of)g Fo(t)d Fn(7!)f Fo(\031)2028 3398 y Fl(')2076 3386 y Fp(\()p Fn(\000)p Fo(t)14 b Fp(log)g Fo(\021)s Fp(\).)37 b(The)28 b(follo)n(wing,)f(th)n(us,)g(is)h(just)456 3485 y(a)e(reform)n(ulation)f(of)i(theorem)g(3.4)f(for)g(the)h(sp)r (ecial)g(case)f(whic)n(h)h(w)n(e)f(will)i(consider)d(later)i(on.)456 3637 y Ft(3.5.)41 b(Corollary)-8 b(.)41 b Fk(L)l(et)c Fo(')h Fk(b)l(e)g(an)g(exp)l(anding)g Fp(\()p Fo(\025;)14 b(\021)s Fp(\))39 b Fk(almost)f(c)l(onformal)h(bund)t(le)g(RDS)d(on)456 3737 y(a)f(c)l(omp)l(act)g(bund)t(le)g Fo(E)k Fk(with)d(\014b)l(ers)e (in)h(a)f(\014nite)h(dimensional)h(R)n(iemannian)f(manifold)h Fo(M)9 b Fk(.)456 3837 y(Assume)532 3956 y Fp(\(i\))661 3889 y Fg(R)730 3956 y Fp(log)15 b(diam)e Fo(E)1106 3968 y Fl(!)1155 3956 y Fo(d)p Fj(P)p Fp(\()p Fo(!)s Fp(\))22 b Fo(<)h Fn(1)p Fk(,)509 4056 y Fp(\(ii\))42 b Fo(')30 b Fk(is)g(str)l(ongly)g(exp)l(ansive.)456 4175 y(Then)44 b(ther)l(e)h(exist)f(unique)f(r)l(e)l(al)i(numb)l(ers)e Fo(t)1944 4187 y Fm(2)2031 4175 y Fn(\025)49 b Fo(t)2175 4187 y Fm(1)2261 4175 y Fn(\025)g Fp(0)44 b Fk(with)g Fo(\031)2702 4187 y Fl(')2751 4175 y Fp(\()p Fn(\000)p Fo(t)2878 4187 y Fm(1)2928 4175 y Fp(log)15 b Fo(\021)s Fp(\))49 b(=)g(0)g(=)456 4274 y Fo(\031)503 4286 y Fl(')551 4274 y Fp(\()p Fn(\000)p Fo(t)678 4286 y Fm(2)729 4274 y Fp(log)14 b Fo(\025)p Fp(\))p Fk(.)39 b(We)30 b(have)1426 4413 y Fo(t)1456 4425 y Fm(1)1517 4413 y Fn(\024)22 b Fp(HD)q(\()p Fo(E)1823 4425 y Fl(!)1872 4413 y Fp(\))h Fn(\024)g Fo(t)2045 4425 y Fm(2)2252 4413 y Fj(P)p Fk(-a.)13 b(s.)555 4565 y Fp(Note)26 b(that)f(this)h(giv)n(es)e(an)h(exact)g (form)n(ula)g(for)g(the)g(Hausdor\013)g(dimension)g(in)h(the)g(confor-) 456 4665 y(mal)h(case,)g(i.)14 b(e.)27 b(when)h Fo(\025)c Fp(=)e Fo(\021)s Fp(.)456 4817 y(3.6)p Fk(.)37 b(R)l(emark.)42 b Fp(A)30 b(\014rst)f(random)f(v)n(ersion)g(of)h(the)g(Bo)n(w)n (en{Ruelle)f(form)n(ula)g(has)h(b)r(een)g(giv)n(en)456 4917 y(b)n(y)i(Kifer)g([Kif96)o(,)h(Theorem)f(4.8].)48 b(Our)30 b(pro)r(of)h(of)h(theorem)f(3.4)f(is)i(based)f(on)g(the)h (ideas)f(of)456 5016 y(Kifer's)h(pro)r(of:)46 b(Using)33 b(co)n(v)n(erings)d(b)n(y)j(random)e(Bo)n(w)n(en)h(balls)g(to)h(pro)n (v)n(e)e(the)i(upp)r(er)g(b)r(ound)456 5116 y(and)27 b(using)g(the)h(Brin-Katok)e(form)n(ula)g(to)i(pro)n(v)n(e)e(the)i(lo)n (w)n(er)e(b)r(ound.)555 5216 y(The)i(main)f(di\013erences)h(b)r(et)n(w) n(een)f(theorem)g(3.4)g(and)h(Kifer)f([Kif96)o(,)g(Theorem)g(4.8])g (are:)p eop %%Page: 13 13 13 12 bop 947 226 a Fm(HA)n(USDORFF)28 b(DIMENSION)g(UNDER)h(RANDOM)g (PER)-5 b(TURBA)g(TION)425 b(13)578 425 y Fn(\017)41 b Fp(W)-7 b(e)28 b(do)g(not)f(assume)g(a)g(mixing)h(condition)f(on)g Fo(')p Fp(.)578 525 y Fn(\017)41 b Fp(Our)27 b(expansiv)n(e)f(constan)n (t)g Fo(\017)h Fp(is)h(allo)n(w)n(ed)d(to)j(b)r(e)f(a)g(random)f(v)-5 b(ariable,)27 b(whic)n(h)g(mak)n(es)f(it)661 624 y(p)r(ossible)k(to)h (w)n(ork)e(with)i(systems)f(with)h(w)n(eak)n(er)d(uniformit)n(y)j(prop) r(erties.)44 b([Consider)661 727 y(the)32 b(follo)n(wing)f(example:)44 b(Let)31 b Fo(')p Fp(\()p Fo(!)s Fp(\))g(:)e Fo(S)1992 697 y Fm(1)2059 727 y Fn(!)g Fo(S)2227 697 y Fm(1)2264 727 y Fo(;)14 b(z)33 b Fn(7!)c Fo(z)2528 697 y Fl(n)p Fm(\()p Fl(!)r Fm(\))2700 727 y Fp(for)i(some)g(un)n(b)r(ounded)661 827 y(random)g(v)-5 b(ariable)31 b Fo(n)f Fn(2)g Fo(L)1508 797 y Fm(1)1545 827 y Fp(\(\012)p Fo(;)14 b Fj(N)t Fp(\),)39 b Fo(n)p Fp(\()p Fo(!)s Fp(\))30 b Fn(\025)g Fp(2.)49 b(Then)32 b(the)g(RDS)g(generated)f(b)n(y)g Fo(')i Fp(is)661 926 y(not)28 b(expansiv)n(e)e(for)h(an)n(y)g(deterministic)h(constan)n (t)f Fo(\016)f(>)d Fp(0.])578 1026 y Fn(\017)41 b Fo(')22 b Fp(is)f(not)f(assumed)h(to)g(b)r(e)g(conformal.)33 b(This)21 b(mak)n(es)f(it)i(p)r(ossible)e(to)h(apply)g(theorem)f(3.4) 661 1126 y(to)h(general)e Fo(C)1100 1096 y Fm(1+)p Fl(\013)1253 1126 y Fp(p)r(erturbations)h(of)g(a)h(conformal)e(rep)r(eller)h (instead)h(of)f(p)r(erturbations)661 1225 y(b)n(y)28 b(conformal)e(maps.)578 1325 y Fn(\017)41 b Fp(Our)24 b(setup)h(allo)n(ws)e(bundles)i(of)f(compact)g(metric)h(spaces)e(and)i (do)r(es)f(not)g(assume)g(that)661 1425 y(the)k(system)g(liv)n(es)f(on) g(a)g(smo)r(oth)g(manifold.)462 1656 y(4.)45 b Fq(St)-6 b(ability)32 b(of)f(the)h(Ha)n(usdorff)f(dimension)h(under)f(random)g (per)-6 b(turba)g(tions)555 1805 y Fp(In)36 b(this)f(section)g(w)n(e)g (apply)g(theorem)g(3.4)f(to)h(random)f(p)r(erturbations)h(of)g(a)g (conformal)456 1905 y(rep)r(eller.)456 2067 y Ft(4.1.)41 b(De\014nition.)f Fp(A)23 b(compact)g(subset)g Fo(J)31 b Fp(of)23 b(a)f(smo)r(oth)h(Riemannian)g(manifold)g Fo(M)32 b Fp(is)23 b(said)456 2167 y(to)30 b(b)r(e)g(a)g Fk(c)l(onformal)j(r)l(ep)l(el)t(ler)f Fp(for)e(a)f Fn(C)1682 2136 y Fm(1+)p Fl(\013)1843 2167 y Fp(map)i Fo(F)39 b Fp(:)27 b Fo(U)36 b Fn(!)28 b Fo(M)9 b Fp(,)30 b(where)g Fo(U)36 b Fn(\032)27 b Fo(M)39 b Fp(is)30 b(an)f(op)r(en)456 2266 y(neigh)n(b)r(orho)r(o)r(d)d(of)i Fo(J)8 b Fp(,)27 b(if)h(the)g(follo)n(wing)f(holds:)532 2394 y(\(i\))42 b Fo(F)12 b(J)31 b Fp(=)23 b Fo(J)8 b Fp(,)509 2494 y(\(ii\))42 b Fo(J)31 b Fp(=)826 2432 y Fg(T)895 2519 y Fl(n)p Fi(\025)p Fm(0)1039 2494 y Fo(F)1104 2464 y Fi(\000)p Fl(n)1201 2494 y Fp(\()p Fo(U)9 b Fp(\),)486 2600 y(\(iii\))42 b Fo(D)r(F)12 b Fp(\()p Fo(x)p Fp(\))27 b(is)e(conformal)f(for)h(ev)n (ery)f Fo(x)g Fn(2)f Fo(J)8 b Fp(,)26 b(i.)14 b(e.)26 b Fo(D)r(F)12 b Fp(\()p Fo(x)p Fp(\))24 b(=)e Fo(\025)p Fp(\()p Fo(x)p Fp(\))p Fo(O)r Fp(\()p Fo(x)p Fp(\),)30 b(where)25 b Fo(\025)p Fp(\()p Fo(x)p Fp(\))f Fn(2)f Fj(R)661 2700 y Fp(and)28 b Fo(O)r Fp(\()p Fo(x)p Fp(\))c Fn(2)g(O)r Fp(\()p Fo(d;)14 b Fj(R)q Fp(\))34 b(is)27 b(orthogonal,)488 2799 y(\(iv\))42 b(There)27 b(exists)h Fo(n)23 b Fn(\025)f Fp(1)27 b(suc)n(h)h(that)f Fn(j)p Fo(\025)p Fp(\()p Fo(F)1895 2769 y Fl(n)p Fi(\000)p Fm(1)2026 2799 y Fp(\()p Fo(x)p Fp(\)\))p Fo(:::\025)p Fp(\()p Fo(x)p Fp(\))p Fn(j)f Fo(>)d Fp(1)k(for)g(ev)n(ery)g Fo(x)c Fn(2)h Fo(J)8 b Fp(.)555 2961 y(Conformal)26 b(rep)r(ellers)h(p) r(ersist)g(under)g(random)f(p)r(erturbations.)36 b(Recall)27 b(that)g Fn(K)i Fp(denotes)456 3061 y(the)f(collection)f(of)g(all)g (compact)h(subsets)f(of)g Fo(M)37 b Fp(endo)n(w)n(ed)26 b(with)j(the)e(Hausdor\013)g(top)r(ology)-7 b(.)456 3223 y Ft(4.2.)41 b(Prop)s(osition.)e Fk(Assume)26 b Fo(J)34 b Fk(is)27 b(a)g(c)l(onformal)h(r)l(ep)l(el)t(ler)g(for)f(a)g Fn(C)2664 3192 y Fm(1+)p Fl(\013)2821 3223 y Fk(map)g Fo(F)35 b Fp(:)23 b Fo(U)32 b Fn(!)23 b Fo(M)9 b Fk(.)456 3322 y(Ther)l(e)24 b(exists)g(a)g Fn(C)1025 3292 y Fm(1)1086 3322 y Fk(neighb)l(orho)l(o)l(d)i Fn(U)8 b Fp(\()p Fo(F)k Fp(\))24 b Fn(\032)f(C)1923 3292 y Fm(1)1959 3322 y Fp(\()p Fo(U;)14 b(M)9 b Fp(\))24 b Fk(of)h Fo(F)35 b Fk(such)24 b(that)g(the)g(fol)t(lowing)i(holds:)532 3450 y Fp(\(i\))42 b Fk(F)-6 b(or)39 b(every)g(r)l(andom)g(variable)h Fo(')f Fp(:)f(\012)h Fn(!)f(U)8 b Fp(\()p Fo(F)k Fp(\))39 b Fk(ther)l(e)f(exists)g(a)h Fn(K)q Fk(-value)l(d)g(r)l(andom)661 3550 y(variable)31 b Fo(J)1017 3562 y Fl(')1093 3550 y Fk(satisfying)e Fo(J)1506 3562 y Fl(')1554 3550 y Fp(\()p Fo(!)s Fp(\))23 b Fn(\032)g Fo(U)37 b Fk(and)29 b Fo(')p Fp(\()p Fo(!)s Fp(\))p Fo(J)2257 3562 y Fl(')2305 3550 y Fp(\()p Fo(!)s Fp(\))23 b(=)g Fo(J)2581 3562 y Fl(')2629 3550 y Fp(\()p Fo(#!)s Fp(\))p Fk(,)29 b(and)g(a)f Fn(C)3130 3520 y Fm(0)3167 3550 y Fp(\()p Fo(U;)14 b(M)9 b Fp(\))p Fk(-)661 3649 y(value)l(d)35 b(r)l(andom)g(variable)h Fo(h)d Fk(such)h(that)g(e)l(ach)h Fo(h)p Fp(\()p Fo(!)s Fp(\))f Fk(is)h(a)f(home)l(omorphism)j(b)l(etwe)l(en)661 3749 y Fo(J)h Fk(and)30 b Fo(J)952 3761 y Fl(')1000 3749 y Fp(\()p Fo(!)s Fp(\))g Fk(and)g Fo(')p Fp(\()p Fo(!)s Fp(\))19 b Fn(\016)f Fo(h)p Fp(\()p Fo(!)s Fp(\))23 b(=)g Fo(h)p Fp(\()p Fo(#!)s Fp(\))c Fn(\016)f Fo(F)41 b Fk(on)30 b Fo(J)8 b Fk(.)509 3849 y Fp(\(ii\))42 b Fk(If)29 b Fp(\()p Fo(')833 3861 y Fl(\017)889 3849 y Fp(:)23 b(\012)g Fn(!)g(U)8 b Fp(\()p Fo(F)k Fp(\)\))1345 3861 y Fl(\017>)p Fm(0)1491 3849 y Fk(is)29 b(a)g(family)h(of)g(r)l(andom)f(variables)i (with)e Fp(lim)2937 3861 y Fl(\017)p Fi(!)p Fm(0)3082 3849 y Fo(')3136 3861 y Fl(\017)3191 3849 y Fp(=)23 b Fo(F)40 b Fk(in)661 3948 y(pr)l(ob)l(ability)28 b(\(with)e(r)l(esp)l(e) l(ct)g(to)f(the)h Fn(C)1812 3918 y Fm(1)1874 3948 y Fk(distanc)l(e\),)i (then)d Fp(lim)2546 3960 y Fl(\017)p Fi(!)p Fm(0)2691 3948 y Fo(h)2739 3960 y Fl(\017)2794 3948 y Fp(=)e(id)i Fk(in)h(pr)l(ob)l(ability)661 4048 y(and)32 b(thus)f Fp(lim)1119 4060 y Fl(\017)p Fi(!)p Fm(0)1264 4048 y Fo(J)1310 4060 y Fl(\017)1368 4048 y Fp(=)25 b Fo(J)39 b Fk(in)32 b(pr)l(ob)l(ability.)45 b(Her)l(e)31 b Fo(h)2333 4060 y Fl(\017)2396 4048 y Fk(and)h Fo(J)2605 4060 y Fl(\017)2668 4048 y Fk(ar)l(e)g(the)f(c)l(orr)l(esp)l(onding)661 4148 y(obje)l(cts)g(asso)l(ciate)l(d)g(to)f Fo(')1473 4160 y Fl(\017)1535 4148 y Fk(by)g(\(i\).)555 4275 y(In)48 b(other)g(wor)l(ds,)53 b(e)l(ach)c(family)g Fn(f)p Fo(')1768 4287 y Fl(\017)1800 4275 y Fn(g)1842 4287 y Fl(\017>)p Fm(0)2006 4275 y Fk(of)f Fn(U)8 b Fp(\()p Fo(F)k Fp(\))p Fk(-value)l(d)49 b(r)l(andom)f(variables)i(with)456 4375 y Fp(lim)571 4387 y Fl(\017)p Fi(!)p Fm(0)716 4375 y Fo(')770 4387 y Fl(\017)831 4375 y Fp(=)29 b Fo(F)45 b Fk(in)34 b(pr)l(ob)l(ability)h(is)f(a)f(structur)l(al)t(ly)g(stable)h (r)l(andom)g(p)l(erturb)l(ation)f(of)h Fo(F)45 b Fk(on)456 4475 y Fo(J)8 b Fk(.)456 4637 y(Pr)l(o)l(of.)43 b Fp(\(i\))35 b(can)f(b)r(e)h(sho)n(wn)e(analogously)f(to)j(the)f(corresp)r(onding)f (assertions)g(in)h(Theorem)456 4736 y(1.1)26 b(of)i([Liu98)o(].)555 4836 y(T)-7 b(o)31 b(pro)n(v)n(e)f(\(ii\))i(one)f(has)g(to)h(observ)n (e)d(that)j Fo(h)p Fp(\()p Fo(!)s Fp(\))e Fn(2)g(C)5 b Fp(\()p Fo(J)o(;)14 b(M)9 b Fp(\))31 b(dep)r(ends)g(con)n(tin)n (uously)g(on)456 4935 y(\()p Fo(')p Fp(\()p Fo(#)623 4905 y Fl(k)664 4935 y Fo(!)s Fp(\)\))783 4947 y Fl(k)q Fi(2)p Ff(Z)931 4935 y Fn(2)23 b(U)8 b Fp(\()p Fo(F)k Fp(\))1198 4905 y Ff(Z)1270 4935 y Fp(\(equipp)r(ed)28 b(with)g(pro)r(duct)f(top)r(ology\).)36 b(This)27 b(follo)n(ws)g(in)g (the)h(same)456 5035 y(w)n(a)n(y)e(as)h(the)h(corresp)r(onding)e (statemen)n(t)h(of)h([Liu98)o(,)g(Theorem)e(1.1].)p 3384 5035 4 57 v 3388 4982 50 4 v 3388 5035 V 3437 5035 4 57 v 456 5216 a(Here)h(is)g(our)g(\014nal)h(result.)p eop %%Page: 14 14 14 13 bop 456 238 a Fm(14)880 b(T.)29 b(BOGENSCH)1880 223 y(\177)1872 238 y(UTZ)h(AND)f(G.)g(OCHS)456 425 y Ft(4.3.)41 b(Theorem.)e Fk(Assume)34 b Fo(J)42 b Fk(is)35 b(a)f(c)l(onformal)i(r)l(ep)l(el)t(ler)f(for)h(a)e Fn(C)2615 395 y Fm(1+)p Fl(\013)2780 425 y Fk(map)h Fo(F)44 b Fp(:)31 b Fo(U)40 b Fn(!)31 b Fo(M)9 b Fk(.)456 525 y(Ther)l(e)32 b(exists)g(a)g Fn(C)1049 495 y Fm(1)1118 525 y Fk(neighb)l(orho)l(o)l (d)i Fn(U)8 b Fp(\()p Fo(F)k Fp(\))27 b Fn(\032)g(C)1970 495 y Fm(1)2007 525 y Fp(\()p Fo(U;)14 b(M)9 b Fp(\))31 b Fk(of)i Fo(F)43 b Fk(such)32 b(that)g(the)g(assertions)h(of)456 624 y(pr)l(op)l(osition)e(4.2)g(hold)g(with)g(the)f(fol)t(lowing)i (additional)g(pr)l(op)l(erty.)555 724 y(If)e Fp(\()p Fo(')728 736 y Fl(\017)784 724 y Fp(:)23 b(\012)g Fn(!)g(U)8 b Fp(\()p Fo(F)k Fp(\))19 b Fn(\\)g(C)1350 694 y Fm(1+)p Fl(\013)1481 724 y Fp(\()p Fo(U;)14 b(M)9 b Fp(\)\))1761 736 y Fl(\017>)p Fm(0)1907 724 y Fk(is)30 b(a)h(r)l(andom)f(p)l(erturb) l(ation)g(of)g Fo(F)42 b Fk(with)1265 881 y Fp(lim)1259 932 y Fl(\017)p Fi(!)p Fm(0)1400 881 y Fo(')1454 893 y Fl(\017)1509 881 y Fp(=)23 b Fo(F)182 b Fk(in)30 b Fj(L)1983 850 y Fm(1)2026 881 y Fp(\(\012)p Fo(;)14 b Fn(C)2204 850 y Fm(1+)p Fl(\013)2335 881 y Fp(\()p Fo(U;)g(M)9 b Fp(\)\))p Fk(,)456 1063 y(then)1318 1219 y Fp(lim)1313 1271 y Fl(\017)p Fi(!)p Fm(0)1453 1219 y Fp(HD)q(\()p Fo(J)1657 1231 y Fl(\017)1689 1219 y Fp(\()p Fo(!)s Fp(\)\))24 b(=)e(HD)q(\()p Fo(J)8 b Fp(\))170 b Fj(P)p Fk(-a.)14 b(s.)456 1406 y(Mor)l(e)l(over,)37 b(if)e Fo(L)30 b Fn(\032)h Fo(J)42 b Fk(is)34 b(c)l(omp)l(act)h(with)g Fo(F)12 b(L)30 b Fp(=)h Fo(L)p Fk(,)k(then)f Fp(lim)2480 1418 y Fl(\017)p Fi(!)p Fm(0)2625 1406 y Fp(HD)q(\()p Fo(h)2831 1418 y Fl(\017)2863 1406 y Fp(\()p Fo(!)s Fp(\))p Fo(L)p Fp(\))d(=)g(HD\()p Fo(L)p Fp(\))456 1506 y Fk(a.)15 b(s.)456 1668 y(Pr)l(o)l(of.)43 b Fp(Cho)r(ose)31 b(an)h(op)r(en)g(neigh)n(b)r(orho)r(o)r(d)f Fo(V)51 b Fp(of)32 b Fo(J)41 b Fp(suc)n(h)31 b(that)2533 1647 y(\026)2521 1668 y Fo(V)51 b Fp(is)32 b(a)g(compact)f(subset)i(of) 456 1768 y Fo(U)9 b Fp(.)36 b(Then)25 b Fn(U)8 b Fp(\()p Fo(F)k Fp(\))27 b(can)e(b)r(e)h(c)n(hosen)f(in)h(suc)n(h)f(a)g(w)n(a)n (y)g(that)g Fo(J)2273 1780 y Fl(\017)2305 1768 y Fp(\()p Fo(!)s Fp(\))f Fn(\032)2548 1747 y Fp(\026)2535 1768 y Fo(V)45 b Fp(for)25 b(ev)n(ery)f Fo(\017)f(>)g Fp(0,)i Fo(!)h Fn(2)d Fp(\012.)555 1867 y(Fix)28 b Fo(\017)23 b(>)f Fp(0.)37 b(F)-7 b(or)27 b(\()p Fo(!)s(;)14 b(x)p Fp(\))23 b Fn(2)h Fp(\012)18 b Fn(\002)1578 1846 y Fp(\026)1565 1867 y Fo(V)47 b Fp(set)837 2025 y Fo(\021)878 2037 y Fl(\017)910 2025 y Fp(\()p Fo(!)s(;)14 b(x)p Fp(\))24 b(:=)e Fn(k)p Fo(D)r(')1414 2037 y Fl(\017)1446 2025 y Fp(\()p Fo(!)s Fp(\)\()p Fo(x)p Fp(\))p Fn(k)29 b Fp(and)e Fo(\025)1956 2037 y Fl(\017)1988 2025 y Fp(\()p Fo(!)s(;)14 b(x)p Fp(\))24 b(:=)f Fn(k)p Fp(\()p Fo(D)r(')2525 2037 y Fl(\017)2557 2025 y Fp(\()p Fo(!)s Fp(\)\()p Fo(x)p Fp(\)\))2819 1991 y Fi(\000)p Fm(1)2909 2025 y Fn(k)2951 1991 y Fi(\000)p Fm(1)3040 2025 y Fo(:)456 2181 y Fp(\(1\))k(W)-7 b(e)28 b(claim)g(that)g Fo(')1188 2193 y Fl(\017)1247 2181 y Fp(is)g(\()p Fo(\025)1411 2193 y Fl(\017)1443 2181 y Fo(;)14 b(\021)1521 2193 y Fl(\017)1553 2181 y Fp(\))28 b(almost)f(conformal.)555 2281 y(Cho)r(ose)g Fo(r)881 2293 y Fm(0)942 2281 y Fo(>)22 b Fp(0)28 b(suc)n(h)f(that)h Fo(A)23 b Fp(:=)g Fn(f)p Fo(x)g Fp(:)g(dist\()p Fo(x;)2083 2260 y Fp(\026)2070 2281 y Fo(V)d Fp(\))j Fn(\024)g Fo(r)2318 2293 y Fm(0)2355 2281 y Fn(g)g(\032)f Fo(U)9 b Fp(.)37 b(De\014ne)556 2487 y Fo(K)627 2499 y Fm(0)664 2487 y Fp(\()p Fo(!)s Fp(\))23 b(:=)g Fn(j)p Fo(D)r(')1065 2499 y Fl(\017)1097 2487 y Fp(\()p Fo(!)s Fp(\))p Fn(j)1239 2499 y Fl(\013;A)1379 2487 y Fp(=)g(sup)1606 2370 y Fg(\032)1678 2431 y Fn(k)p Fo(D)r(')1845 2443 y Fl(\017)1876 2431 y Fp(\()p Fo(!)s Fp(\)\()p Fo(x)p Fp(\))d Fn(\000)e Fo(D)r(')2334 2443 y Fl(\017)2366 2431 y Fp(\()p Fo(!)s Fp(\)\()p Fo(y)s Fp(\))p Fn(k)p 1678 2468 958 4 v 2015 2544 a Fo(d)p Fp(\()p Fo(x;)c(y)s Fp(\))2250 2520 y Fl(\013)2668 2487 y Fp(:)24 b Fo(x;)14 b(y)26 b Fn(2)d Fo(A;)14 b(x)24 b Fn(6)p Fp(=)e Fo(y)3245 2370 y Fg(\033)3321 2487 y Fo(:)456 2702 y Fp(W)-7 b(e)27 b(restrict)g(ourselv)n(es)e(to)i(the)h(case)f(when)g Fo(M)36 b Fp(is)27 b(an)g(op)r(en)g(subset)h(of)f Fj(R)2790 2672 y Fl(d)2835 2702 y Fp(.)37 b(F)-7 b(or)26 b(the)i(general)456 2801 y(case,)g(similar)f(calculations)h(can)g(b)r(e)h(done)f(via)g(lo)r (cal)g(co)r(ordinates.)39 b(W)-7 b(e)29 b(let)g Fn(j)19 b(\001)g(j)29 b Fp(denote)f(the)456 2901 y(Euclidian)f(norm)g(on)g Fj(R)1212 2871 y Fl(d)1284 2901 y Fp(and)h(write)f Fo(')h Fp(instead)g(of)f Fo(')2175 2913 y Fl(\017)2207 2901 y Fp(\()p Fo(!)s Fp(\))h(for)f(con)n(v)n(enience.)555 3003 y(F)-7 b(or)27 b Fo(x;)14 b(y)26 b Fn(2)e Fo(J)980 3015 y Fl(\017)1012 3003 y Fp(\()p Fo(!)s Fp(\))j(with)i(0)22 b Fo(<)h Fn(j)p Fo(x)c Fn(\000)f Fo(y)s Fn(j)23 b Fo(<)f(r)1886 3015 y Fm(0)1952 3003 y Fp(w)n(e)27 b(put)h Fo(e)23 b Fp(:=)2428 2966 y Fl(y)r Fi(\000)p Fl(x)p 2408 2984 165 4 v 2408 3032 a Fi(j)p Fl(y)r Fi(\000)p Fl(x)p Fi(j)2611 3003 y Fp(and)k(obtain)706 3257 y Fn(j)p Fo(')p Fp(\()p Fo(x)p Fp(\))20 b Fn(\000)e Fo(')p Fp(\()p Fo(y)s Fp(\))p Fn(j)84 b Fp(=)1413 3111 y Fg(\014)1413 3161 y(\014)1413 3211 y(\014)1413 3261 y(\014)1413 3311 y(\014)1441 3144 y(Z)1524 3164 y Fi(j)p Fl(y)r Fi(\000)p Fl(x)p Fi(j)1487 3332 y Fm(0)1707 3257 y Fo(D)r(')p Fp(\()p Fo(x)19 b Fp(+)f Fo(te)p Fp(\))p Fo(edt)2226 3111 y Fg(\014)2226 3161 y(\014)2226 3211 y(\014)2226 3261 y(\014)2226 3311 y(\014)2277 3257 y Fn(\024)2364 3144 y Fg(Z)2447 3164 y Fi(j)p Fl(y)r Fi(\000)p Fl(x)p Fi(j)2411 3332 y Fm(0)2630 3257 y Fn(k)p Fo(D)r(')p Fp(\()p Fo(x)h Fp(+)f 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Fn(j)p Fo(y)21 b Fn(\000)d Fo(x)p Fn(j)811 3965 y Fg(\014)811 4015 y(\014)811 4065 y(\014)811 4115 y(\014)811 4165 y(\014)839 3998 y(Z)922 4018 y Fi(j)p Fl(y)r Fi(\000)p Fl(x)p Fi(j)885 4186 y Fm(0)1105 4111 y Fo(D)r(')p Fp(\()p Fo(x)h Fp(+)f Fo(te)p Fp(\))p Fo(edt)1624 3965 y Fg(\014)1624 4015 y(\014)1624 4065 y(\014)1624 4115 y(\014)1624 4165 y(\014)633 4310 y Fn(\025)23 b Fp(inf)7 b Fn(fj)p Fo(Ae)p Fn(j)22 b Fp(:)i Fo(A)f Fn(2)g Fp(\\con)n(v)n(ex)j(h)n(ull)i(of)f Fo(D)r(')p Fp(\()p Fo(x)20 b Fp(+)e Fo(te)p Fp(\))p Fo(;)c Fp(0)22 b Fn(\024)h Fo(t)g Fn(\024)f(j)p Fo(y)g Fn(\000)c Fo(x)p Fn(j)p Fp(")p Fn(g)633 4439 y(\025)23 b(j)p Fo(D)r(')p Fp(\()p Fo(x)p Fp(\))p Fo(e)p Fn(j)698 4569 y(\000)18 b Fp(sup)p Fn(fj)p Fo(Ae)p Fn(j)k Fp(:)h Fo(A)h Fn(2)f Fp(\\con)n(v)n(ex)j(h)n(ull)h(of)h(\()p Fo(D)r(')p Fp(\()p Fo(x)20 b Fp(+)e Fo(te)p Fp(\))g Fn(\000)g Fo(D)r(')p Fp(\()p Fo(x)p Fp(\)\))p Fo(;)c Fp(0)24 b Fn(\024)f Fo(t)g Fn(\024)g(j)p Fo(y)e Fn(\000)d Fo(x)p Fn(j)p Fp(")p Fn(g)633 4698 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Fp(0)f Fo(<)h(\031)2048 4117 y Fl(\017)2080 4105 y Fp(\()p Fn(\000)p Fp(\()p Fo(t)2239 4117 y Fm(0)2295 4105 y Fn(\000)18 b Fo(\016)s Fp(\))c(log)g Fo(\025)2633 4117 y Fl(\017)2665 4105 y Fp(\))116 b(and)1037 4229 y Fo(\031)1084 4241 y Fl(\017)1116 4229 y Fp(\()p Fn(\000)p Fp(\()p Fo(t)1275 4241 y Fm(0)1331 4229 y Fp(+)18 b Fo(\016)s Fp(\))c(log)g Fo(\021)1662 4241 y Fl(\017)1694 4229 y Fp(\))23 b Fo(<)g Fp(0)g Fo(<)f(\031)2036 4241 y Fl(\017)2068 4229 y Fp(\()p Fn(\000)p Fp(\()p Fo(t)2227 4241 y Fm(0)2283 4229 y Fn(\000)c Fo(\016)s Fp(\))c(log)g Fo(\021)2614 4241 y Fl(\017)2646 4229 y Fp(\))p Fo(:)456 4366 y Fp(Th)n(us)27 b Fo(t)696 4378 y Fm(0)752 4366 y Fn(\000)18 b Fo(\016)26 b(<)c(t)1015 4336 y Fl(\017)1015 4387 y Fm(1)1076 4366 y Fn(\024)g Fp(HD)q(\()p Fo(h)1369 4378 y Fl(\017)1401 4366 y Fp(\()p Fo(!)s Fp(\))p Fo(L)p Fp(\))h Fn(\024)g Fo(t)1750 4336 y Fl(\017)1750 4387 y Fm(2)1810 4366 y Fo(<)g(t)1928 4378 y Fm(0)1983 4366 y Fp(+)18 b Fo(\016)s Fp(,)28 b(\014nishing)g (the)g(pro)r(of.)p 3384 4366 4 57 v 3388 4313 50 4 v 3388 4366 V 3437 4366 4 57 v 456 4518 a(4.4)p Fk(.)37 b(R)l(emark.)42 b Fp(Ruelle)27 b([Rue83])g(sho)n(w)n(ed)e(that)j(if)f Fo(J)35 b Fp(is)27 b(a)f(mixing)h(conformal)f(rep)r(eller)g(for)g Fo(F)456 4618 y Fp(and)g Fo(t)d Fp(=)g(HD\()p Fo(J)8 b Fp(\))28 b(then)f(the)f(equilibrium)h(state)f Fo(m)h Fp(of)f Fn(\000)p Fo(t)14 b Fp(log)g Fn(j)p Fo(D)r(F)e Fn(j)27 b Fp(is)f(the)h(unique)g(in)n(v)-5 b(arian)n(t)456 4717 y(probabilit)n(y)26 b(measure)h(whic)n(h)g(is)h(equiv)-5 b(alen)n(t)27 b(to)h(the)g Fo(t)p Fp({dimensional)e(Hausdor\013)h (measure.)555 4817 y(Corollary)h(2.4)g(implies)i(that)g Fo(m)g Fp(is)g(sto)r(c)n(hastically)e(stable)i(in)g(the)g(follo)n(wing) e(sense.)43 b(Let)456 4917 y(\()p Fo(')542 4929 y Fl(\017)574 4917 y Fp(\))606 4929 y Fl(\017>)p Fm(0)753 4917 y Fp(satisfy)30 b(the)g(assumptions)g(of)g(theorem)f(3.4)h(and)g(let)g Fo(\026)2530 4929 y Fl(\017)2592 4917 y Fp(b)r(e)h(an)e(equilibrium)i (state)456 5016 y(for)c Fn(\000)p Fo(t)678 4986 y Fl(\017)678 5037 y Fm(1)728 5016 y Fp(log)14 b Fo(\021)890 5028 y Fl(\017)950 5016 y Fp(\(or)27 b Fn(\000)p Fo(t)1179 4986 y Fl(\017)1179 5037 y Fm(2)1230 5016 y Fp(log)14 b Fo(\025)1399 5028 y Fl(\017)1431 5016 y Fp(\))28 b(with)g(resp)r(ect)f(to)h Fo(')2120 5028 y Fl(\017)2152 5016 y Fp(.)37 b(Then)28 b(lim)2544 5028 y Fl(\017)p Fi(!)p Fm(0)2689 5016 y Fo(\026)2739 5028 y Fl(\017)2794 5016 y Fp(=)23 b Fj(P)17 b Fn(\012)h Fo(m)p Fp(.)555 5116 y(If)35 b(in)f(addition)g(the)g Fo(')1285 5128 y Fl(\017)1317 5116 y Fp(\()p Fo(!)s Fp(\))h(are)e (conformal)g(then)h Fo(\021)2244 5128 y Fl(\017)2310 5116 y Fp(=)g Fo(\025)2457 5128 y Fl(\017)2489 5116 y Fp(.)56 b(\\Randomizing")32 b(Ruelle's)456 5216 y(pro)r(of)25 b(one)g(should)h(b)r(e)g(able)f(to)h(sho)n(w)f(that)h Fo(\026)1899 5228 y Fl(\017)1956 5216 y Fp(is)g(the)g(unique)g Fo(')2501 5228 y Fl(\017)2559 5216 y Fp(in)n(v)-5 b(arian)n(t)24 b(measure)h(whose)p eop %%Page: 16 16 16 15 bop 456 238 a Fm(16)880 b(T.)29 b(BOGENSCH)1880 223 y(\177)1872 238 y(UTZ)h(AND)f(G.)g(OCHS)456 425 y Fp(disin)n(tegrations)j(on)i(the)g(\014b)r(ers)g Fo(J)1567 437 y Fl(\017)1599 425 y Fp(\()p Fo(!)s Fp(\))g(are)f(almost)h(surely)f (equiv)-5 b(alen)n(t)34 b(to)g(the)g Fo(t)3109 395 y Fl(\017)3109 446 y Fm(1)3180 425 y Fp(=)g Fo(t)3309 395 y Fl(\017)3309 446 y Fm(2)3380 425 y Fp(=)456 525 y(HD\()p Fo(J)659 537 y Fl(\017)691 525 y Fp(\()p Fo(!)s Fp(\)\))29 b(dimensional)e(Hausdor\013)g(measure.)456 674 y(4.5)p Fk(.)37 b(R)l(emark.)42 b Fp(Com)n(bining)23 b(tec)n(hniques)g(of)h (McClusk)n(ey)e(and)h(Manning)g([MM83])g(with)h(that)456 774 y(of)34 b(this)g(pap)r(er)g(it)h(should)e(b)r(e)i(p)r(ossible)f(to) g(deriv)n(e)f(sto)r(c)n(hastic)h(stabilit)n(y)f(of)i(the)f(Hausdor\013) 456 874 y(dimension)23 b(of)h(\\horsesho)r(es",)e(where)h(a)g(horsesho) r(e)f(is)i(mean)n(t)f(to)h(b)r(e)g(a)f(h)n(yp)r(erb)r(olic)h(in)n(v)-5 b(arian)n(t)456 973 y(set)22 b(of)g(a)g(surface)f(di\013eomorphism)h (with)h(one{dimensional)d(stable)i(and)g(unstable)g(manifold,)456 1073 y(resp)r(ectiv)n(ely)-7 b(.)1708 1242 y Fq(References)456 1375 y Fs([Arn98])59 b(Ludwig)27 b(Arnold,)h Fh(R)l(andom)j(dynamic)l (al)f(systems)p Fs(,)e(Springer,)g(New)f(Y)-6 b(ork)28 b(Berlin,)f(1998.)h(T)-6 b(o)27 b(ap-)745 1458 y(p)r(ear.)456 1541 y([Bal97])74 b(Viviane)28 b(Baladi,)h Fh(Corr)l(elation)i(sp)l(e)l (ctrum)g(of)f(quenche)l(d)h(and)f(anne)l(ale)l(d)i(e)l(quilibrium)e (states)g(for)745 1624 y(r)l(andom)d(exp)l(anding)g(maps)p Fs(,)d(Comm)n(un.)e(Math.)h(Ph)n(ys.)h Fb(186)e Fs(\(1997\),)k (671{700.)456 1707 y([BKS96])35 b(Viviane)18 b(Baladi,)g(Ab)r(delaziz)h (Kondah,)g(and)g(Bernard)e(Sc)n(hmitt,)i Fh(R)l(andom)j(c)l(orr)l (elations)g(for)e(smal)t(l)745 1790 y(p)l(erturb)l(ations)27 b(of)e(exp)l(anding)i(maps)p Fs(,)e(Random)e(Comput.)g(Dynamics)g Fb(4)g Fs(\(1996\),)i(85{98.)456 1873 y([Bog92])59 b(Thomas)31 b(Bogensc)n(h)r(\177)-37 b(utz,)35 b Fh(Entr)l(opy,)h(pr)l(essur)l(e,)g (and)e(a)g(variational)g(principle)f(for)h(r)l(andom)h(dy-)745 1956 y(namic)l(al)27 b(systems)p Fs(,)c(Random)g(Comput.)g(Dynamics)g Fb(1)g Fs(\(1992\),)i(99{116.)456 2039 y([Bog93])59 b(Thomas)25 b(Bogensc)n(h)r(\177)-37 b(utz,)29 b Fh(Equilibrium)g(states)f(for)g(r) l(andom)i(dynamic)l(al)g(systems)p Fs(,)c(Ph.D.)g(thesis,)745 2122 y(Univ)n(ersit\177)-35 b(at)24 b(Bremen,)e(1993.)456 2205 y([Bog96])59 b(Thomas)26 b(Bogensc)n(h)r(\177)-37 b(utz,)31 b Fh(Sto)l(chastic)e(stability)g(of)g(e)l(quilibrium)g (states)p Fs(,)f(Random)f(Comput.)g(Dy-)745 2288 y(namics)22 b Fb(4)h Fs(\(1996\),)j(85{98.)456 2371 y([Bo)n(w79])45 b(Rufus)31 b(Bo)n(w)n(en,)k Fh(Hausdor\013)g(dimension)f(of)g (quasi-cir)l(cles)p Fs(,)g(Publ.)d(Math.)h(I.H.E.S.)f Fb(50)h Fs(\(1979\),)745 2454 y(11{26.)456 2537 y([CF98])82 b(Hans)30 b(Crauel)h(and)g(F)-6 b(ranco)31 b(Flandoli,)h Fh(Hausdor\013)h(dimension)g(of)f(invariant)g(sets)g(for)g(r)l(andom) 745 2620 y(dynamic)l(al)27 b(systems)p Fs(,)c(1998.)h(T)-6 b(o)24 b(app)r(ear)g(in:)31 b(J.)23 b(Dynamics)g(Di\013eren)n(tial)h (Equations.)456 2703 y([Cra95])65 b(Hans)26 b(Crauel,)g Fh(R)l(andom)k(pr)l(ob)l(ability)f(me)l(asur)l(es)g(on)g(Polish)g(sp)l (ac)l(es)p Fs(,)f(Habilitationssc)n(hrift,)e(Uni-)745 2786 y(v)n(ersit\177)-35 b(at)24 b(Bremen,)e(1995.)456 2869 y([GR70])72 b(M.)15 b(L.)i(Gromo)n(v)f(and)h(V.)f(A.)g(Rokhlin,)h Fh(Emb)l(e)l(ddings)k(and)f(immersions)g(in)f(Riemannian)h(ge)l(ometry) p Fs(,)745 2952 y(Russian)j(Math.)h(Surv)n(eys)g Fb(25)f Fs(\(1970\),)i(1{57.)456 3035 y([GK98])69 b(V)-6 b(olk)n(er)23 b(Matthias)h(Gundlac)n(h)h(and)g(Y)-6 b(uri)22 b(Kifer,)g Fh(R)l(andom)28 b(so\014c)e(systems)p Fs(,)d(1998,)h(Preprin)n(t.)456 3118 y([Kif96])82 b(Y)-6 b(uri)26 b(Kifer,)g Fh(F)-5 b(r)l(actal)29 b(dimensions)h(and)f(r)l(andom)i(tr)l(ansformations)p Fs(,)e(T)-6 b(rans.)26 b(Amer.)f(Math.)i(So)r(c.)745 3201 y Fb(348)22 b Fs(\(1996\),)j(2003{2038.)456 3284 y([Liu98])76 b(P)n(ei{Dong)21 b(Liu,)f Fh(R)l(andom)k(p)l(erturb)l (ations)g(of)e(Axiom)h(A)f(b)l(asic)h(sets)p Fs(,)d(J.)g(of)f(Stat.)i (Ph)n(ys.)f Fb(90)g Fs(\(1998\),)745 3367 y(467{490.)456 3451 y([MM83])49 b(Heather)21 b(McClusk)n(ey)g(and)g(An)n(ton)n(y)h (Manning,)f Fh(Hausdor\013)j(dimension)g(for)f(horsesho)l(es)p Fs(,)g(Ergo)r(dic)745 3534 y(Theory)h(Dynamical)f(Systems)g Fb(3)g Fs(\(1983\),)i(251{260.)456 3617 y([Oc)n(h95])56 b(Gun)n(ter)20 b(Oc)n(hs,)g Fh(Hyp)l(erb)l(olische)j(Limesmengen)e(b)l (ei)h(Iter)l(ation)g(algebr)l(aischer)h(F)-5 b(unktionen)p Fs(,)20 b(Ph.D.)745 3700 y(thesis,)j(Univ)n(ersit\177)-35 b(at)24 b(Bremen,)e(1995.)456 3783 y([Rue82])57 b(Da)n(vid)28 b(Ruelle,)i Fh(R)l(ep)l(el)t(lers)h(for)f(r)l(e)l(al)h(analytic)g(maps) p Fs(,)f(Ergo)r(dic)f(Theory)g(Dynamical)f(Systems)g Fb(2)745 3866 y Fs(\(1982\),)d(99{107.)456 3949 y([Rue83])57 b(Da)n(vid)22 b(Ruelle,)f Fh(Bowen)-7 b('s)25 b(formula)g(for)f(the)g (Hausdor\013)h(dimension)g(of)f(self-similar)g(sets)p Fs(,)e(Scaling)745 4032 y(and)36 b(Self-similarit)n(y)c(in)j(Ph)n (ysics,)i(Progrss)e(in)g(Ph)n(ysics,)j(v)n(ol.)d(7,)j(Birkh\177)-35 b(auser,)37 b(Boston,)i(1983,)745 4115 y(pp.)23 b(351{358.)456 4198 y([W)-6 b(al82])58 b(P)n(eter)24 b(W)-6 b(alters,)24 b Fh(A)n(n)h(intr)l(o)l(duction)i(to)f(er)l(go)l(dic)g(the)l(ory)p Fs(,)e(Springer,)f(New)g(Y)-6 b(ork)24 b(Berlin,)e(1982.)456 4281 y([Y)-6 b(ou82])58 b(Lai{Sang)19 b(Y)-6 b(oung,)19 b Fh(Dimension,)j(entr)l(opy)f(and)g(Lyapunov)g(exp)l(onents)p Fs(,)e(Ergo)r(dic)f(Theory)g(Dynam-)745 4364 y(ical)23 b(Systems)g Fb(2)g Fs(\(1982\),)j(109{129.)555 4519 y Fr(F)-8 b(a)o(chbereich)28 b(6)g(Ma)l(thema)l(tik,)g(Universit)1841 4513 y(\177)1839 4519 y(at)g(Gesamthochschule)g(Essen,)i(Postf)-6 b(a)o(ch)29 b(103)11 b(764,)456 4602 y(D-45117)23 b(Essen,)k(Federal)e (Republic)g(of)g(Germany)555 4685 y Fh(E-mail)h(addr)l(ess)5 b Fs(:)33 b Fa(matb10@aixrs1.hrz.uni-)q(ess)q(en.d)q(e)555 4827 y Fr(Institut)k(f)906 4821 y(\177)904 4827 y(ur)f(D)n(ynamische)g (Systeme,)k(Universit)2110 4821 y(\177)2108 4827 y(at)d(Bremen,)i (Postf)-6 b(a)o(ch)36 b(330)12 b(440,)39 b(D-28334)456 4910 y(Bremen,)25 b(Federal)g(Republic)g(of)g(Germany)555 4993 y Fh(E-mail)h(addr)l(ess)5 b Fs(:)33 b Fa(gunter@math.uni-bremen)q (.de)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF