Content-Type: multipart/mixed; boundary="-------------9911241208882" This is a multi-part message in MIME format. ---------------9911241208882 Content-Type: text/plain; name="99-443.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-443.keywords" decay of correlation; multi-dimensional dynamics; equilibrium states; tower extensions; thermodynamical formalism; piecewise expanding maps; less-than-Holder smoothness; transfer operator; topological pressure ---------------9911241208882 Content-Type: application/postscript; name="correlations.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="correlations.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: correlations.dvi %%Pages: 23 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips -o correlations.ps correlations %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX 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FC(X)1678 4147 y FF(')c(d\026)k FE(\001)1896 4035 y FD(Z)1941 4221 y FC(X)2015 4147 y FF( )f(d\026)2177 4028 y FD(\014)2177 4077 y(\014)2177 4126 y(\014)2177 4175 y(\014)2227 4147 y FE(\024)23 b FF(C)h FE(\001)18 b FF(e)2475 4113 y FB(\000)p FC(\013n)305 4354 y FG(p)r(our)35 b(toutes)g(les)g (fonctions)h(H\177)-41 b(older)35 b FF(';)14 b( )38 b FG(:)d FF(X)41 b FE(!)34 b FA(R)q FG(,)42 b(tout)34 b FF(n)h FE(\025)g FG(0)f(et)h(des)g(constan)n(tes)305 4455 y FF(C)29 b(<)22 b FE(1)p FG(,)27 b FF(\013)c(>)f FG(0.)427 4557 y(Une)e(dimin)n(ution)d(de)i(la)f(r)n(\023)-39 b(egularit)n(\023)g(e)19 b(fait)e(appara)-9 b(^)-32 b(\020tre)19 b(des)h(vitesses)f(sous-exp)r(onen)n(tielles.)p 85 4628 1319 3 v 220 4711 a Fz(AMS)26 b(classi\014ction)h(1991:)33 b FI(Primary)28 b(58F11)85 4800 y Fz(key)22 b(wor)l(ds:)29 b FI(absolutely)21 b(con)n(tin)n(uous)e(in)n(v)l(arian)n(t)h(measures;) i(equilibrium)f(states;)h(deca)n(y)e(of)e(correlations;)24 b(transfer)85 4890 y(op)r(erator;)i(m)n(ulti-dimensional)j(dynamics;)d (to)n(w)n(er)g(extension.)1711 5114 y Fy(1)p eop %%Page: 2 2 2 1 bop 0 26 a Fy(2)745 b Fx(J)n(\023)-41 b(er^)d(ome)29 b(Buzzi)g(&)g(V)n(\023)-41 b(eronique)28 b(Maume-Desc)n(hamps)1233 302 y Fw(Intr)m(oduction)132 532 y Fy(Let)h(\()p Fv(X)r(;)15 b Fu(Z)7 b Fv(;)15 b(T)7 b(;)15 b(g)s Fy(\))32 b(b)r(e)d(a)g Ft(weigthe)l(d)j Fv(d)p Ft(-dimensional)g(pie)l(c)l(ewise)g(invertible) h(map)p Fy(,)c(i.e.:)131 644 y Fu(\017)43 b Fv(X)32 b Fy(=)p 417 566 300 4 v 417 576 a Fs(S)492 670 y FC(Z)t FB(2Z)651 644 y Fv(Z)f Fu(\032)25 b Fr(R)896 611 y FC(d)938 644 y Fy(.)131 844 y Fu(\017)43 b(Z)37 b Fy(is)28 b(a)h(\014nite)g (collection)e(of)i(pairwise)f(disjoin)n(t,)f(b)r(ounded)h(and)h(op)r (en)g(subsets)f(of)h Fr(R)3080 811 y FC(d)3122 844 y Fy(.)131 1044 y Fu(\017)43 b Fy(eac)n(h)29 b(restriction)f Fv(T)12 b Fu(j)p Fv(Z)6 b Fy(,)31 b Fv(Z)g Fu(2)25 b(Z)7 b Fy(,)30 b(is)e(the)h(restriction)f(of)h(a)g(homeomorphism)f Fv(T)2890 1057 y FC(Z)2967 1044 y Fy(:)d Fv(U)34 b Fu(!)25 b Fv(V)219 1152 y Fy(with)k Fv(U;)15 b(V)49 b Fy(op)r(en)29 b(sets)g(in)f Fr(R)1163 1120 y FC(d)1235 1152 y Fy(suc)n(h)g(that)h Fv(U)34 b Fu(\033)1830 1130 y Fy(\026)1812 1152 y Fv(Z)6 b Fy(,)30 b Fv(V)44 b Fu(\033)p 2122 1074 199 4 v 25 w Fv(T)12 b Fy(\()p Fv(Z)6 b Fy(\))q(.)131 1352 y Fu(\017)43 b Fv(g)29 b Fy(:)c Fv(X)32 b Fu(!)24 b Fr(R)s Fy(.)0 1553 y(Suc)n(h)42 b(systems)h(ha)n(v)n(e)f(usually)f(plen)n(t)n(y)h(of) h(in)n(v)-5 b(arian)n(t)41 b(probabilit)n(y)f(measures,)47 b(some)42 b(of)h(them)0 1661 y(rather)38 b(irrelev)-5 b(an)n(t)38 b(as)f(the)i(ones)e(living)g(on)h(an)n(y)g(p)r(erio)r(dic)f (orbit.)66 b(A)38 b(classical)e(w)n(a)n(y)i(to)g(select)0 1769 y(\\in)n(teresting")d(measures)i(is)g(through)g(the)h(follo)n (wing)d(v)-5 b(ariational)36 b(principle.)62 b(One)38 b(considers)0 1878 y Ft(e)l(quilibrium)i(states)p Fy(,)g(i.e.,)g(in)n (v)-5 b(arian)n(t)36 b(probabilit)n(y)f(measures)i(whic)n(h)f(maximize) h(the)h Ft(me)l(asur)l(e-)0 1986 y(the)l(or)l(etic)31 b(pr)l(essur)l(e)p Fy(:)1252 2143 y Fv(h)p Fy(\()p Fv(\026;)15 b(T)d Fy(\))21 b Fu(\000)1637 2021 y Fs(Z)1686 2224 y FC(Y)1756 2143 y Fy(log)15 b Fv(g)k(d\026)0 2359 y Fy(\()p Fv(h)p Fy(\()p Fv(\026;)c(T)d Fy(\))30 b(is)e(the)i(en)n(trop)n(y)e(of) h Fv(\026)g Fy(|)g(see)g([DKS)q(]\).)132 2471 y(In)e(this)g Ft(pie)l(c)l(ewise,)j(multidimensional)e Fy(setting,)g(existence)f(and) g(uniqueness)f(of)i(these)f(mea-)0 2579 y(sures)g(ha)n(v)n(e)f(b)r(een) i(studied)d(in)i([Bu2])h(and)f([BPS].)39 b(F)-7 b(urthermore)28 b(in)e([Bu2)q(])h(suc)n(h)f(measures)h(ha)n(v)n(e)0 2688 y(b)r(een)h(giv)n(en)f(the)i(alternativ)n(e)d(and)i(more)h (\\geometric")e(c)n(haracterization)g(of)g(b)r(eing)h(exactly)g(the)0 2796 y(measures)22 b(absolutely)f(con)n(tin)n(uous)g(w.r.t.)j(the)f (conformal)f(measure)h(of)g(the)g(system)f(\(see)h(b)r(elo)n(w\).)132 2908 y(The)29 b(crucial)f(condition)f(in)h(these)h(pap)r(ers)g(is)f (the)h(follo)n(wing)e(inequalit)n(y)g(on)i(the)g(top)r(ological)0 3016 y(pressure)f(of)h(the)h(b)r(oundary)f(of)g(the)g(pieces)f(of)h (the)g(map)g(\(see)g(b)r(elo)n(w)g(for)g(the)g(de\014nitions\):)1237 3225 y Fv(P)12 b Fy(\()p Fv(@)5 b Fu(Z)i Fv(;)15 b(T)d Fy(\))26 b Fv(<)g(P)12 b Fy(\()p Fv(X)r(;)j(T)d Fy(\))p Fv(:)132 3436 y Fy(In)33 b(this)g(pap)r(er,)i(w)n(e)f(sho)n(w)f(in)g(a) h(similar)e(setting)h(that)g(these)h(equilibrium)d(states)j(ha)n(v)n(e) f(ex-)0 3545 y(p)r(onen)n(tial)39 b(deca)n(y)i(of)g(correlations)f(in)g (the)h(case)g(of)g(H\177)-44 b(older)40 b(smo)r(othness)g(of)h Fv(g)k Fy(and)40 b(uniform)0 3653 y(expansion.)e(F)-7 b(or)29 b(less-than-H\177)-44 b(older)27 b(smo)r(othness)i(w)n(e)g(get) g(sub)r(exp)r(onen)n(tial)e(sp)r(eeds.)0 3870 y Fq(Ac)m(kno)m (wledgemen)m(ts:)37 b Fy(The)27 b(authors)f(are)g(grateful)f(to)i Ft(pr)l(o)l(gr)l(am)g(ESF/PR)n(OD)n(YN)h Fy(whic)n(h)d(has)0 3978 y(partially)f(supp)r(orted)i(the)g Ft(International)j(Confer)l (enc)l(e)g(on)g(Dynamic)l(al)f(Systems,)h(A)n(bb)l(ey)f(of)g(\\L)l(a)0 4086 y(Bussi)n(\022)-43 b(er)l(e")29 b Fy(where)g(most)g(of)g(this)g(w) n(ork)g(w)n(as)g(carried)g(through.)0 4367 y Fp(Statemen)m(t)41 b(of)f(Results)0 4541 y Fy(W)-7 b(e)33 b(use)f(indi\013eren)n(tly)f Fu(Z)876 4508 y FC(n)952 4541 y Fy(to)i(denote)g(the)g(set)g(of)f(w)n (ords)h Fv(A)2071 4554 y Fo(0)2121 4541 y Fv(:)15 b(:)g(:)h(A)2305 4554 y FC(n)p FB(\000)p Fo(1)2462 4541 y Fy(of)32 b(length)g Fv(n)i Fy(on)f Fu(Z)40 b Fy(and)0 4649 y(the)29 b(set)g(of)g Fv(n)p Fy(-cylinders:)906 4890 y([)p Fv(A)996 4903 y Fo(0)1047 4890 y Fv(:)15 b(:)g(:)h(A)1231 4903 y FC(n)p FB(\000)p Fo(1)1354 4890 y Fy(])1404 4840 y FG(def)1423 4890 y Fy(=)44 b Fv(A)1601 4903 y Fo(0)1656 4890 y Fu(\\)20 b Fv(:)15 b(:)g(:)21 b Fu(\\)f Fv(T)2002 4853 y FB(\000)p FC(n)p Fo(+1)2175 4890 y Fv(A)2241 4903 y FC(n)p FB(\000)p Fo(1)2365 4890 y Fv(:)p eop %%Page: 3 3 3 2 bop 832 26 a Fx(Piecewise)28 b(in)n(v)n(ertible)f(maps)i(in)f (higher)g(dimensions)745 b Fy(3)85 302 y(The)30 b Ft(top)l(olo)l(gic)l (al)g(pr)l(essur)l(e)f Fy(of)g(a)g(subset)f Fv(S)35 b Fy(of)29 b Fv(X)36 b Fy(is:)1026 536 y Fv(P)12 b Fy(\()p Fv(S;)j(T)d Fy(\))26 b(=)f(lim)15 b(sup)1490 607 y FC(n)p FB(!1)1735 475 y Fy(1)p 1731 515 53 4 v 1731 598 a Fv(n)1807 536 y Fy(log)1980 451 y Fs(X)1965 632 y Fn(A)p Fm(2Z)2087 619 y Fn(n)1959 672 y Fl(\026)1945 683 y Fn(A)p Fm(\\)p Fn(S)r Fm(6)p Fl(=)p Fk(?)2169 536 y Fv(g)2214 499 y Fo(\()p FC(n)p Fo(\))2306 536 y Fy(\()p Fv(A)p Fy(\))85 876 y(where)30 b Fv(g)384 844 y Fo(\()p FC(n)p Fo(\))476 876 y Fy(\()p Fv(A)p Fy(\))25 b(=)g(sup)860 897 y FC(x)p FB(2)p FC(A)1004 876 y Fv(g)s Fy(\()p Fv(x)p Fy(\))p Fv(g)s Fy(\()p Fv(T)12 b(x)p Fy(\))j Fv(:)g(:)g(:)j(g)s Fy(\()p Fv(T)1670 844 y FC(n)p FB(\000)p Fo(1)1794 876 y Fv(x)p Fy(\).)217 985 y(As)28 b(in)h(pap)r(ers)g([Bu2],)h([BPS])g (our)f(main)f(condition)g(is:)1329 1139 y Fv(P)12 b Fy(\()p Fv(@)5 b Fu(Z)i Fv(;)15 b(T)d Fy(\))26 b Fv(<)g(P)12 b Fy(\()p Fv(X)r(;)j(T)d Fy(\))1113 b(\()p Fu(\003)p Fy(\))85 1292 y(where)30 b Fv(@)5 b Fu(Z)32 b Fy(=)580 1225 y Fs(S)655 1318 y FC(Z)t FB(2Z)814 1292 y Fv(@)5 b(Z)h Fy(.)217 1401 y(The)45 b(mo)r(dulus)e(of)i(con)n(tin)n(uit)n(y)d (of)j Fv(g)j Fy(will)43 b(b)r(e)i(con)n(trolled)e(b)n(y)i(the)g(follo)n (wing,)h(dynamics-)85 1509 y(dep)r(enden)n(t)28 b(sequence:)1209 1663 y Fv(!)1264 1676 y FC(n)1306 1663 y Fy(\()p Fv(g)s Fy(\))d(=)49 b(sup)1538 1736 y FC(Z)t FB(2Z)1678 1719 y Fn(n)1753 1663 y Fy(sup)1731 1736 y FC(x;y)r FB(2)p FC(Z)1922 1663 y Fy(log)2060 1602 y Fv(g)s Fy(\()p Fv(x)p Fy(\))p 2060 1642 164 4 v 2062 1724 a Fv(g)s Fy(\()p Fv(y)s Fy(\))2232 1663 y Fv(:)217 1876 y Fy(W)-7 b(e)38 b(shall)f(also)h(assume)g(\\the)h(discon)n(tin)n(uities)34 b(to)k(b)r(e)h(in)f(general)g(p)r(osition",)i(i.e.,)i(for)d(all)85 1984 y Fv(d)20 b Fy(+)g(1-uples)28 b(of)h(in)n(tegers)f(0)c Fu(\024)h Fv(k)1171 1997 y Fo(1)1231 1984 y Fv(<)g(k)1370 1997 y Fo(2)1430 1984 y Fv(<)h(:)15 b(:)g(:)26 b(<)f(k)1791 1997 y FC(d)p Fo(+1)1908 1984 y Fy(,)30 b(for)f(all)f Fv(Z)j Fu(2)25 b(Z)2466 1952 y FC(k)2499 1963 y Fn(d)p Fl(+1)2601 1952 y Fo(+1)2686 1984 y Fy(:)1399 2107 y FC(d)p Fo(+1)1406 2134 y Fs(\\)1404 2323 y FC(i)p Fo(=1)1528 2219 y Fv(T)1591 2179 y FB(\000)p FC(k)1673 2188 y Fn(i)1579 2244 y FC(Z)1701 2219 y Fv(@)5 b(Z)1812 2232 y FC(k)1845 2241 y Fn(i)1898 2219 y Fy(=)25 b Fr(?)1182 b Fy(\()p Fu(y)p Fy(\))85 2459 y(where)37 b Fv(T)409 2426 y FC(k)397 2482 y(Z)487 2459 y Fy(=)h Fv(T)644 2472 y FC(Z)687 2483 y Fn(k)q Fm(\000)p Fl(1)821 2459 y Fu(\016)25 b Fv(T)942 2472 y FC(Z)985 2483 y Fn(k)q Fm(\000)p Fl(2)1120 2459 y Fu(\016)g Fv(:)15 b(:)g(:)26 b Fu(\016)f Fv(T)1439 2472 y FC(Z)1482 2480 y Fl(0)1517 2459 y Fy(.)63 b(Remark)37 b(that)g(this)f(is)g(basically)e(a)j(transv)n(ersalit)n(y)85 2567 y(assumption.)j(Th)n(us,)30 b(one)g(exp)r(ects)g(this)f(condition) f(to)i(b)r(e)h(almost)e(alw)n(a)n(ys)f(satis\014ed)h(\(see)h([Bu1])85 2676 y(and)f([Co]\).)217 2784 y(Finally)-7 b(,)25 b(w)n(e)g(shall)f (need)h(a)h Ft(c)l(onformal)h(me)l(asur)l(e)e Fv(\027)6 b Fy(,)27 b(i.e.,)g(a)e(not)h(necessarily)e(in)n(v)-5 b(arian)n(t,)24 b(prob-)85 2893 y(abilit)n(y)39 b(measure)i(on)f Fv(X)7 b Fy(,)45 b(suc)n(h)40 b(that)h(the)g(Jacobian)e(of)i Fv(T)54 b Fy(w.r.t.)43 b(this)d(measure)h(is)f(equal)g(to)85 3001 y Fv(e)126 2968 y FB(\000)p FC(P)9 b Fo(\()p FC(X)q(;T)g Fo(\))p FC(g)427 3001 y Fy(.)47 b(By)32 b([Bu2)q(],)h(the)f Fv(\027)6 b Fy(-absolutely)30 b(con)n(tin)n(uous)f(in)n(v)-5 b(arian)n(t)30 b(probabilit)n(y)g(measures)h(coin-)85 3109 y(cide)e(with)f(the)i(equilibrium)c(states.)217 3218 y(Replacing)h Fv(g)32 b Fy(b)n(y)d Fv(g)23 b Fu(\000)d Fv(P)12 b Fy(\()p Fv(X)r(;)j(T)d Fy(\),)31 b(w)n(e)e(ma)n(y)-7 b(,)29 b(and)g(shall,)f(assume)g(that)h Fv(P)12 b Fy(\()p Fv(X)r(;)j(T)d Fy(\))26 b(=)g(0.)163 3417 y Fj(Main)47 b(Theorem.)78 b Fx(Let)42 b Fy(\()p Fv(X)r(;)15 b Fu(Z)7 b Fv(;)15 b(T)7 b(;)15 b(g)s Fy(\))45 b Fx(b)r(e)d(a)g(w)n(eigh)n(ted)f (piecewise)g(in)n(v)n(ertible)f(dynamical)85 3526 y(system.)f(Assume:) 127 3710 y(H1.)44 b Fv(T)37 b Fx(is)24 b(non-con)n(tracting:)35 b Fv(d)p Fy(\()p Fv(T)12 b(x;)j(T)d(y)s Fy(\))27 b Fu(\025)e Fv(d)p Fy(\()p Fv(x;)15 b(y)s Fy(\))25 b Fx(for)f(all)g Fv(x;)15 b(y)28 b Fx(in)c(the)h(same)f(elemen)n(t)g(of)h Fu(Z)7 b Fx(;)27 b Fv(T)305 3818 y Fu(Z)7 b Fx(-separates:)43 b Fy(lim)948 3831 y FC(n)p FB(!1)1132 3818 y Fx(diam)o Fy(\()p Fu(Z)1428 3786 y FC(n)1471 3818 y Fy(\))29 b Fu(!)f Fy(0)k Fx(and)f Fv(T)44 b Fx(satis\014es)29 b(the)j(transv)n (ersalit)n(y)d(condition)305 3926 y(\()p Fu(y)p Fx(\).)127 4111 y(H2.)44 b Fv(g)29 b Fx(is)d(suc)n(h)f(that)h(the)h(pressure)e (condition)g(\()p Fu(\003)p Fx(\))i(holds)e(and)g(it)h(is)g(smo)r(oth)g (enough)g(w.r.t.)i(the)305 4219 y(expansion)f(of)i Fv(T)42 b Fx(in)29 b(the)g(sense)f(that:)1480 4336 y Fs(X)1485 4525 y FC(n)p FB(\025)p Fo(1)1625 4421 y Fv(!)1680 4434 y FC(n)1722 4421 y Fy(\()p Fv(g)s Fy(\))d Fv(<)g Fu(1)p Fv(:)1176 b Fx(\()p Fu(z)p Fx(\))127 4782 y(H3.)44 b(there)d(is)e(a)i (conformal)f(measure)h Fv(\027)47 b Fx(suc)n(h)39 b(that)i Fv(T)2062 4749 y FC(n)2105 4782 y Fv(X)52 b Fu(\032)44 b Fy(supp)o(\()p Fv(\027)6 b Fy(\))41 b(\(mo)r(d)k(0\))c Fx(for)g(large)305 4890 y(enough)28 b Fv(n)p Fx(.)p eop %%Page: 4 4 4 3 bop 0 26 a Fy(4)745 b Fx(J)n(\023)-41 b(er^)d(ome)29 b(Buzzi)g(&)g(V)n(\023)-41 b(eronique)28 b(Maume-Desc)n(hamps)0 302 y(Then,)44 b Fv(T)54 b Fx(admits)40 b(an)h(in)n(v)-5 b(arian)n(t)39 b(measure,)45 b(absolutely)39 b(con)n(tin)n(uous)f(with) j(resp)r(ect)g(to)h Fv(\027)47 b Fx(\()p Fv(\027)6 b Fx(-)0 411 y(a.c.i.m.)41 b(for)30 b(short\).)40 b(F)-7 b(or)29 b(an)n(y)h(ergo)r(dic)f Fv(\027)6 b Fx(-a.c.i.m.,)31 b Fv(\026)p Fx(,)f(up)f(to)h(replacing)e Fv(T)42 b Fx(b)n(y)29 b(some)h(iterate,)f Fv(\026)0 519 y Fx(is)f Fy(\()p Fv(u)172 532 y FC(n)215 519 y Fy(\))249 532 y FC(n)p FB(2)p Fi(N)377 519 y Fx(-mixing,)g(i.e.,)i(for)f(all)f Fv(n)e Fu(\025)e Fy(1)p Fx(,)245 639 y Fs(\014)245 693 y(\014)245 747 y(\014)245 800 y(\014)275 647 y(Z)325 850 y FC(X)400 769 y Fv(')c Fu(\016)g Fv(T)606 732 y FC(n)669 769 y Fu(\001)f 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Fv(@)5 b Fu(Z)i Fv(;)15 b(T)d Fy(\).)p eop %%Page: 5 5 5 4 bop 832 26 a Fx(Piecewise)28 b(in)n(v)n(ertible)f(maps)i(in)f (higher)g(dimensions)745 b Fy(5)217 302 y(W)-7 b(e)38 b(then)f(state)h(a)g(v)n(ersion)f(of)h(our)f(Main)h(Theorem)g(for)g (abstract)g(to)n(w)n(ers)f(and)h(c)n(hec)n(k)f(its)85 411 y(h)n(yp)r(othesis)28 b(under)g(the)i(assumptions)d(of)i(the)g (Main)g(Theorem.)217 520 y(W)-7 b(e)27 b(\014nally)e(recall)h(standard) g(facts)h(ab)r(out)f(Birkho\013)h(metrics)g(and)f(put)h(them)g(to)g (use)g(for)f(the)85 629 y(pro)r(of)j(of)g(the)g(abstract)g(Theorem.)85 896 y Fp(Commen)m(ts)216 1064 y Fu(\017)44 b Fy(Our)38 b(Main)f(Theorem)h(generalizes)e(w)n(ell-kno)n(wn)h(results)g(from)g (the)h Ft(glob)l(al)t(ly)j(exp)l(anding)305 1173 y Fy(case)28 b(\(see,)i(e.g.,)g([Bo)q(]\))f(to)h(our)f Ft(pie)l(c)l(ewise)g Fy(setting.)216 1361 y Fu(\017)44 b Fy(Ev)n(en)29 b(in)h(the)g (one-dimensional)e(case,)j(our)f(assumptions)e(are)i(essen)n(tially)e (standard)i(\(see)305 1470 y([LSV)o(])i(for)g(recen)n(t)g (references\).)46 b(The)32 b(exception)f(is)g(the)h(transv)n(ersalit)n (y)d(assumption)h(\()p Fu(y)p Fy(\))305 1578 y(\(whic)n(h)k(in)g(the)i (one-dimensional)c(case)j(is)f(the)h(exclusion)f(of)h(p)r(erio)r(dic)f (critical)g(p)r(oin)n(ts\).)305 1686 y(W)-7 b(e)29 b(think)f(that)i(it) e(is)h(tec)n(hnical)e(but)i(w)n(e)g(could)f(not)h(remo)n(v)n(e)g(it.) 216 1875 y Fu(\017)44 b Fy(W)-7 b(e)24 b(b)r(eliev)n(e)f(our)h(result)f (to)h(b)r(e)h(the)f(\014rst)g(considering)e(less-than-H\177)-44 b(older)22 b(smo)r(othness)h Ft(out-)305 1984 y(side)i(of)h(the)g (Markov)g(setting)f Fy(\(see)e([KMS)q(])g(where)h(non-H\177)-44 b(older)22 b(p)r(oten)n(tials)f(are)j(considered)305 2092 y(in)30 b(the)i(Mark)n(o)n(v)g(setting\).)46 b(Also)30 b(w)n(e)i(pro)n(vide)e(an)i(answ)n(er)f(to)h(a)f(question)g(of)g(L.S.)h (Y)-7 b(oung)305 2200 y([Y2])34 b(ab)r(out)g Ft(c)l(ontr)l(ol)t(ling)k (the)e(upp)l(er)f(\015o)l(ors)e Fy(of)h(the)g(to)n(w)n(er:)49 b(it)34 b(is)f(done)g(b)n(y)h(a)g(thermo)r(dy-)305 2309 y(namical)27 b(estimate)i(\(Prop)r(osition)e(1.1\))j(in)f(the)g(spirit) e(of)i([Bu2)q(].)216 2498 y Fu(\017)44 b Fy(As)38 b(w)n(e)g(go)h(to)g (a)f(Mark)n(o)n(v)h(extension,)i(w)n(e)e(a)n(v)n(oid)e(ha)n(ving)g(to)i (deal)f(with)g(spaces)g(of)g(dis-)305 2606 y(con)n(tin)n(uous)d (functions,)k(i.e.,)i(spaces)c(of)h(functions)e(with)i(b)r(ounded)f Ft(multi-dimensional)305 2714 y(variation)p Fy(.)i(This)28 b(is)h(a)g(signi\014can)n(t)d(tec)n(hnical)i(gain.)216 2903 y Fu(\017)44 b Fy(Our)35 b(abstract)f(theorem)h(is)f(m)n(uc)n(h)g (in)f(the)i(spirit)f(of)g(L.-S.)i(Y)-7 b(oung)33 b(\(see)i([Y1]\).)57 b(But)34 b(our)305 3011 y(pro)r(of)j(is)f(quite)h(di\013eren)n(t,)h (based)f(on)g(Birkho\013)g(cone)g(tec)n(hniques)e(and)i(previous)f(w)n (orks)305 3120 y([KMS],)30 b([M)q(].)678 3426 y Fh(1)193 b Fw(Constr)m(uction)51 b(of)e(the)h(to)m(wer)85 3640 y Fy(W)-7 b(e)20 b(de\014ne)f(a)h(Mark)n(o)n(v)g(to)n(w)n(er)f (extension)g(\()1513 3617 y(^)1487 3640 y Fv(X)7 b(;)1624 3617 y Fy(^)1606 3640 y Fv(T)13 b Fy(\))20 b(in)f(the)h(v)n(ein)f(of)h (F.)g(Hofbauer)f([Ho])i(and)e(G.)h(Keller)85 3748 y([Ke)q(])i(and)g (then)g(pro)n(v)n(e)g(that)h(the)f(pressure)f(of)i(the)f(b)r(oundary)g (pro)n(vides)f(an)h(exp)r(onen)n(tial)f(estimate)85 3857 y(on)29 b(the)g(tail)f(statistics)g(of)h(the)g(return)g(times)g(to)g (the)g(basis.)217 3966 y(Let)39 b(\()p Fv(X)r(;)15 b Fu(Z)7 b Fv(;)15 b(T)7 b(;)15 b(g)s(;)g(\027)6 b Fy(\))41 b(b)r(e)f(as)e(in)g(the)h(Main)g(Theorem.)68 b(Let)2331 3944 y(^)2305 3966 y Fv(X)2378 3979 y Fo(0)2455 3966 y Fy(=)42 b Fu(f)p Fy(\()p Fv(x;)15 b(Z)6 b Fy(\))42 b(:)g Fv(x)f Fu(2)g Fv(Z)46 b Fy(and)85 4075 y Fv(Z)31 b Fu(2)25 b(Z)7 b(g)p Fy(.)40 b(F)-7 b(or)29 b Fv(x)c Fu(2)f Fv(Y)40 b Fu(\\)19 b Fv(T)991 4042 y FB(\000)p Fo(1)1077 4075 y Fv(Y)49 b Fy(and)28 b Fv(A)d Fu(\032)g Fv(X)7 b Fy(,)29 b(set:)1171 4253 y(^)1154 4276 y Fv(T)12 b Fy(\()p Fv(x;)j(A)p Fy(\))26 b(=)f(\()p Fv(T)12 b(x;)j(T)d Fy(\()p Fv(A)p Fy(\))21 b Fu(\\)f(Z)7 b Fy([)p Fv(T)12 b(x)p Fy(]\))85 4478 y(where)30 b Fu(Z)7 b Fy([)p Fv(x)p Fy(])30 b(is)e(that)h(elemen)n(t)g(of)g Fu(Z)36 b Fy(whic)n(h)28 b(con)n(tains)f Fv(x)p Fy(.)85 4586 y(Set)262 4563 y(^)236 4586 y Fv(X)32 b Fy(=)434 4519 y Fs(S)509 4612 y FC(n)p FB(\025)p Fo(0)665 4563 y Fy(^)647 4586 y Fv(T)710 4553 y FC(n)753 4586 y Fy(\()813 4563 y(^)787 4586 y Fv(X)860 4599 y Fo(0)896 4586 y Fy(\).)39 b(Remark)30 b(that)619 4794 y(^)594 4817 y Fv(X)i Fy(=)816 4732 y Fs([)792 4922 y FC(D)r FB(2D)954 4817 y Fv(D)22 b Fu(\002)e(f)p Fv(D)r Fu(g)29 b Fy(with)g Fu(D)f Fy(=)d Fu(f)p Fv(T)1831 4780 y FC(n)1874 4817 y Fv(Z)31 b Fy(:)26 b Fv(n)f Fu(\025)g Fy(0)k(and)f Fv(Z)k Fu(2)25 b(Z)2679 4780 y FC(n)p Fo(+1)2802 4817 y Fu(g)p Fv(:)p eop %%Page: 6 6 6 5 bop 0 26 a Fy(6)745 b Fx(J)n(\023)-41 b(er^)d(ome)29 b(Buzzi)g(&)g(V)n(\023)-41 b(eronique)28 b(Maume-Desc)n(hamps)0 302 y Fy(Th)n(us,)272 280 y(^)247 302 y Fv(X)36 b Fy(is)28 b(a)h(coun)n(table,)f(disjoin)n(t)f(union)h(of)h(\(sets)g(naturally)f (isomorphic)f(to\))i(subsets)f(of)h Fv(X)7 b Fy(.)0 413 y Fj(Pr)n(oposition)34 b(1.1)42 b Fx(F)-7 b(or)30 b(eac)n(h)e Fv(\016)g(>)d Fy(0)p Fx(,)30 b(if)e Fv(N)1511 426 y FB(\003)1572 413 y Fv(<)d Fu(1)k Fx(is)g(large)f(enough,)h(then)g(for)1418 594 y Fy(^)1393 617 y Fv(X)1466 630 y FB(\003)1526 617 y Fy(=)1637 594 y(^)1619 617 y Fv(T)1682 580 y FC(N)1732 587 y Fm(\003)1795 594 y Fy(^)1770 617 y Fv(X)1843 630 y Fo(0)1878 617 y Fv(;)0 821 y Fx(the)g(return)g(time)g(to)752 798 y Fy(^)727 821 y Fv(X)800 834 y FB(\003)835 821 y Fx(,)908 798 y Fy(^)889 821 y Fv(R)p Fy(\()5 b(^)-49 b Fv(x)p Fy(\))25 b(=)h(inf)5 b Fu(f)p Fv(n)25 b Fu(\025)g Fy(1)f(:)1651 798 y(^)1634 821 y Fv(T)1697 788 y FC(n)1740 821 y Fy(\()5 b(^)-49 b Fv(x)p Fy(\))25 b Fu(2)1993 798 y Fy(^)1967 821 y Fv(X)2040 834 y FB(\003)2076 821 y Fu(g)p Fx(,)k(satis\014es,)f(for)h(eac)n(h)g Fv(D)e Fu(2)e(D)r Fx(,)456 1049 y Fv(\027)520 950 y Fs(\020)573 1049 y Fu(f)p Fv(x)g Fu(2)g Fv(D)i Fy(:)946 1026 y(^)927 1049 y Fv(R)q Fy(\()p Fv(x;)15 b(D)r Fy(\))25 b Fv(>)h(n)p Fu(g)1443 950 y Fs(\021)1521 1049 y Fu(\024)f Fx(const)19 b Fu(\001)h Fy(exp[)p Fu(\000)p Fv(n)p Fy(\()p Fu(\000)p Fv(P)12 b Fy(\()p Fv(@)5 b Fu(Z)i Fv(;)15 b(T)d Fy(\))22 b Fu(\000)e Fv(\016)s Fy(\)])0 1273 y Fx(for)29 b(some)g(p)r(ositiv)n (e)f(n)n(um)n(b)r(er)g(const)d Fv(<)g Fu(1)p Fx(.)0 1449 y Ft(Pr)l(o)l(of)18 b Fy(:)132 1559 y(Let)32 b Fv(\016)i(>)d Fy(0.)49 b(Observ)n(e)33 b(that)f Fv(D)j Fy(b)r(eing)d(of)h(the)f(form) h Fv(T)1995 1527 y FC(k)2034 1559 y Fy(\()p Fv(A)2134 1572 y Fo(1)2191 1559 y Fu(\\)22 b Fv(:)15 b(:)g(:)24 b Fu(\\)e Fv(T)2544 1527 y FB(\000)p FC(k)2632 1559 y Fv(A)2698 1572 y FC(k)2736 1559 y Fy(\),)34 b Fv(P)12 b Fy(\()p Fv(@)5 b(D)r(;)15 b(T)d Fy(\))33 b Fu(\024)0 1667 y Fv(P)12 b Fy(\()p Fv(@)5 b Fu(Z)i Fv(;)15 b(T)d Fy(\).)41 b(In)28 b(particular,)714 1802 y Fs(X)699 1984 y Fn(Z)s Fm(2Z)821 1971 y Fn(n)669 2023 y Fl(\026)658 2034 y Fn(Z)s Fm(\\)p Fn(@)s(D)q Fm(6)p Fl(=)p Fk(?)924 1887 y Fv(g)969 1850 y Fo(\()p FC(n)p Fo(\))1060 1887 y Fy(\()p Fv(Z)6 b Fy(\))26 b Fu(\024)f Fy(const)19 b Fu(\001)h Fv(e)1621 1850 y Fo(\()p FC(P)9 b Fo(\()p FC(@)t FB(Z)c FC(;T)k Fo(\)+)p FC(\016)r(=)p Fo(10\))p FC(n)2312 1887 y Fy(\()p Fu(8)p Fv(n)25 b Fu(\025)f Fy(0\))0 2217 y(where)30 b(const)e(dep)r(ends)h(only)f(on)h Fv(D)r Fy(.)132 2328 y(T)-7 b(ak)n(e)29 b(an)g(in)n(teger)f Fv(N)826 2341 y Fo(0)891 2328 y Fy(so)h(large)f(that)i(the)f(follo)n (wing)e(holds:)680 2457 y Fs(X)665 2639 y Fn(A)p Fm(2Z)787 2626 y Fn(n)639 2679 y Fl(\026)625 2690 y Fn(A)q Fm(\\)p Fn(@)s Fm(Z)t(6)p Fl(=)p Fk(?)889 2542 y Fv(g)934 2505 y Fo(\()p FC(n)p Fo(\))1026 2542 y Fy(\()p Fv(A)p Fy(\))e Fu(\024)f Fv(e)1320 2505 y Fo(\()p FC(\016)r(=)p Fo(10)p FC(d)p Fo(\))p FC(N)1579 2513 y Fl(0)1613 2542 y Fv(e)1654 2505 y Fo(\()p FC(P)9 b Fo(\()p FC(@)t FB(Z)c FC(;T)k Fo(\)+)p FC(\016)r(=)p Fo(10\))p FC(n)2345 2542 y Fy(\()p Fu(8)p Fv(n)24 b Fu(\025)h Fy(0\))637 2946 y Fv(d)698 2820 y Fs(\022)800 2886 y Fv(n)p 773 2926 106 4 v 773 3008 a(N)843 3021 y Fo(0)908 2946 y Fy(+)20 b(2)1040 2820 y Fs(\023)15 b(\022)1401 2886 y Fv(n)1196 3008 y(d)p Fy(\()p Fv(n=)-5 b(N)1437 3021 y Fo(0)1493 3008 y Fy(+)20 b(1\))1668 2820 y Fs(\023)1759 2946 y Fu(\024)k Fv(e)1894 2909 y FC(\016)r(n=)p Fo(10)2062 2946 y Fv(:)176 b Fy(\()p Fu(8)p Fv(n)24 b Fu(\025)h Fv(N)2588 2959 y Fo(0)2624 2946 y Fy(\))132 3159 y(Observ)n(e)36 b(that)h(the)g(\\general)f(p)r (osition")f(assumption)g(implies)g(that)i(there)g(exists)f Fv(")3072 3172 y FB(\003)3145 3159 y Fv(>)i Fy(0)0 3267 y(suc)n(h)26 b(that)i(there)f(is)g(no)g(\\)p Fv(")899 3280 y FB(\003)934 3267 y Fy(-near,)i Fv(d)15 b Fy(+)i(1-order)28 b(in)n(tersection)d(up)i(to)g(time)h Fv(N)2597 3280 y Fo(0)2632 3267 y Fy(")g(in)e(the)i(follo)n(wing)0 3375 y(sense:)38 b(for)30 b(all)e Fv(x)c Fu(2)675 3308 y Fs(T)750 3332 y FC(N)800 3340 y Fl(0)831 3332 y FB(\000)p Fo(2)750 3401 y FC(k)r Fo(=0)931 3375 y Fv(T)994 3343 y FB(\000)p FC(k)1083 3375 y Fy(\()p Fv(X)j Fu(n)19 b Fv(@)5 b Fu(Z)i Fy(\),)905 3579 y(#)p Fu(f)p Fy(0)25 b Fu(\024)f Fv(k)k(<)d(N)1423 3592 y Fo(0)1484 3579 y Fy(:)g Fv(d)p Fy(\()p Fv(T)1676 3542 y FC(k)1715 3579 y Fv(x;)15 b(@)5 b Fu(Z)i Fy(\))25 b Fv(<)h(")2121 3592 y FB(\003)2156 3579 y Fu(g)f(\024)g Fv(d:)724 b Fy(\(1.1\))0 3783 y(Fix)38 b Fv(N)236 3796 y FB(\003)311 3783 y Fv(<)i Fu(1)d Fy(so)h(large)f(that)h(the)g (diameter)f(of)h(an)n(y)f Fv(N)1972 3796 y FB(\003)2008 3783 y Fy(-cylinder)g(is)g(smaller)f(than)i Fv(")3043 3796 y FB(\003)3078 3783 y Fy(.)65 b(F)-7 b(or)0 3892 y Fv(n)25 b Fu(\025)g Fy(1,)30 b(set:)603 4006 y Fv(\032)648 4019 y FC(n)691 4006 y Fy(\()p Fv(x)p Fy(\))25 b(=)g(sup)o Fu(f)p Fv(\032)g Fu(2)g Fy([0)p Fv(;)15 b(")1407 4019 y FB(\003)1443 4006 y Fy(])25 b(:)h Fv(B)5 b Fy(\()p Fv(T)1711 3969 y FC(n)1753 4006 y Fv(x;)15 b(\032)p Fy(\))26 b Fu(\032)e Fv(T)2104 3969 y FC(n)2147 4006 y Fy(\()p Fv(D)e Fu(\\)e(Z)2428 3969 y FC(n)2471 4006 y Fy(\()p Fv(x)p Fy(\)\))p Fu(g)p Fv(:)0 4170 y Fy(Recall)28 b(that)h(a)g(Bo)n(w) n(en)g(ball)e(is:)525 4374 y Fv(B)592 4387 y FC(n)634 4374 y Fy(\()p Fv(x;)15 b(")798 4387 y FB(\003)834 4374 y Fy(\))25 b(:=)h Fu(f)p Fv(y)h Fu(2)e Fv(X)32 b Fy(:)25 b Fu(8)p Fv(k)i Fy(=)e(0)p Fv(;)15 b(:)g(:)g(:)h(;)f(n)21 b Fu(\000)e Fy(1)25 b Fv(d)p Fy(\()p Fv(T)2196 4337 y FC(k)2235 4374 y Fv(x;)15 b(T)2387 4337 y FC(k)2426 4374 y Fv(y)s Fy(\))25 b Fv(<)g(")2665 4387 y FB(\003)2701 4374 y Fu(g)p Fv(:)0 4578 y Fy(Remark)i(that)g Fv(\032)561 4591 y FC(n)603 4578 y Fy(\()p Fv(x)p Fy(\))g(is)f(also)f(the)i(in)n (terior)e(diameter)h(at)h Fv(T)1994 4545 y FC(n)2037 4578 y Fv(x)f Fy(of)h Fv(T)2274 4545 y FC(n)2317 4578 y Fy(\()p Fv(D)17 b Fu(\\)e(Z)2588 4545 y FC(n)2630 4578 y Fy(\()p Fv(x)p Fy(\))g Fu(\\)g Fv(B)2905 4591 y FC(n)p Fo(+1)3027 4578 y Fy(\()p Fv(x;)g(")3191 4591 y FB(\003)3227 4578 y Fy(\)\))0 4686 y(as)29 b Fv(T)42 b Fy(is)28 b(non-con)n (tracting.)38 b(Also,)1257 4867 y(^)1238 4890 y Fv(R)p Fy(\()p Fv(x;)15 b(D)r Fy(\))26 b Fu(\024)f Fv(R)q Fy(\()p Fv(x)p Fy(\))19 b(+)i Fv(N)2022 4903 y FB(\003)p eop %%Page: 7 7 7 6 bop 832 26 a Fx(Piecewise)28 b(in)n(v)n(ertible)f(maps)i(in)f (higher)g(dimensions)745 b Fy(7)85 329 y(with)29 b Fv(R)q Fy(\()p Fv(x)p Fy(\))496 278 y FG(def)515 329 y Fy(=)44 b(min)p Fu(f)p Fv(n)25 b Fu(\025)g Fy(1)f(:)i Fv(\032)1154 342 y FC(n)1196 329 y Fy(\()p Fv(x)p Fy(\))f(=)g Fv(")1473 342 y FB(\003)1509 329 y Fu(g)p Fy(.)39 b(Indeed,)322 469 y(^)305 491 y Fv(T)368 454 y FC(R)p Fo(\()p FC(x)p Fo(\)+)p FC(N)601 461 y Fm(\003)638 491 y Fy(\()p Fv(x;)15 b(D)r Fy(\))75 b(=)1039 419 y Fs(\000)1080 491 y Fv(T)1143 459 y FC(R)p Fo(\()p FC(x)p Fo(\)+)p FC(N)1376 466 y Fm(\003)1413 491 y Fv(x;)15 b(T)1565 459 y FC(N)1615 466 y Fm(\003)1652 491 y Fy(\()p Fu(Z)1758 459 y FC(N)1808 466 y Fm(\003)1845 491 y Fy([)p Fv(T)1932 459 y FC(R)p Fo(\()p FC(x)p Fo(\))2070 491 y Fv(x)p Fy(])20 b Fu(\\)g Fv(T)2307 459 y FC(R)p Fo(\()p FC(x)p Fo(\))2444 491 y Fy(\()p Fv(D)j Fu(\\)d(Z)2726 459 y FC(R)p Fo(\()p FC(x)p Fo(\))2863 491 y Fy([)p Fv(x)p Fy(]\)\))3029 419 y Fs(\001)1420 630 y Fy(=)25 b(\()p Fv(T)1610 598 y FC(R)p Fo(\()p FC(x)p Fo(\)+)p FC(N)1843 605 y Fm(\003)1880 630 y Fv(x;)15 b(T)2032 598 y FC(N)2082 605 y Fm(\003)2119 630 y Fu(Z)2191 598 y FC(N)2241 605 y Fm(\003)2278 630 y Fy([)p Fv(T)2365 598 y FC(R)p Fo(\()p FC(x)p Fo(\))2503 630 y Fv(x)p Fy(]\))1220 761 y(=)1330 738 y(^)1313 761 y Fv(T)1376 728 y FC(N)1426 735 y Fm(\003)1463 761 y Fy(\()p Fv(T)1560 728 y FC(R)p Fo(\()p FC(x)p Fo(\))1698 761 y Fv(x;)g Fu(Z)7 b Fy([)p Fv(T)1946 728 y FC(R)p Fo(\()p FC(x)p Fo(\))2084 761 y Fv(x)p Fy(]\))26 b Fu(2)e Fv(T)2365 728 y FC(N)2415 735 y Fm(\003)2478 738 y Fy(^)2452 761 y Fv(X)2525 774 y Fo(0)2586 761 y Fy(=)2704 738 y(^)2679 761 y Fv(X)2752 774 y FB(\003)2787 761 y Fv(:)85 924 y Fy(Th)n(us)k(it)h(is)g(enough)f(to)h(pro)n(v)n(e)g(the)g(prop)r (osition)e(for)j Fv(R)q Fy(\()p Fv(x)p Fy(\))24 b Fv(>)h(n)30 b Fy(instead)e(of)2684 901 y(^)2665 924 y Fv(R)p Fy(\()p Fv(x;)15 b(D)r Fy(\))26 b Fv(>)f(n)p Fy(.)217 1032 y(The)i Fv(n)p Fx(-sk)n(eleton)f Fy(of)g(a)h(p)r(oin)n(t)e Fv(x)g Fu(2)g Fv(D)k Fy(with)d Fv(R)q Fy(\()p Fv(x)p Fy(\))f Fv(>)g(n)i Fy(will)e(b)r(e)i(the)f(\014nite)g(decreasing)f(in)n(teger) 85 1140 y(sequence)32 b(\()p Fv(s)527 1153 y FC(i)553 1140 y Fy(\))587 1153 y FC(i)p Fo(=0)p FC(;:::)n(;r)812 1140 y Fy(,)i Fv(r)e Fu(2)e Fr(N)7 b Fy(,)39 b(de\014ned)32 b(as)g(follo)n(ws.)47 b(Set)32 b Fv(s)2098 1153 y Fo(0)2164 1140 y Fy(=)f Fv(n)i Fy(and)e(de\014ne)h(inductiv)n(ely)e Fv(s)3273 1153 y FC(i)p Fo(+1)85 1249 y Fy(to)f(b)r(e)h(the)f(smallest) f(in)n(teger)g(0)d Fu(\024)f Fv(s)h(<)h(s)1455 1262 y FC(i)1510 1249 y Fy(\(if)i(it)h(exists\))g(suc)n(h)f(that:)p 1034 1333 485 4 v 1034 1411 a([)p Fv(A)1124 1424 y FC(s)1172 1411 y Fv(:)15 b(:)g(:)h(A)1356 1424 y FC(s)1385 1433 y Fn(i)1409 1424 y FB(\000)p Fo(1)1494 1411 y Fy(])21 b Fu(\\)e Fv(@)5 b Fu(Z)28 b(\\)p 1841 1333 403 4 v 19 w Fv(B)5 b Fy(\()p Fv(T)2010 1385 y FC(s)2043 1411 y Fv(x;)15 b(")2173 1424 y FB(\003)2209 1411 y Fy(\))25 b Fu(6)p Fy(=)g Fr(?)85 1574 y Fy(where)30 b Fv(A)405 1587 y FC(i)455 1574 y Fy(=)25 b Fu(Z)7 b Fy(\()p Fv(T)717 1542 y FC(i)743 1574 y Fv(x)p Fy(\).)40 b(Finally)27 b(set)i Fv(r)j Fy(to)d(b)r(e)g(maxim)n(um)g(suc)n(h)f(that)h Fv(s)2463 1587 y Fo(0)2499 1574 y Fv(;)15 b(:)g(:)g(:)i(;)e(s)2737 1587 y FC(r)2801 1574 y Fy(exist.)217 1682 y(W)-7 b(e)29 b(claim)f(that:)p 1266 1713 495 4 v 1266 1791 a([)p Fv(A)1356 1804 y Fo(0)1407 1791 y Fv(:)15 b(:)g(:)h(A)1591 1804 y FC(s)1620 1811 y Fn(r)1651 1804 y FB(\000)p Fo(1)1736 1791 y Fy(])21 b Fu(\\)e Fv(@)5 b(D)28 b Fu(6)p Fy(=)d Fr(?)p Fv(:)85 1933 y Fy(If)f Fv(s)209 1946 y FC(r)268 1933 y Fy(=)h(0,)g(the)f(claim)f(reduces)h(to)g Fv(@)5 b(D)27 b Fu(6)p Fy(=)e Fr(?)p Fy(:)37 b(it)23 b(is)g(clear.)37 b(So,)25 b(assume)f(that)g Fv(s)2688 1946 y FC(r)2747 1933 y Fv(>)h Fy(0)f(and)f(that)h(the)85 2041 y(ab)r(o)n(v)n(e)29 b(in)n(tersection)f(is)h(empt)n(y)-7 b(.)41 b(Hence)29 b(p)r(oin)n(ts)f(in)h(the)h(b)r(oundary)g(of)f Fv(T)2531 2008 y FC(s)2560 2015 y Fn(r)2596 2041 y Fy(\()p Fv(D)23 b Fu(\\)d Fy([)p Fv(A)2896 2054 y Fo(0)2947 2041 y Fv(:)15 b(:)g(:)h(A)3131 2054 y FC(s)3160 2061 y Fn(r)3191 2054 y FB(\000)p Fo(1)3276 2041 y Fy(])21 b Fu(\\)85 2149 y Fv(B)152 2162 y FC(s)181 2169 y Fn(r)212 2162 y Fo(+1)297 2149 y Fy(\()p Fv(x;)15 b(")461 2162 y FB(\003)497 2149 y Fy(\)\))41 b(do)f(not)g(come)g(from)h Fv(@)5 b(D)r Fy(.)73 b(They)41 b(do)f(not)g(come)h(either)e(from)i Fv(@)5 b Fy([)p Fv(A)2976 2162 y Fo(0)3027 2149 y Fv(:)15 b(:)g(:)h(A)3211 2162 y FC(s)3240 2169 y Fn(r)3271 2162 y FB(\000)p Fo(1)3356 2149 y Fy(])85 2258 y(b)r(ecause)29 b(of)g(the)g(de\014n)n(tion)e(of)i Fv(r)r Fy(.)40 b(Hence)29 b(this)f(b)r(oundary)h(is)f(a)h(subset)g(of)g Fv(@)5 b(T)2702 2225 y FC(s)2731 2232 y Fn(r)2767 2258 y Fv(B)2834 2271 y FC(s)2863 2278 y Fn(r)2894 2271 y Fo(+1)2979 2258 y Fy(\()p Fv(x;)15 b(")3143 2271 y FB(\003)3179 2258 y Fy(\).)217 2366 y Fv(T)42 b Fy(b)r(eing)28 b(non-con)n(tracting,)g Fv(T)1265 2333 y FC(s)1294 2340 y Fn(r)1330 2366 y Fv(B)1397 2379 y FC(s)1426 2386 y Fn(r)1458 2379 y Fo(+1)1542 2366 y Fy(\()p Fv(x;)15 b(")1706 2379 y FB(\003)1742 2366 y Fy(\))25 b(=)h Fv(B)5 b Fy(\()p Fv(T)2064 2333 y FC(s)2093 2340 y Fn(r)2128 2366 y Fv(x;)15 b(")2258 2379 y FB(\003)2294 2366 y Fy(\).)39 b(Hence,)30 b Fv(\032)2722 2379 y FC(s)2751 2386 y Fn(r)2786 2366 y Fy(\()p Fv(x)p Fy(\))25 b(=)h Fv(")3064 2379 y FB(\003)3099 2366 y Fy(.)40 b(Th)n(us,)85 2474 y Fv(R)q Fy(\()p Fv(x)p Fy(\))25 b Fu(\024)f Fv(s)431 2487 y FC(r)r Fo(+1)572 2474 y Fu(\024)h Fv(n)p Fy(,)30 b(a)f(con)n(tradiction.)217 2583 y(Observ)n(e)g(that)g(eq.)h(\(1.1\))g (implies)1398 2797 y Fv(r)d Fu(\024)e Fv(d)1621 2671 y Fs(\022)1722 2736 y Fv(n)p 1695 2776 106 4 v 1695 2858 a(N)1765 2871 y Fo(0)1830 2797 y Fy(+)20 b(1)1962 2671 y Fs(\023)2043 2797 y Fv(:)217 3006 y Fy(W)-7 b(e)29 b(compute,)g(using)f(the)h(conformalit)n(y)f(of)h Fv(\027)6 b Fy(:)403 3180 y Fv(\027)g Fy(\()p Fu(f)p Fv(x)25 b Fu(2)g Fv(D)56 b Fy(:)26 b Fv(R)q Fy(\()p Fv(x)p Fy(\))e Fu(\025)h Fv(n)p Fu(g)p Fy(\))g Fu(\024)1631 3094 y Fs(X)1615 3276 y Fn(A)p Fm(2Z)1737 3263 y Fn(n)1435 3343 y Fm(9)p Fn(x)p Fm(2)p Fn(A)30 b Fy(s.t.)39 b Fn(R)p Fl(\()p Fn(x)p Fl(\))p Fm(\025)p Fn(n)1980 3180 y Fv(e)2021 3143 y FB(\000)p FC(nP)9 b Fo(\()p FC(X)q(;T)g Fo(\))2328 3180 y Fv(g)2373 3143 y Fo(\()p FC(n)p Fo(\))2464 3180 y Fy(\()p Fv(A)p Fy(\))478 3566 y Fu(\024)75 b Fv(e)664 3529 y FB(\000)p FC(nP)9 b Fo(\()p FC(X)q(;T)g Fo(\))985 3447 y FC(d)p Fo(\()p FC(n=)l(N)1160 3455 y Fl(0)1191 3447 y Fo(+1\))1076 3481 y Fs(X)1085 3669 y FC(r)r Fo(=0)1497 3481 y Fs(X)1341 3669 y Fo(0)p FC()g(`)p Fu(g)p Fv(:)0 2381 y Fy(The)e(to)n(w)n(er)f(\001)j Fu(\032)f Fv(X)13 b Fu(\002)6 b(D)i(\002)e Fr(N)35 b Fy(is)21 b(the)i(disjoin)n(t)d(union)h(of)h(the) g(\001)2079 2394 y FC(`)2109 2381 y Fy('s.)37 b(W)-7 b(e)22 b(will)f(denote)h(b)n(y)g(\001)2996 2394 y FC(`;j)3098 2381 y Fu(\032)i Fy(\001)3265 2394 y FC(`)0 2490 y Fy(the)33 b(set)f(of)g(\()p Fv(x;)15 b(`)p Fy(\))33 b(suc)n(h)f(that)g(\()p Fv(x;)15 b Fy(0\))33 b(b)r(elongs)f(to)h(\001)1757 2503 y Fo(0)p FC(;j)1839 2490 y Fy(.)49 b(The)33 b(\001)2169 2503 y FC(`;j)2246 2490 y Fy('s)g(for)f Fv(`)f Fu(2)f Fr(N)7 b Fy(,)40 b Fv(j)c Fu(2)30 b Fr(N)7 b Fy(,)40 b(form)32 b(a)0 2598 y(partition)c(of)h(\001)g(whic)n(h)f(w)n(e)i(will) e(denote)g(b)n(y)h Fu(P)7 b Fy(.)132 2706 y(Let)29 b Fv(F)37 b Fy(:)25 b(\001)g Fu(\000)-15 b(!)25 b Fy(\001)k(b)r(e)h (de\014ned)e(in)g(the)i(follo)n(wing)c(w)n(a)n(y:)855 2819 y Fs(\()964 2915 y Fv(F)12 b Fy(\()p Fv(x;)j(`)p Fy(\))75 b(=)g(\()p Fv(x;)15 b(`)21 b Fy(+)f(1\))29 b(if)g Fv(`)19 b Fy(+)i(1)j Fv(<)2228 2892 y Fy(^)2208 2915 y Fv(R)q Fy(\()p Fv(x)p Fy(\))1302 3037 y(=)75 b(\()1497 3014 y(^)1479 3037 y Fv(T)1557 2989 y Fo(^)1543 3004 y FC(R)p Fo(\()p FC(x)p Fo(\))1680 3037 y Fv(x;)15 b Fy(0\))30 b(otherwise)o Fv(:)0 3243 y Fy(F)-7 b(or)33 b Fv(x;)15 b(y)34 b Fu(2)d Fy(\001)i(the)g Ft(sep)l(ar)l(ation)g(time)g Fv(s)p Fy(\()p Fv(x;)15 b(y)s Fy(\))34 b(is)e(the)h(largest)f(in)n (teger)g Fv(n)f Fu(\025)g Fy(0)h(suc)n(h)g(that)h(for)g(all)0 3351 y Fv(j)d Fu(\024)25 b Fv(n)p Fy(,)30 b Fv(F)336 3319 y FC(j)369 3351 y Fy(\()p Fv(x)p Fy(\))f(and)g Fv(F)756 3319 y FC(j)789 3351 y Fy(\()p Fv(y)s Fy(\))g(b)r(elong)g(to)g(the)g (same)g(atom)h(of)f(the)g(partition)f Fu(P)7 b Fy(.)40 b(Set)553 3559 y Fv(d)599 3572 y Fo(0)634 3559 y Fy(\()p Fv(x;)15 b(y)s Fy(\))26 b(=)f Fv(C)h Fu(\001)1188 3474 y Fs(X)1090 3668 y FC(j)s FB(\025)p FC(s)p Fo(\()p FC(x;y)r Fo(\)+1)1430 3559 y Fv(!)1485 3572 y FC(j)1518 3559 y Fy(\()p Fv(g)s Fy(\))176 b(with)58 b Fv(C)31 b Fy(=)25 b(exp)2372 3474 y Fs(X)2382 3664 y FC(j)s FB(\025)p Fo(1)2517 3559 y Fv(!)2572 3572 y FC(j)2604 3559 y Fy(\()p Fv(g)s Fy(\))p Fv(:)0 3846 y Fy(Note)38 b(that)g(this)f(metric)g(is)g (designed)f(so)i(that)g(the)g(family)f(of)g(functions)f(log)16 b Fv(g)2762 3859 y FC(n)2804 3846 y Fy(,)40 b Fv(n)g Fu(\025)g Fy(1,)g(are)0 3954 y(uniformly)28 b(Lipsc)n(hitz)f(w.r.t.)k (it)e(\(and)g(the)g(constan)n(t)f Fv(C)35 b Fy(ensures)29 b(\(A.I)r(I)r(I\))g(b)r(elo)n(w\).)132 4063 y(Remark)36 b(that)h(w)n(e)f(do)h(not)f(assume)g(that)g(the)h(images)e Fv(F)2127 4030 y FC(R)2179 4063 y Fy(\001)2252 4076 y Fo(0)p FC(;j)2371 4063 y Fy(are)i(the)f(whole)g(basis)f(\001)3235 4076 y Fo(0)3271 4063 y Fy(.)0 4171 y(That)28 b(is)f(wh)n(y)h(w)n(e)g (ha)n(v)n(e)f(to)h(in)n(tro)r(duce)f(the)h(\014nite)f(partition)f Fu(f)p Fv(B)2162 4184 y Fo(1)2197 4171 y Fv(;)15 b(:)g(:)g(:)h(;)f(B) 2460 4184 y FC(p)2496 4171 y Fu(g)28 b Fy(of)f(\001)2740 4184 y Fo(0)2804 4171 y Fy(generated)h(b)n(y)0 4279 y Fu(f)p Fv(F)114 4247 y FC(R)166 4279 y Fy(\001)239 4292 y Fo(0)p FC(;j)346 4279 y Fy(:)d Fv(j)31 b Fu(2)24 b Fr(N)7 b Fu(g)p Fy(.)45 b(Also)28 b Fv(B)984 4292 y FC(j;`)1082 4279 y Fu(\032)d Fy(\001)1250 4292 y FC(`)1309 4279 y Fy(is)j(the)h(subset)f(of)h(p)r(oin)n(ts)f(\()p Fv(x;)15 b(`)p Fy(\))30 b(with)f Fv(x)c Fu(2)f Fv(B)2831 4292 y FC(j)2864 4279 y Fy(.)132 4388 y(Finally)-7 b(,)27 b(w)n(e)j(lift)e(the)h(probabilit)n(y)e(measure)i Fv(\027)35 b Fy(on)29 b Fv(X)36 b Fy(to)29 b(a)g(measure)34 b(^)-49 b Fv(\027)36 b Fy(on)29 b(\001)g(de\014ned)f(b)n(y:)1150 4580 y(^)-49 b Fv(\027)6 b Fy(\()p Fv(S)25 b Fu(\002)20 b(f)p Fv(D)r Fu(g)f(\002)h(f)p Fv(`)p Fu(g)p Fy(\))25 b(=)g Fv(\027)6 b Fy(\()p Fv(S)f Fy(\))p Fv(:)0 4782 y Fy(Since)39 b(\001)313 4795 y Fo(0)390 4782 y Fy(is)g(a)h(\014nite)g (union)f(of)1193 4759 y Fs(b)1175 4782 y Fv(D)s Fy(,)k Fv(D)k Fu(2)c(D)r Fy(,)50 b(^)-49 b Fv(\027)6 b Fy(\(\001)1837 4795 y Fo(0)1873 4782 y Fy(\))44 b Fv(<)g Fu(1)p Fy(.)73 b(Also,)42 b(b)n(y)e(the)h(c)n(hoice)e(of)h Fv(N)3235 4795 y FB(\003)3271 4782 y Fy(,)0 4890 y(supp)o(\()5 b(^)-49 b Fv(\027)6 b Fy(\))25 b(=)g(\001.)p eop %%Page: 9 9 9 8 bop 832 26 a Fx(Piecewise)28 b(in)n(v)n(ertible)f(maps)i(in)f (higher)g(dimensions)745 b Fy(9)217 302 y(Let)29 b(us)g(summarize)f (the)h(crucial)f(prop)r(erties)h(of)g(the)g(to)n(w)n(er.)70 414 y(\(A.I\))45 b Fq(Exp)s(onen)m(tially)32 b(small)e(upp)s(er)k (\015o)s(ors.)40 b Fy(By)30 b(Prop)r(osition)d(1.1,)1129 648 y(^)-49 b Fv(\027)1188 548 y Fs(\020)1242 648 y Fu(f)p Fv(x)24 b Fu(2)h Fy(\001)1519 661 y Fo(0)1584 648 y Fv(=)1676 625 y Fy(^)1657 648 y Fv(R)p Fy(\()p Fv(x)p Fy(\))h Fv(>)f(`)p Fu(g)2043 548 y Fs(\021)2121 648 y Fu(\024)g Fy(const)19 b Fu(\001)h Fv(\015)2532 611 y FC(`)305 888 y Fy(for)41 b(some)g(0)k Fv(<)g(\015)50 b(<)c Fy(1.)75 b(In)40 b(particular,)49 b(^)-49 b Fv(\027)6 b Fy(\(\001\))45 b(=)2153 821 y Fs(P)2248 914 y FC(`)2297 888 y Fy(^)-49 b Fv(\027)6 b Fy(\(\001)2448 901 y FC(`)2478 888 y Fy(\))46 b Fu(\024)f Fy(const)2888 821 y Fs(P)2983 914 y FC(`)3027 888 y Fv(\015)3078 855 y FC(`)3153 888 y Fv(<)g Fu(1)p Fy(.)305 996 y(Moreo)n(v)n(er,)29 b(the)g(supp)r(ort)f(of)34 b(^)-49 b Fv(\027)35 b Fy(is)28 b(\001:)39 b(this)28 b(follo)n(ws)f(from)i(our)g(construction)e(and)i (the)f(fact)305 1104 y(that)36 b(the)h(supp)r(ort)g(of)f Fv(\027)43 b Fy(con)n(tains)35 b Fv(T)1602 1072 y FC(N)1652 1079 y Fm(\003)1689 1104 y Fv(X)7 b Fy(.)62 b(In)36 b(what)h(follo)n (ws,)h(w)n(e)e(assume)g(that)42 b(^)-49 b Fv(\027)43 b Fy(has)305 1213 y(b)r(een)29 b(normalized)f(i.e.)44 b(^)-49 b Fv(\027)6 b Fy(\(\001\))25 b(=)g(1.)36 1414 y(\(A.I)r(I\))45 b Fq(Generating)37 b(P)m(artition.)49 b Fy(The)33 b(partition)e Fu(P)41 b Fy(generates)32 b(under)g Fv(F)45 b Fy(i.e.:)h(the)33 b(partition)305 1455 y Fs(W)379 1478 y FB(1)379 1548 y FC(n)p Fo(=0)518 1522 y Fv(F)587 1490 y FB(\000)p FC(n)678 1522 y Fu(P)38 b Fy(is)29 b(the)h(partition)e (in)n(to)h(p)r(oin)n(ts,)g(this)g(follo)n(ws)e(directly)i(from)h(H1.)41 b(In)30 b(partic-)305 1630 y(ular,)e Fv(d)555 1643 y Fo(0)620 1630 y Fy(de\014nes)g(a)h(metric)g(on)g(\001.)2 1831 y(\(A.I)r(I)r(I\))45 b Fq(Bounded)22 b(distortion.)36 b Fy(Let)19 b Fv(J)8 b(F)32 b Fy(b)r(e)19 b(the)h(Jacobian)e(of)h Fv(F)32 b Fy(with)19 b(resp)r(ect)h(to)25 b(^)-49 b Fv(\027)6 b Fy(.)36 b(Ob)n(viously)-7 b(,)305 1940 y Fv(J)8 b(F)k Fy(\()p Fv(x;)j(`)p Fy(\))29 b(=)h(1)i(if)g Fv(`)21 b Fy(+)h(1)30 b Fv(<)1251 1917 y Fy(^)1232 1940 y Fv(R)p Fy(\()p Fv(x)p Fy(\))j(and)e Fv(J)8 b(F)k Fy(\()p Fv(x;)1891 1917 y Fy(^)1872 1940 y Fv(R)q Fy(\()p Fv(x)p Fy(\))22 b Fu(\000)f Fy(1\))30 b(=)g Fv(g)2433 1951 y Fo(^)2419 1965 y FC(R)p Fo(\()p FC(x)p Fo(\))2556 1940 y Fy(\()p Fv(x)p Fy(\).)48 b(If)32 b Fv(x;)15 b(y)35 b Fy(are)d(in)f(the)305 2048 y(same)f Fv(B)593 2061 y FC(j)625 2048 y Fy(,)h(w)n(e)g(can)f (de\014ned)f(their)h(paired)f(pre-images)h Fv(x)2247 2015 y FB(0)2298 2048 y Fy(and)g Fv(y)2516 2015 y FB(0)2567 2048 y Fy(as)g(follo)n(ws.)41 b Fv(F)3071 2015 y FC(n)3113 2048 y Fv(x)3163 2015 y FB(0)3211 2048 y Fy(=)27 b Fv(x)p Fy(,)305 2156 y Fv(F)374 2124 y FC(n)416 2156 y Fv(y)462 2124 y FB(0)508 2156 y Fy(=)e Fv(y)32 b Fy(and)c Fv(F)915 2124 y FC(k)953 2156 y Fy(\()p Fv(x)1037 2124 y FB(0)1058 2156 y Fy(\))h(and)f Fv(F)1360 2124 y FC(k)1398 2156 y Fy(\()p Fv(y)1478 2124 y FB(0)1499 2156 y Fy(\))h(b)r(elong)f(to)h (the)f(same)g(elemen)n(t)g(of)g Fu(P)37 b Fy(for)28 b(0)d Fu(\024)g Fv(k)i(<)f(n)p Fy(.)305 2265 y(W)-7 b(e)29 b(ha)n(v)n(e:)1359 2283 y Fs(\014)1359 2337 y(\014)1359 2390 y(\014)1359 2444 y(\014)1398 2352 y Fv(J)8 b(F)1524 2320 y FC(n)1566 2352 y Fy(\()p Fv(x)1650 2320 y FB(0)1671 2352 y Fy(\))p 1398 2392 308 4 v 1400 2474 a Fv(J)g(F)1526 2448 y FC(n)1568 2474 y Fy(\()p Fv(y)1648 2448 y FB(0)1669 2474 y Fy(\))1734 2413 y Fu(\000)20 b Fy(1)1868 2283 y Fs(\014)1868 2337 y(\014)1868 2390 y(\014)1868 2444 y(\014)1922 2413 y Fu(\024)25 b Fv(d)2063 2426 y Fo(0)2098 2413 y Fy(\()p Fv(x;)15 b(y)s Fy(\))p Fv(:)5 2680 y Fy(\(A.IV\))44 b Fq(Large)35 b(image)e(and)j(Mark)m(o)m(v)g(prop)s(erties.)42 b Ft(L)l(ar)l(ge)33 b(image:)41 b Fy(there)31 b(exists)e Fv(\021)i(>)c Fy(0)k(suc)n(h)305 2788 y(that)1456 2907 y(inf)1451 2964 y FC(j)s FB(2)p Fi(N)1586 2907 y Fy(^)-49 b Fv(\027)6 b Fy(\()p Fv(F)1747 2855 y Fo(^)1733 2870 y FC(R)1785 2907 y Fy(\(\001)1892 2920 y Fo(0)p FC(;j)1975 2907 y Fy(\)\))25 b Fu(\025)g Fv(\021)s(:)305 3142 y Fy(Indeed,)f Fv(F)692 3094 y Fo(^)679 3109 y FC(R)731 3142 y Fy(\(\001)838 3155 y Fo(0)p FC(;j)920 3142 y Fy(\))g(tak)n(es)g (only)f(\014nitely)g(man)n(y)h(distinct)e(v)-5 b(alues,)24 b(none)f(zero)h(as)g(supp)o(\()5 b(^)-49 b Fv(\027)6 b Fy(\))25 b(=)305 3250 y(\001.)305 3405 y Ft(Markov)31 b(pr)l(op)l(erty:)37 b Fv(F)1082 3357 y Fo(^)1068 3372 y FC(R)1120 3405 y Fy(\001)1193 3418 y Fo(0)p FC(;j)1305 3405 y Fy(is)28 b(a)h(union)f(of)h(some)g(\001)2103 3418 y Fo(0)p FC(;p)2189 3405 y Fy(.)85 3606 y(Let)22 b Fq(L)p Fy(\(\001\))h(b)r(e)f(the)h(space)e(of)h(b)r(ounded)g(functions)e(on)i (\001)h(that)f(are)h Ft(lo)l(c)l(al)t(ly)i(Lipschitz)d Fy(with)g(resp)r(ect)85 3714 y(to)29 b(the)h(metric)f Fv(d)663 3727 y Fo(0)698 3714 y Fy(,)g(i.e.,)h(for)f(all)f Fv(x)p Fy(,)i Fv(y)i Fy(in)d(the)g(same)g Fv(B)1893 3727 y FC(j;`)1965 3714 y Fy(,)1161 3923 y Fu(j)p Fv(')p Fy(\()p Fv(x)p Fy(\))20 b Fu(\000)g Fv(')p Fy(\()p Fv(y)s Fy(\))p Fu(j)k(\024)h Fv(K)6 b Fy(\()p Fv(')p Fy(\))p Fv(d)2041 3936 y Fo(0)2076 3923 y Fy(\()p Fv(x;)15 b(y)s Fy(\))p Fv(;)85 4131 y Fy(for)39 b(some)h Fv(K)6 b Fy(\()p Fv(')p Fy(\))39 b(whic)n(h)f(is)h(assumed)f(to)i(b)r(e)g(the)f(smallest)f(p)r (ositiv)n(e)g(n)n(um)n(b)r(er)h(suc)n(h)f(that)h(the)85 4240 y(ab)r(o)n(v)n(e)29 b(inequalit)n(y)e(is)h(satis\014ed.)38 b(Let)29 b Fu(k)p Fv(')p Fu(k)1506 4254 y Ff(L)p Fo(\(\001\))1680 4240 y Fy(=)d Fv(K)6 b Fy(\()p Fv(')p Fy(\))19 b(+)50 b Fu(k)p Fv(')p Fu(k)2266 4253 y FB(1)2362 4240 y Fy(b)r(e)29 b(the)g(norm)h(on)e Fq(L)p Fy(\(\001\))q(.)217 4352 y Fu(P)43 b Fy(has)35 b(a)g(natural)f(graph)h(structure:)52 b Fv(P)47 b Fu(!)35 b Fv(Q)g Fy(i\013)g Fv(F)12 b Fy(\()p Fv(P)g Fy(\))35 b Fu(\033)h Fv(Q)p Fy(.)57 b(Its)35 b(\(restricted\))g Ft(sp)l(e)l(ctr)l(al)85 4472 y(de)l(c)l(omp)l(osition)28 b Fy(is)g Fu(P)k Fy(=)26 b Fu(P)984 4485 y FC(t)1031 4472 y Fu([)1110 4405 y Fs(S)1185 4498 y FC(i)1225 4405 y Fs(S)1300 4429 y FC(p)1332 4438 y Fn(i)1356 4429 y FB(\000)p Fo(1)1300 4498 y FC(j)s Fo(=0)1456 4472 y Fu(P)1525 4426 y Fo(\()p FC(i)p Fo(\))1518 4495 y FC(j)1600 4472 y Fy(,)k(where:)216 4673 y Fu(\017)44 b(P)367 4686 y FC(t)423 4673 y Fy(is)29 b(the)h(set)f(of)h(transien)n(t)e(elemen)n(ts) h(of)g Fu(P)7 b Fy(,)31 b(i.e.,)f(elemen)n(ts)f Fv(P)41 b Fy(suc)n(h)29 b(that)h(there)g(exists)e(a)305 4782 y(path)33 b(from)h Fv(P)45 b Fy(going)33 b(to)h(some)f Fv(Q)f Fu(2)g(P)42 b Fy(and)33 b(there)h(is)f(no)g(path)g(from)h Fv(Q)f Fy(to)h Fv(P)46 b Fy(\(observ)n(e)305 4890 y(that)29 b(w)n(e)g(don't)g(separate)g(this)f(part)h(in)n(to)f(irreducible)g (subsets\).)p eop %%Page: 10 10 10 9 bop 0 26 a Fy(10)745 b Fx(J)n(\023)-41 b(er^)d(ome)29 b(Buzzi)g(&)g(V)n(\023)-41 b(eronique)28 b(Maume-Desc)n(hamps)131 303 y Fu(\017)43 b Fy(for)27 b(eac)n(h)f Fv(i)p Fy(,)628 236 y Fs(S)703 260 y FC(p)735 269 y Fn(i)759 260 y FB(\000)p Fo(1)703 329 y FC(j)s Fo(=0)858 303 y Fu(P)927 258 y Fo(\()p FC(i)p Fo(\))920 326 y FC(j)1030 303 y Fy(is)g(the)h(set)f(of)h Fv(P)37 b Fu(2)24 b(P)35 b Fy(suc)n(h)26 b(that)h(there)g(exist)f (paths)g(from)h Fv(P)39 b Fy(to)27 b Fv(Q)219 412 y Fy(and)k Fv(Q)f Fy(to)h Fv(P)12 b Fy(,)31 b(for)g(some)g(\014xed)f Fv(Q)h Fy(\(i.e.,)g(these)g(unions)e(are)i(the)g(irreducible)d(comp)r (onen)n(ts)219 520 y(of)h Fu(P)37 b Fy(whic)n(h)29 b(are)g(maximal)f (w.r.t.)k(the)d(precedence)g(ordering\).)131 708 y Fu(\017)43 b Fy(if)29 b(there)g(is)g(an)g(arro)n(w)g(from)g Fu(P)1257 662 y Fo(\()p FC(i)p Fo(\))1250 731 y FC(j)1362 708 y Fy(to)g Fu(P)1538 662 y Fo(\()p FC(k)r Fo(\))1531 733 y FC(l)1656 708 y Fy(then)g Fv(k)f Fy(=)d Fv(i)k Fy(and)g Fv(l)e Fy(=)e Fv(j)g Fy(+)20 b(1)30 b(mo)r(d)f Fv(p)2863 721 y FC(i)2889 708 y Fy(.)0 892 y(Finally)-7 b(,)33 b(\001)394 846 y Fo(\()p FC(i)p Fo(\))394 915 y FC(j)503 892 y Fy(is)g(the)h(subset)e(of)i(\001)g(de\014ned)f(b)n(y)g Fu(P)1739 846 y Fo(\()p FC(i)p Fo(\))1732 915 y FC(j)1815 892 y Fy(.)53 b(Observ)n(e)33 b(that,)i(up)f(to)f(trivialities,)g(it)g (is)0 1025 y(enough)28 b(to)h(study)g(the)g(dynamics)f(of)g 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Fv(')c Fu(2)h Fq(L)p Fy(\(\001\))k Fx(and)g Fv( )f Fu(2)d Fv(L)1683 1720 y FB(1)1750 1753 y Fy(\(\001\))p Fx(,)145 1845 y Fs(\014)145 1899 y(\014)145 1953 y(\014)145 2007 y(\014)145 2060 y(\014)145 2114 y(\014)175 1907 y(Z)173 2162 y Fo(\001)279 2029 y Fv(')20 b Fu(\016)g Fv(F)491 1992 y FC(n)553 2029 y Fu(\001)g Fv( )e(d)7 b Fy(^)-51 b Fv(\026)19 b Fu(\000)881 1907 y Fs(Z)879 2162 y Fo(\001)986 2029 y Fv(')c(d)7 b Fy(^)-51 b Fv(\026)1171 1907 y Fs(Z)1170 2162 y Fo(\001)1276 2029 y Fv( )18 b(d)7 b Fy(^)-51 b Fv(\026)1450 1845 y Fs(\014)1450 1899 y(\014)1450 1953 y(\014)1450 2007 y(\014)1450 2060 y(\014)1450 2114 y(\014)1504 2029 y Fu(\024)25 b Fv(C)h Fu(\001)20 b(k)p Fv(')p Fu(k)1881 2043 y Ff(L)p Fo(\(\001\))2030 2029 y Fu(k)p Fv( )s Fu(k)2180 2044 y FC(L)2223 2027 y Fl(1)2255 2044 y Fo(\()5 b(^)-37 b FC(\026)o Fo(\))2366 2029 y Fu(\001)20 b Fv(u)2461 2042 y FC(n)2678 2029 y Fx(for)30 b(all)e Fv(n)d Fu(\025)f Fy(0)0 2303 y Fx(for)29 b(some)g Fv(C)i(<)25 b Fu(1)k Fx(and)g(a)g(sequence)g Fv(u)g Fx(con)n(v)n (erging)f(to)h(zero)h(and)e(whic)n(h)g(can)h(b)r(e)h(explicited:)131 2463 y Fu(\017)43 b Fx(if)29 b Fv(!)355 2476 y FC(n)397 2463 y Fy(\()p Fv(g)s Fy(\))c(=)h Fv(O)r Fy(\()p Fv(\032)777 2431 y FC(n)820 2463 y Fy(\))j Fx(for)g(some)g Fy(0)c Fv(<)g(\032)g(<)h Fy(1)p Fx(,)j(then)g Fv(u)1910 2476 y FC(n)1977 2463 y Fy(=)d Fv(\024)2122 2431 y FC(n)2193 2463 y Fx(for)j(some)g Fy(0)c Fv(<)g(\024)f(<)h Fy(1)p Fx(,)131 2639 y Fu(\017)43 b Fx(if)29 b Fv(!)355 2652 y FC(n)397 2639 y Fy(\()p Fv(g)s Fy(\))c(=)h Fv(O)r Fy(\()p Fv(n)784 2606 y FB(\000)p FC(\013)879 2639 y Fy(\))j Fx(for)g(some)g Fv(\013)c(>)h Fy(1)p Fx(,)j(then)g Fv(u)1818 2652 y FC(n)1885 2639 y Fy(=)c Fv(O)r Fy(\()p Fv(n)2133 2606 y FB(\000)p FC(\013)p Fo(+1)2310 2639 y Fy(\))p Fx(.)132 2799 y Fy(The)k(Main)g(Theorem)h(follo)n(ws)d(from)i(Theorem)h (2.1)f(using)f(the)h(follo)n(wing)e(observ)-5 b(ations.)132 2907 y(First,)25 b(the)g(existence)f(of)g(a)g Fv(\027)6 b Fy(-a.c.i.m.)26 b(for)f Fv(T)37 b Fy(follo)n(ws)23 b(from)h(the)h(existence)f(of)g(a)30 b(^)-49 b Fv(\027)6 b Fy(-a.c.i.m.)25 b(for)0 3015 y Fv(F)38 b Fy(whic)n(h)25 b(can)h(b)r(e)g(pro)5 b(jected)26 b(in)f(the)h(follo)n(wing)e(w)n(a)n (y)-7 b(.)38 b(De\014ne)25 b Fv(\031)j Fy(:)e(\001)f Fu(!)f Fv(T)2492 2983 y FC(N)2542 2990 y Fm(\003)2580 3015 y Fv(X)33 b Fy(b)n(y:)k Fv(\031)s Fy(\()p Fv(x;)15 b(D)r(;)g(`)p Fy(\))26 b(=)0 3124 y Fv(T)63 3091 y FC(`)93 3124 y Fv(x)p Fy(.)38 b Fv(\031)26 b Fy(is)c(surjectiv)n(e)h(and)f Fv(T)f Fu(\016)8 b Fv(\031)27 b Fy(=)f Fv(\031)11 b Fu(\016)d Fv(F)k Fy(.)37 b(If)29 b(^)-51 b Fv(\026)24 b Fy(is)e(a)28 b(^)-49 b Fv(\027)6 b Fy(-a.c.i.m.,)26 b(it)d(pro)5 b(jects)23 b(b)n(y)g Fv(\031)j Fy(to)e(a)f Fv(\027)6 b Fy(-a.c.i.m.)132 3232 y(Second,)26 b(to)h(study)f(a)h(giv)n(en)e Fv(\027)6 b Fy(-a.c.i.m.)28 b(w)n(e)e(lift)g(it)g(to)g(a)32 b(^)-49 b Fv(\027)6 b Fy(-a.c.i.m.)39 b(as)26 b(follo)n(ws.)37 b(If)26 b Fv(\026)e Fy(=)i Fv(h)3126 3245 y Fo(0)3161 3232 y Fv(\027)32 b Fy(is)0 3340 y(a)e Fv(\027)6 b Fy(-a.c.i.m.,)31 b(let)610 3317 y Fs(b)612 3340 y Fv(h)663 3353 y Fo(0)724 3340 y Fy(=)26 b(const)19 b Fu(\001)i Fv(h)1135 3353 y Fo(0)1190 3340 y Fu(\016)f Fv(\031)33 b Fy(with)c(const)h(suc)n(h)e (that)35 b(^)-49 b Fv(\027)6 b Fy(\()2239 3317 y Fs(b)2241 3340 y Fv(h)2292 3353 y Fo(0)2327 3340 y Fy(\))26 b(=)g(1.)41 b(Let)2746 3317 y Fs(b)2748 3340 y Fv(h)29 b Fy(b)r(e)h(an)n(y)g(limit) 0 3459 y(p)r(oin)n(t)g(of)h(1)p Fv(=n)488 3391 y Fs(P)583 3415 y FC(n)p FB(\000)p Fo(1)583 3485 y FC(i)p Fo(=0)722 3459 y Fu(L)784 3426 y FC(i)784 3481 y Fo(0)817 3435 y Fs(b)819 3459 y Fv(h)870 3472 y Fo(0)936 3459 y Fy(where)h Fu(L)1254 3472 y Fo(0)1321 3459 y Fy(is)e(the)h(transfer)g(op)r(erator) h(de\014ned)e(in)h(section)f(4,)i(suc)n(h)e(a)0 3581 y(limit)g(p)r(oin)n(t)g(exists)g(and)h(satis\014es)j(^)-49 b Fv(\027)6 b Fy(\()1266 3557 y Fs(b)1268 3581 y Fv(h)p Fy(\))28 b(=)h(1)i(b)n(y)f(Lemma)i(4.2.)45 b(Let)38 b(^)-51 b Fv(\026)28 b Fy(=)2508 3557 y Fs(b)2510 3581 y Fv(h)t Fy(^)-49 b Fv(\027)6 b Fy(,)39 b(^)-51 b Fv(\026)31 b Fy(is)f(a)37 b(^)-49 b Fv(\027)6 b Fy(-a.c.i.m.,)0 3689 y(it)29 b(pro)5 b(jects)28 b(b)n(y)h Fv(\031)k Fy(on)c Fv(\026)p Fy(.)132 3797 y(Finally)-7 b(,)29 b(let)h Fv(';)15 b( )30 b Fy(:)d Fv(X)34 b Fu(!)27 b Fr(R)39 b Fy(b)r(e)31 b(as)f(in)f(the)i(Main)f(Theorem.)43 b(Observ)n(e)30 b(that)h Fv(')20 b Fu(\016)h Fv(\031)30 b Fu(2)c Fq(L)p Fy(\(\001\))0 3906 y(and)20 b Fv( )5 b Fu(\016)r Fv(\031)28 b Fu(2)d Fv(L)494 3873 y FB(1)561 3906 y Fy(.)36 b(Finally)20 b(remark)h(that)f(the)h(in)n(tegrals)d(o)n(v)n(er)i(\001)h(in)n(v)n (olving)d(these)i(lifted)f(functions)0 4014 y(are)29 b(equal)g(to)g(the)h(in)n(tegrals)d(in)h(the)h(Main)g(Theorem.)0 4200 y Fq(Remark)k(2.2)45 b Fx(W)-7 b(e)29 b(could)f(replace)g(in)h (the)g(theorem)g(ab)r(o)n(v)n(e)f(assumption)g(\(A.I\))h(\(the)g(exp)r (onen-)0 4309 y(tially)f(small)g(upp)r(er)h(\015o)r(ors\))g(b)n(y)g (the)g(w)n(eak)n(er)g(summabilit)n(y)e(assumption)2509 4242 y Fs(P)2604 4335 y FC(`)p FB(2)p Fi(N)2739 4309 y Fy(^)-49 b Fv(\027)6 b Fy(\(\001)2890 4322 y FC(`)2920 4309 y Fy(\))25 b Fv(<)h Fu(1)p Fx(.)132 4417 y(W)-7 b(e)29 b(w)n(ould)f(get)h Fv(u)p Fx(-mixing)g(with)g Fv(u)g Fx(giv)n(en)f(b)n(y:)1134 4588 y Fv(C)e Fu(\001)20 b Fy(max[)p Fv(u)1505 4601 y FC(n)1548 4588 y Fv(;)45 b(\024)1668 4551 y FC(n)1710 4588 y Fv(;)f(")1819 4551 y FB(\000)p Fo(1)1819 4610 y FC(n)1909 4588 y Fy(^)-49 b Fv(\027)6 b Fy(\(\001)2060 4601 y FC(n)2103 4588 y Fy(\)])0 4759 y Fx(where)38 b Fv(u)312 4772 y FC(n)392 4759 y Fx(is)e(from)i(Theorem)g(2.1,)i Fy(0)f Fv(<)g(\024)f(<)i Fy(1)d Fx(and)g Fv(")1947 4772 y FC(n)2027 4759 y Fx(is)f(an)n(y)h (decreasing)g(and)f(summable)0 4867 y(sequence.)j(\(see)29 b([M)q(]\).)p eop %%Page: 11 11 11 10 bop 832 26 a Fx(Piecewise)28 b(in)n(v)n(ertible)f(maps)i(in)f (higher)g(dimensions)745 b Fy(11)275 302 y Fh(3)145 b Fw(Birkhoff's)49 b(cones)h(and)f(pr)m(ojective)g(metrics)85 476 y Fy(The)20 b(main)g(to)r(ol)f(for)h(the)g(pro)r(of)h(of)e(the)h 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Fy(a)c(cone)g(with)g(the)g(follo)n(wing)e(prop)r(erties.)216 1452 y Fu(\017)44 b Fv(C)35 b Fy(is)28 b(con)n(v)n(ex,)216 1627 y Fu(\017)44 b Fv(C)26 b Fu(\\)19 b(\000)p Fv(C)31 b Fy(=)25 b Fu(f)p Fy(0)p Fu(g)p Fy(,)216 1802 y Fu(\017)44 b Fy(if)29 b Fv(\013)442 1815 y FC(n)514 1802 y Fy(is)h(a)g(sequence)g (of)g(real)f(n)n(um)n(b)r(ers)g(suc)n(h)g(that)i Fv(\013)2117 1815 y FC(n)2185 1802 y Fu(!)26 b Fv(\013)31 b Fy(and)f Fv(x)20 b Fu(\000)h Fv(\013)2777 1815 y FC(n)2819 1802 y Fv(y)29 b Fu(2)e Fv(C)36 b Fu(8)p Fv(n)30 b Fy(then)305 1910 y Fv(x)19 b Fu(\000)h Fv(\013)q(y)27 b Fu(2)e Fv(C)6 b Fy(.)40 b(This)28 b(prop)r(ert)n(y)i(is)e(called)g(\\in)n(tegral)f (closure".)85 2070 y(F)-7 b(or)21 b(suc)n(h)e(a)h(cone,)j(the)d (pseudo-metric)g Fv(\016)1444 2083 y FC(C)1517 2070 y Fy(on)g Fv(C)27 b Fy(is)20 b(de\014ned)f(in)h(the)g(follo)n(wing)e(w)n (a)n(y)-7 b(.)36 b(F)-7 b(or)21 b Fv(x;)15 b(y)28 b Fu(2)d Fv(C)6 b Fy(,)899 2240 y Fv(\026)p Fy(\()p Fv(x;)15 b(y)s Fy(\))25 b(=)g(inf)6 b Fu(f)p Fv(\014)29 b(>)c Fy(0)k(suc)n(h)f(that)h Fv(\014)5 b(x)20 b Fu(\000)f Fv(y)28 b Fu(2)d Fv(C)6 b Fu(g)p Fv(:)85 2411 y Fy(with)35 b(the)g(con)n(v)n(en)n(tion:)50 b Fv(\026)p Fy(\()p Fv(x;)15 b(y)s Fy(\))35 b(=)h Fu(1)f Fy(if)f(the)h(corresp)r(onding)f(set)h(is)g(empt)n(y)-7 b(.)57 b(Let)35 b Fv(\016)3020 2424 y FC(C)3073 2411 y Fy(\()p Fv(x;)15 b(y)s Fy(\))36 b(=)85 2519 y(log)15 b Fv(\026)p Fy(\()p Fv(x;)g(y)s Fy(\))p Fv(\026)p Fy(\()p Fv(y)s(;)g(x)p Fy(\).)55 b Fv(\016)843 2532 y FC(C)930 2519 y Fy(is)33 b(not)h(necessarily)e(\014nite.)52 b(Moreo)n(v)n(er,)36 b(it)e(is)f(pro)5 b(jectiv)n(e:)47 b(if)34 b Fv(x)f Fy(and)h Fv(x)3345 2532 y Fo(1)85 2628 y Fy(are)29 b(prop)r(ortional)f(then)h (for)g(an)n(y)g Fv(y)f Fu(2)d Fv(C)6 b Fy(,)30 b Fv(\016)1554 2641 y FC(C)1607 2628 y Fy(\()p Fv(x;)15 b(y)s Fy(\))25 b(=)g Fv(\016)1967 2641 y FC(C)2020 2628 y Fy(\()p Fv(x)2104 2641 y Fo(1)2140 2628 y Fv(;)15 b(y)s Fy(\).)217 2736 y(The)29 b(follo)n(wing)e(t)n(w)n(o)i(results)f(rev)n(eal)g(the)i (usefulness)c(of)j(pro)5 b(jectiv)n(e)29 b(metrics.)217 2844 y(Let)35 b Fv(C)42 b Fy(and)35 b Fv(C)731 2812 y FB(0)788 2844 y Fy(b)r(e)h(t)n(w)n(o)f(cones,)i Fv(P)48 b Fy(a)35 b(linear)f(op)r(erator)i Fv(P)48 b Fy(:)36 b Fv(C)42 b Fu(!)35 b Fv(C)2611 2812 y FB(0)2633 2844 y Fy(.)58 b(Let)36 b(\000)f(denote)g(the)85 2953 y(diameter)29 b(of)g Fv(P)12 b(C)35 b Fy(in)29 b Fv(C)890 2920 y FB(0)911 2953 y Fy(:)1209 3061 y(\000)c(=)46 b(sup)1382 3135 y FC(f)s(;g)r FB(2)p FC(C)1572 3061 y Fv(\016)1611 3074 y FC(C)1660 3057 y Fm(0)1684 3061 y Fy(\()p Fv(P)12 b(f)5 b(;)15 b(P)d(g)s Fy(\))26 b Fu(\024)e(1)p Fv(:)85 3308 y Fj(Theorem)34 b(3.1)43 b Fy([Bi])30 b Fx(F)-7 b(or)29 b(an)n(y)g Fv(f)5 b(;)15 b(g)33 b Fx(in)28 b Fv(C)6 b Fx(,)30 b(w)n(e)f(ha)n(v)n(e:)1084 3530 y Fv(\016)1123 3543 y FC(C)1172 3527 y Fm(0)1196 3530 y Fy(\()p Fv(P)12 b(f)5 b(;)15 b(P)d(g)s Fy(\))25 b Fu(\024)g Fy(tanh)1845 3404 y Fs(\022)1919 3469 y Fy(\000)p 1919 3510 55 4 v 1924 3592 a(4)1983 3404 y Fs(\023)2064 3530 y Fv(\016)2103 3543 y FC(C)2156 3530 y Fy(\()p Fv(f)5 b(;)15 b(g)s Fy(\))p Fv(:)85 3752 y Fy(This)29 b(theorem)i(implies)d(that)i Fv(P)38 b Fy(:)27 b Fv(C)33 b Fu(!)26 b Fv(C)1546 3720 y FB(0)1598 3752 y Fy(is)j(alw)n(a)n(ys)g(a)h(con)n(traction)f(\(in)g (the)i(w)n(eak)f(sense\))f(for)85 3861 y(the)g(pro)5 b(jectiv)n(e)29 b(metrics.)38 b(If)29 b(\000)c Fv(<)g Fu(1)k Fy(then)g(it)g(is)f(a)h(strict)g(con)n(traction.)217 3969 y(The)37 b(follo)n(wing)e(result)h(sho)n(ws)g(that)i(the)f(pro)5 b(jectiv)n(e)36 b(metric)h Fv(\016)2389 3982 y FC(C)2479 3969 y Fy(dominates)f(norms)h(on)g Fv(B)85 4077 y Fy(whic)n(h)25 b(are)h(w)n(ell-b)r(eha)n(v)n(ed:)36 b(A)26 b(norm)g Fu(k)f(k)h Fy(on)g Fv(B)k Fy(is)25 b Ft(adapte)l(d)i(to)i Fv(C)j Fy(if)25 b(for)h Fv(f)36 b Fy(and)25 b Fv(g)k Fy(in)c Fv(B)31 b Fy(suc)n(h)24 b(that)85 4186 y(b)r(oth)j Fv(f)e Fy(+)16 b Fv(g)30 b Fy(and)d Fv(f)e Fu(\000)15 b Fv(g)30 b Fy(b)r(elong)c(to)h Fv(C)34 b Fy(then)26 b Fu(k)p Fv(g)s Fu(k)f(\024)g(k)p Fv(f)10 b Fu(k)p Fy(.)38 b Fv(\032)25 b Fy(:)h Fv(C)31 b Fu(!)24 b Fr(R)2432 4199 y Fo(+)2518 4186 y Fy(is)i(a)h Ft(homo)l(gene)l(ous)i(form)85 4294 y(adapte)l(d)35 b(to)i Fv(C)j Fy(if,)c(for)e(an)n(y)g Fv(\025)g(>)h Fy(0)f(and)g Fv(f)44 b Fu(2)33 b Fv(C)6 b Fy(,)37 b Fv(\032)p Fy(\()p Fv(\025f)10 b Fy(\))34 b(=)h Fv(\025\032)p Fy(\()p Fv(f)10 b Fy(\))35 b(and)f(if)f Fv(f)h Fu(\000)23 b Fv(g)37 b Fu(2)d Fv(C)41 b Fy(implies)85 4402 y Fv(\032)p Fy(\()p Fv(g)s Fy(\))26 b Fu(\024)e Fv(\032)p Fy(\()p Fv(f)10 b Fy(\).)85 4589 y Fj(Theorem)34 b(3.2)43 b Fy([Bi],)32 b([Li].)45 b Fx(Let)31 b Fv(C)38 b Fx(b)r(e)31 b(a)g(cone,)g(let)g Fu(k)g(k)g Fx(and)f Fv(\032)i Fx(b)r(e)f(adapted)f(to)i Fv(C)6 b Fx(.)45 b(F)-7 b(or)31 b(an)n(y)f Fv(f)85 4697 y Fx(and)f Fv(g)j Fx(in)d Fv(C)35 b Fx(suc)n(h)28 b(that)h Fv(\032)p Fy(\()p Fv(f)10 b Fy(\))26 b(=)f Fv(\032)p Fy(\()p Fv(g)s Fy(\))h Fu(6)p Fy(=)f(0)k Fx(w)n(e)g(ha)n(v)n(e:)1024 4867 y Fu(k)p Fv(f)h Fu(\000)19 b Fv(g)s Fu(k)25 b(\024)g Fy(\()p Fv(e)1516 4830 y FC(\016)r Fo(\()p FC(f)s(;g)r Fo(\))1703 4867 y Fu(\000)20 b Fy(1\))15 b(min)o(\()p Fu(k)p Fv(f)10 b Fu(k)p Fv(;)15 b Fu(k)p Fv(g)s Fu(k)p Fy(\))p Fv(:)p eop %%Page: 12 12 12 11 bop 0 26 a Fy(12)745 b Fx(J)n(\023)-41 b(er^)d(ome)29 b(Buzzi)g(&)g(V)n(\023)-41 b(eronique)28 b(Maume-Desc)n(hamps)781 302 y Fh(4)194 b Fw(Pr)m(oof)48 b(of)h(Theorem)h(2.1)0 489 y Fy(As)29 b(usual,)f(the)h Ft(tr)l(ansfer)i(op)l(er)l(ator)d Fy(acting)h(on)g(b)r(ounded)f(functions)g(is)g(de\014ned)g(b)n(y:)1138 706 y Fu(L)1200 719 y Fo(0)1236 706 y Fv(f)10 b Fy(\()p Fv(x)p Fy(\))25 b(=)1544 621 y Fs(X)1525 811 y FC(F)9 b(y)r Fo(=)p FC(x)1814 646 y Fy(1)p 1716 686 241 4 v 1716 768 a Fv(J)f(F)k Fy(\()p Fv(y)s Fy(\))1965 706 y Fv(f)e Fy(\()p Fv(y)s Fy(\))p Fv(:)0 975 y Fy(The)28 b(measure)k(^)-49 b Fv(\027)33 b Fy(is)27 b(conformal)g(for)g Fu(L)1282 988 y Fo(0)1345 975 y Fy(in)g(the)g(follo)n(wing)e(sense:)38 b(for)27 b(an)n(y)g(b)r(ounded)g(function)f Fv(f)10 b Fy(,)1276 1073 y Fs(Z)1380 1196 y Fu(L)1442 1209 y Fo(0)1478 1196 y Fv(f)g(d)5 b Fy(^)-49 b Fv(\027)30 b Fy(=)1743 1073 y Fs(Z)1848 1196 y Fv(f)10 b(d)5 b Fy(^)-49 b Fv(\027)5 b(:)0 1421 y Fy(First,)29 b(w)n(e)g(are)g(going)f(to)h(construct)g(a)34 b(^)-49 b Fv(\027)6 b Fy(-a.c.i.m.)40 b(and)29 b(discuss)e(its)h (mixing)g(prop)r(erties)g(\(section)0 1529 y(4.1\).)38 b(Then)23 b(will)f(construct)h(a)g(sequence)g(of)g(cones)g Fv(C)1782 1542 y FC(j)1838 1529 y Fy(and)f(a)i(sequence)e(of)i(in)n (tegers)d Fv(k)2886 1542 y FC(j)2942 1529 y Fy(suc)n(h)h(that)0 1638 y Fu(L)62 1605 y FC(k)95 1614 y Fn(j)163 1638 y Fy(maps)34 b Fv(C)461 1651 y FC(j)s FB(\000)p Fo(1)609 1638 y Fy(in)n(to)g Fv(C)856 1651 y FC(j)923 1638 y Fy(with)h (uniformly)e(b)r(ounded)h(diameter)h(where)g Fu(L)f Fy(is)g(the)h (normalized)0 1746 y(tranfer)29 b(op)r(erator)h(\(sections)e(4.2,)i (4.3\).)0 1880 y Fp(4.1)120 b(Construction)41 b(of)f(a)46 b FJ(^)-59 b Fe(\027)7 b Fp(-a.c.i.m.)53 b(and)40 b(mixing)i(prop)s (erties)0 2047 y Fy(F)-7 b(or)43 b Fv(s)48 b Fu(2)f Fr(N)7 b Fy(,)52 b(the)43 b Fv(s)p Ft(-cylinders)h Fy(are)f(the)f(non)g(empt)n (y)h(sets)f(of)g(the)h(form:)2643 1979 y Fs(T)2718 2003 y FC(s)p FB(\000)p Fo(1)2718 2073 y FC(i)p Fo(=0)2847 2047 y Fv(F)2916 2014 y FB(\000)p FC(i)2991 2047 y Fv(A)3057 2060 y FC(i)3125 2047 y Fy(with)0 2155 y Fv(A)66 2168 y FC(i)122 2155 y Fu(2)31 b(P)7 b Fy(.)50 b(Recall)31 b(that)i Fu(P)40 b Fy(is)32 b(obtained)g(b)n(y)g(translation)f(from)i (the)f(partition)g(of)g(\001)2861 2168 y Fo(0)2929 2155 y Fy(according)0 2263 y(to)d(the)h(return)f(time)746 2241 y(^)727 2263 y Fv(R)h Fy(and)e Fu(Z)1080 2216 y Fo(^)1066 2231 y FC(R)1118 2263 y Fy(-itinerary)-7 b(.)132 2372 y(F)g(or)29 b Fv(k)f Fu(2)c Fr(N)42 b Fy(and)29 b Fv(x)c Fu(2)f Fy(\001,)30 b Fv(C)1061 2385 y FC(k)1099 2372 y Fy(\()p Fv(x)p Fy(\))g(denotes)e(the)i Fv(k)s Fy(-cylinder)e(whic)n(h)g(con)n(tains)f Fv(x)p Fy(.)132 2480 y(The)i(follo)n(wing)e(lemmas)h(are)i(tec)n(hnical)d(to)r(ols)i (to)g(study)g Fu(L)2139 2448 y FC(n)2139 2502 y Fo(0)2181 2480 y Fy(.)0 2671 y Fj(Lemma)34 b(4.1)43 b Fx(There)31 b(exists)f Fv(C)j(<)28 b Fu(1)i Fx(suc)n(h)g(that)g(for)h(an)n(y)f Fv(`)d Fu(2)g Fr(N)44 b Fx(and)30 b(an)n(y)g Fv(x)e Fu(2)f Fy(\001)2835 2684 y FC(`)2895 2671 y Fx(and)j Fv(k)g Fu(2)e Fr(N)0 2780 y Fx(with)h Fv(F)269 2747 y FC(k)307 2780 y Fv(x)c Fu(2)g Fy(\001)540 2793 y Fo(0)576 2780 y Fx(,)917 2923 y Fv(C)986 2886 y FB(\000)p Fo(1)1076 2923 y Fy(^)-49 b Fv(\027)6 b Fy(\()p Fv(C)1217 2936 y FC(k)1256 2923 y Fy(\()p Fv(x)p Fy(\)\))25 b Fu(\024)1656 2862 y Fy(1)p 1537 2902 283 4 v 1537 2984 a Fv(J)8 b(F)1663 2958 y FC(k)1701 2984 y Fy(\()p Fv(x)p Fy(\))1853 2923 y Fu(\024)25 b Fv(C)11 b Fy(^)-49 b Fv(\027)6 b Fy(\()p Fv(C)2163 2936 y FC(k)2201 2923 y Fy(\()p Fv(x)p Fy(\)\))p Fv(:)0 3145 y Ft(Pr)l(o)l(of)18 b Fy(:)32 b(Let)g Fv(x)e Fu(2)f Fy(\001)681 3158 y FC(`)742 3145 y Fy(suc)n(h)i(that)h Fv(F)1205 3112 y FC(k)1243 3145 y Fy(\()p Fv(x)p Fy(\))e Fu(2)f Fy(\001)1553 3158 y Fo(0)1589 3145 y Fy(.)47 b(The)32 b(Mark)n(o)n(v)g(prop)r(ert)n(y)g(and)g(the)g(large)f(image)0 3253 y(prop)r(ert)n(y)36 b(\(A.IV\))g(imply)f(that)41 b(^)-49 b Fv(\027)6 b Fy(\()p Fv(F)1259 3220 y FC(k)1297 3253 y Fv(C)1360 3266 y FC(k)1398 3253 y Fy(\()p Fv(x)p Fy(\)\))37 b Fu(\025)f Fv(\021)j(>)e Fy(0.)59 b(The)36 b(b)r(ounded)f(distortion)f(prop)r(ert)n(y)0 3361 y(\(A.I)r(I)r(I\))c (giv)n(es:)754 3593 y Fv(C)823 3556 y FB(\000)p Fo(1)976 3532 y Fy(^)-49 b Fv(\027)6 b Fy(\()p Fv(C)1117 3545 y FC(k)1156 3532 y Fy(\()p Fv(x)p Fy(\)\))p 917 3572 445 4 v 922 3654 a(^)-49 b Fv(\027)6 b Fy(\()p Fv(F)1069 3628 y FC(k)1108 3654 y Fv(C)1171 3667 y FC(k)1209 3654 y Fy(\()p Fv(x)p Fy(\)\))1445 3593 y Fu(\024)1668 3532 y Fy(1)p 1549 3572 283 4 v 1549 3654 a Fv(J)i(F)1675 3628 y FC(k)1713 3654 y Fy(\()p Fv(x)p Fy(\))1865 3593 y Fu(\024)75 b Fv(C)2147 3532 y Fy(^)-49 b Fv(\027)6 b Fy(\()p Fv(C)2288 3545 y FC(k)2326 3532 y Fy(\()p Fv(x)p Fy(\)\))p 2088 3572 445 4 v 2093 3654 a(^)-49 b Fv(\027)6 b Fy(\()p Fv(F)2240 3628 y FC(k)2278 3654 y Fv(C)2341 3667 y FC(k)2380 3654 y Fy(\()p Fv(x)p Fy(\)\))862 3834 y Fv(C)931 3797 y FB(\000)p Fo(1)1030 3774 y Fy(^)-49 b Fv(\027)6 b Fy(\()p Fv(C)1171 3787 y FC(k)1209 3774 y Fy(\()p Fv(x)p Fy(\)\))p 1025 3814 337 4 v 1171 3896 a(1)1445 3834 y Fu(\024)1668 3774 y Fy(1)p 1549 3814 283 4 v 1549 3896 a Fv(J)i(F)1675 3870 y FC(k)1713 3896 y Fy(\()p Fv(x)p Fy(\))1865 3834 y Fu(\024)75 b Fv(C)2093 3774 y Fy(^)-49 b Fv(\027)6 b Fy(\()p Fv(C)2234 3787 y FC(k)2272 3774 y Fy(\()p Fv(x)p Fy(\)\))p 2088 3814 337 4 v 2233 3896 a Fv(\021)0 4147 y Fy(The)29 b(Lemma)h(is)e(pro)n(v)n (ed.)2363 b Fg(\004)0 4338 y Fj(Lemma)34 b(4.2)43 b Fx(There)29 b(exists)g Fv(K)i(<)25 b Fu(1)k Fx(suc)n(h)f(that:)131 4507 y Fu(\017)43 b Fx(for)30 b(all)e Fv(x)c Fu(2)h Fy(\001)p Fx(,)30 b(all)e Fv(n)d Fu(2)g Fr(N)7 b Fx(,)35 b Fu(L)1226 4475 y FC(n)1226 4529 y Fo(0)1268 4507 y Fq(1)p Fy(\()p Fv(x)p Fy(\))26 b Fu(\024)e Fv(K)6 b Fx(.)131 4686 y Fu(\017)43 b Fx(for)30 b(all)e Fv(x;)15 b(y)32 b Fx(in)c(a)i(giv)n(en)e Fv(B)1112 4699 y FC(j;`)1214 4686 y Fx(and)g(all)g Fv(n)e Fu(2)e Fr(N)7 b Fx(,)q(:)1152 4867 y Fu(jL)1239 4830 y FC(n)1239 4890 y Fo(0)1281 4867 y Fq(1)p Fy(\()p Fv(x)p Fy(\))21 b Fu(\000)e(L)1621 4830 y FC(n)1621 4890 y Fo(0)1663 4867 y Fq(1)p Fy(\()p Fv(y)s Fy(\))p Fu(j)26 b(\024)f Fv(K)6 b(d)2100 4880 y Fo(0)2135 4867 y Fy(\()p Fv(x;)15 b(y)s Fy(\))p Fv(:)753 b Fy(\(4.1\))p eop %%Page: 13 13 13 12 bop 832 26 a Fx(Piecewise)28 b(in)n(v)n(ertible)f(maps)i(in)f (higher)g(dimensions)745 b Fy(13)85 302 y Ft(Pr)l(o)l(of)18 b Fy(:)217 411 y(The)29 b(upp)r(er)g(b)r(ound)g Fu(L)981 378 y FC(n)981 433 y Fo(0)1023 411 y Fq(1)d Fu(\024)e Fv(K)36 b Fy(follo)n(ws)27 b(from)i(Lemma)h(4.1,)f(writing:)718 622 y Fu(L)780 585 y FC(n)780 645 y Fo(0)822 622 y Fq(1)p Fy(\()p Fv(x)p Fy(\))d(=)1177 537 y Fs(X)1109 732 y FC(x)1145 715 y Fm(0)1165 732 y FB(2)p FC(F)1256 715 y Fm(\000)p Fn(n)1338 732 y FC(x)1529 562 y Fy(1)p 1397 602 308 4 v 1397 684 a Fv(J)8 b(F)1523 658 y FC(n)1566 684 y Fy(\()p Fv(x)1650 658 y FB(0)1671 684 y Fy(\))1739 622 y Fu(\024)24 b Fv(C)1985 537 y Fs(X)1917 732 y FC(x)1953 715 y Fm(0)1973 732 y FB(2)p FC(F)2064 715 y Fm(\000)p Fn(n)2146 732 y FC(x)2202 622 y Fy(^)-49 b Fv(\027)6 b Fy(\()p Fv(C)2343 635 y FC(n)2385 622 y Fy(\()p Fv(x)2469 585 y FB(0)2490 622 y Fy(\)\))25 b Fu(\024)g Fv(C)459 b Fy(\(4.2\))85 882 y(Let)33 b Fv(x)h Fy(and)e Fv(y)37 b Fy(b)r(elong)32 b(to)i(one)f Fv(B)1212 895 y FC(j;`)1284 882 y Fy(.)52 b(Their)33 b(preimages)f(b)n(y)i Fv(F)2213 849 y FC(n)2288 882 y Fy(are)g(paired,)f(i.e.,)i(if)e Fv(F)3061 849 y FC(n)3103 882 y Fv(x)3153 849 y FB(0)3206 882 y Fy(=)f Fv(x)p Fy(,)85 990 y(there)d(is)g(exactly)g(one)g Fv(y)907 958 y FB(0)953 990 y Fu(2)c Fv(C)1101 1003 y FC(n)1143 990 y Fy(\()p Fv(x)1227 958 y FB(0)1248 990 y Fy(\))k(suc)n(h)f(that)h Fv(F)1768 958 y FC(n)1811 990 y Fv(y)1857 958 y FB(0)1903 990 y Fy(=)c Fv(y)s Fy(.)39 b(So,)30 b(using)d(\(A.IV\),)j(w)n(e)f (get:)547 1241 y Fu(jL)634 1204 y FC(n)634 1263 y Fo(0)676 1241 y Fq(1)p Fy(\()p Fv(x)p Fy(\))21 b Fu(\000)e(L)1016 1204 y FC(n)1016 1263 y Fo(0)1058 1241 y Fq(1)p Fy(\()p Fv(y)s Fy(\))p Fu(j)76 b Fy(=)1467 1084 y Fs(\014)1467 1138 y(\014)1467 1191 y(\014)1467 1245 y(\014)1467 1299 y(\014)1546 1156 y(X)1497 1347 y FC(F)1546 1330 y Fn(n)1584 1347 y FC(x)1620 1330 y Fm(0)1640 1347 y Fo(=)p FC(x)1740 1241 y Fv(J)8 b(F)1866 1204 y FC(n)1908 1241 y Fy(\()p Fv(x)1992 1204 y FB(0)2013 1241 y Fy(\))2047 1204 y FB(\000)p Fo(1)2152 1241 y Fu(\000)2291 1156 y Fs(X)2242 1347 y FC(F)2291 1330 y Fn(n)2329 1347 y FC(x)2365 1330 y Fm(0)2385 1347 y Fo(=)p FC(x)2485 1241 y Fv(J)g(F)2611 1204 y FC(n)2653 1241 y Fy(\()p Fv(y)2733 1204 y FB(0)2755 1241 y Fy(\))2789 1204 y FB(\000)p Fo(1)2874 1084 y Fs(\014)2874 1138 y(\014)2874 1191 y(\014)2874 1245 y(\014)2874 1299 y(\014)1323 1543 y Fy(=)1467 1386 y Fs(\014)1467 1440 y(\014)1467 1493 y(\014)1467 1547 y(\014)1467 1601 y(\014)1546 1458 y(X)1497 1649 y FC(F)1546 1632 y Fn(n)1584 1649 y FC(x)1620 1632 y Fm(0)1640 1649 y Fo(=)p FC(x)1740 1543 y Fv(J)g(F)1866 1506 y FC(n)1908 1543 y Fy(\()p Fv(x)1992 1506 y FB(0)2013 1543 y Fy(\))2047 1506 y FB(\000)p Fo(1)2147 1416 y Fs(\022)2222 1482 y Fv(J)g(F)2348 1450 y FC(n)2390 1482 y Fy(\()p Fv(x)2474 1450 y FB(0)2495 1482 y Fy(\))p 2222 1522 V 2224 1604 a Fv(J)g(F)2350 1578 y FC(n)2392 1604 y Fy(\()p Fv(y)2472 1578 y FB(0)2494 1604 y Fy(\))2558 1543 y Fu(\000)20 b Fy(1)2692 1416 y Fs(\023)2758 1386 y(\014)2758 1440 y(\014)2758 1493 y(\014)2758 1547 y(\014)2758 1601 y(\014)1323 1770 y Fu(\024)74 b(L)1529 1733 y FC(n)1529 1792 y Fo(0)1571 1770 y Fq(1)p Fy(\()p Fv(x)p Fy(\)\()p Fv(e)1814 1733 y FC(d)1847 1741 y Fl(0)1879 1733 y Fo(\()p FC(x;y)r Fo(\))2039 1770 y Fu(\000)19 b Fy(1\))1323 1900 y Fu(\024)74 b Fv(K)6 b(d)1594 1913 y Fo(0)1629 1900 y Fy(\()p Fv(x;)15 b(y)s Fy(\))85 2072 y(\(4.1\))30 b(is)e(pro)n(v)n(ed.)2647 b Fg(\004)85 2297 y Fj(Cor)n(ollar)-6 b(y)34 b(4.3)43 b Fv(F)e Fx(admits)29 b(a)34 b Fy(^)-49 b Fv(\027)35 b Fx(a.c.i.m.)85 2459 y Ft(Pr)l(o)l(of)18 b Fy(:)71 b(By)23 b(Lemma)f(4.2,)j(the)d(sequence)1514 2424 y Fo(1)p 1511 2439 39 4 v 1511 2490 a FC(n)1573 2392 y Fs(P)1668 2416 y FC(n)p FB(\000)p Fo(1)1668 2485 y FC(i)p Fo(=0)1806 2459 y Fu(L)1868 2427 y FC(i)1868 2481 y Fo(0)1903 2459 y Fq(1)h Fy(is)e(relativ)n(ely)g(compact)h(for)g(the)h(top)r(ology)85 2568 y(of)i(uniform)f(con)n(v)n(ergence)g(on)h(compact)f(subsets)g (\(this)g(is)g(Arzela-Ascoli)f(theorem)i(on)g(separable)85 2676 y(spaces\).)78 b(Eac)n(h)42 b(limit)f(p)r(oin)n(t)h Fv(h)g Fy(of)g(this)f(sequence)i(is)e(a)i(non)f(zero)g(\014xed)h(p)r (oin)n(t)e(for)i Fu(L)3176 2689 y Fo(0)3253 2676 y Fy(\(b)n(y)85 2784 y(Leb)r(esgue's)29 b(dominated)f(theorem,)35 b(^)-49 b Fv(\027)6 b Fy(\()p Fv(h)p Fy(\))24 b(=)i(1\),)j(so)g(that)36 b(^)-51 b Fv(\026)25 b Fy(=)g Fv(h)5 b Fy(^)-49 b Fv(\027)35 b Fy(is)28 b(a)35 b(^)-49 b Fv(\027)6 b Fy(-a.c.i.m.)455 b Fg(\004)122 2930 y Fy(The)37 b(system)f(\(\001)p Fv(;)15 b(F)r(;)21 b Fy(^)-50 b Fv(\027)7 b(;)15 b Fu(P)7 b Fy(\))38 b(has)e(a)h(Mark)n(o)n(v)g(structure)f(in)g(the)h(sense)e(that)i(for)g (eac)n(h)f Fv(P)49 b Fu(2)37 b(P)7 b Fy(,)85 3038 y Fv(F)12 b Fy(\()p Fv(P)g Fy(\))29 b(is)g(a)g(union)f(of)h(atom)g(of)g Fu(P)7 b Fy(.)40 b(According)28 b(to)i([ADU],)g(w)n(e)f(will)f(sa)n(y)g (that)i Fv(F)41 b Fy(is)28 b Ft(ap)l(erio)l(dic)g Fy(if:)573 3210 y Fu(8)g Fv(P)12 b(;)45 b(P)859 3173 y FB(0)904 3210 y Fu(2)25 b(P)37 b(9)p Fv(N)e Fu(2)24 b Fr(N)42 b Fy(suc)n(h)28 b(that)34 b(^)-49 b Fv(\027)7 b Fy(\()p Fv(F)1962 3173 y FB(\000)p FC(n)2053 3210 y Fv(P)32 b Fu(\\)20 b Fv(P)2291 3173 y FB(0)2312 3210 y Fy(\))25 b Fv(>)g Fy(0)k Fu(8)f Fv(n)d Fu(\025)g Fv(N)9 b(:)309 b Fy(\(4.3\))85 3382 y(The)32 b(existence)e(of)i(a)k(^)-49 b Fv(\027)6 b Fy(-a.c.i.m.)47 b(implies)29 b(that)i(the)h(recuren)n(t)f (part)g(is)g(non)g(empt)n(y)g(\(it)g(con)n(tains)85 3491 y(the)f(supp)r(ort)f(of)36 b(^)-50 b Fv(\026)o Fy(\).)41 b(Up)30 b(to)f(decomp)r(osition)f(in)n(to)h(ap)r(erio)r(dic)g(comp)r (onen)n(ts,)g Ft(we)k(may)e(and)g(shal)t(l)85 3599 y(assume)h(that)h Fv(F)45 b Ft(is)32 b(ap)l(erio)l(dic)p Fy(.)43 b(Remark)31 b(that)f(ap)r(erio)r(dicit)n(y)g(implies)e(that)j(an)n(y)f Fv(s)p Fy(-cylinder)g(has)85 3707 y(p)r(ositiv)n(e)25 b(^)-49 b Fv(\027)6 b Fy(-measure.)37 b(The)22 b(follo)n(wing)d(lemma)i (implies)e(that)j(an)n(y)f Fv(s)p Fy(-cylinder)g(also)f(has)28 b(^)-51 b Fv(\026)22 b Fy(p)r(ositiv)n(e)85 3816 y(measure.)85 4003 y Fj(Lemma)34 b(4.4)43 b Fx(Ap)r(erio)r(dicit)n(y)d(of)i Fv(F)54 b Fx(implies)40 b(that)i Fy(inf)1952 4016 y FC(n)p FB(2)p Fi(N)2095 4003 y Fy(inf)2201 4016 y FC(x)p FB(2)p Fo(\001)2351 4003 y Fu(L)2413 3970 y FC(n)2413 4025 y Fo(0)2455 4003 y Fq(1)p Fy(\()p Fv(x)p Fy(\))47 b Fv(>)g Fy(0)p Fx(.)77 b(So)42 b(w)n(e)g(ha)n(v)n(e)85 4111 y Fy(0)25 b Fv(<)g Fy(inf)20 b Fv(h)25 b Fu(\024)f Fy(sup)14 b Fv(h)25 b(<)g Fu(1)p Fx(.)85 4274 y Ft(Pr)l(o)l(of)18 b Fy(:)78 b(Let)30 b Fv(K)35 b Fy(b)r(e)30 b(giv)n(en)e(b)n(y)h(Lemma)g (4.2.)40 b(Remark)29 b(that)g(for)h Fv(n)p Fy(,)g Fv(k)d Fu(2)e Fr(N)7 b Fy(,)35 b Fv(x)25 b Fu(2)g Fy(\001,)1305 4446 y Fu(L)1367 4409 y FC(n)p Fo(+)p FC(k)1367 4468 y Fo(0)1494 4446 y Fq(1)p Fy(\()p Fv(x)p Fy(\))g Fu(\024)g Fv(K)6 b Fu(L)1925 4409 y FC(n)1925 4468 y Fo(0)1967 4446 y Fq(1)p Fy(\()p Fv(x)p Fy(\))p Fv(:)1041 b Fy(\(4.4\))85 4618 y(Moreo)n(v)n(er,)30 b(w)n(e)f(ha)n(v)n(e)f(for)i Fv(j)g Fy(=)25 b(1)p Fv(;)15 b(:)g(:)g(:)h(;)f(p)p Fy(,)30 b(an)n(y)f Fv(n)c Fu(2)g Fr(N)7 b Fy(,)35 b Fv(x;)15 b(y)29 b Fu(2)24 b Fv(B)2216 4631 y FC(j)2249 4618 y Fy(,)451 4808 y Fu(L)513 4771 y FC(n)513 4830 y Fo(0)555 4808 y Fq(1)p Fy(\()p Fv(x)p Fy(\))i(=)891 4723 y Fs(X)842 4914 y FC(F)891 4897 y Fn(n)929 4914 y FC(x)965 4897 y Fm(0)985 4914 y Fo(=)p FC(x)1085 4808 y Fv(J)8 b(F)k Fy(\()p Fv(x)1295 4771 y FB(0)1316 4808 y Fy(\))1350 4771 y FB(\000)p Fo(1)1460 4808 y Fu(\024)25 b Fv(K)6 b(d)1682 4821 y Fo(0)1717 4808 y Fy(\()p Fv(x;)15 b(y)s Fy(\))1983 4723 y Fs(X)1935 4914 y FC(F)1984 4897 y Fn(n)2023 4914 y FC(y)2056 4897 y Fm(0)2076 4914 y Fo(=)p FC(y)2174 4808 y Fv(J)8 b(F)k Fy(\()p Fv(y)2380 4771 y FB(0)2401 4808 y Fy(\))2435 4771 y FB(\000)p Fo(1)2545 4808 y Fu(\024)25 b Fv(K)6 b Fu(L)2783 4771 y FC(n)2783 4830 y Fo(0)2825 4808 y Fq(1)p Fy(\()p Fv(y)s Fy(\))p Fv(:)187 b Fy(\(4.5\))p eop %%Page: 14 14 14 13 bop 0 26 a Fy(14)745 b Fx(J)n(\023)-41 b(er^)d(ome)29 b(Buzzi)g(&)g(V)n(\023)-41 b(eronique)28 b(Maume-Desc)n(hamps)0 302 y Fy(So,)33 b(for)f(eac)n(h)f Fv(j)5 b Fy(,)33 b(either)f(sup)969 324 y FC(B)1017 333 y Fn(j)1065 302 y Fu(L)1127 270 y FC(n)1127 325 y Fo(0)1169 302 y Fq(1)e Fu(!)f Fy(0)j(or)g Fu(L)1616 270 y FC(n)1616 325 y Fo(0)1658 302 y Fq(1)p Fy(\()p Fv(x)p Fy(\))e Fu(\025)f Fv(\013)k Fy(for)f(an)n(y)f Fv(x)f Fu(2)f Fv(B)2585 315 y FC(j)2618 302 y Fy(,)k Fv(n)d Fu(2)f Fr(N)7 b Fy(,)39 b(for)32 b(some)0 411 y Fv(\013)27 b(>)h Fy(0.)43 b(On)30 b(the)h(one)f(hand,)g(assume)g Fu(L)1364 378 y FC(n)1364 433 y Fo(0)1406 411 y Fq(1)p Fy(\()p Fv(x)p Fy(\))d Fu(\025)g Fv(\013)k Fy(for)f(an)n(y)g Fv(x)d Fu(2)g Fv(B)2318 424 y FC(j)2351 411 y Fy(,)k Fv(n)c Fu(2)g Fr(N)7 b Fy(,)37 b(for)30 b(some)g Fv(\013)e(>)f Fy(0,)0 519 y(let)32 b(us)h(consider)e Fu(L)654 486 y FC(n)654 541 y Fo(0)696 519 y Fq(1)p Fy(\()p Fv(x)p Fy(\))i(for)g Fv(x)e Fu(2)g Fv(B)1274 532 y FC(q)1308 519 y Fy(.)50 b(The)33 b(ap)r(erio)r(dicit)n(y)e(implies)f(that)j(there)g(exists)f Fv(n)3073 532 y Fo(0)3140 519 y Fu(2)f Fr(N)0 627 y Fy(suc)n(h)i(that)h (for)g Fv(n)g Fu(\025)e Fv(n)777 640 y Fo(0)813 627 y Fy(,)k Fv(C)936 640 y FC(n)978 627 y Fy(\()p Fv(z)t Fy(\))d Fu(\032)g Fv(F)1296 595 y FB(\000)p FC(n)1388 627 y Fv(B)1455 640 y FC(j)1511 627 y Fu(\\)23 b Fv(B)1661 640 y FC(q)1729 627 y Fy(for)34 b(some)g Fv(z)j Fu(2)c Fv(B)2331 640 y FC(q)2399 627 y Fy(i.e.)53 b Fv(F)2633 595 y FC(n)2676 627 y Fv(z)2717 640 y Fo(0)2785 627 y Fy(=)34 b Fv(z)k Fy(for)c(some)0 736 y Fv(z)41 749 y Fo(0)101 736 y Fu(2)25 b Fv(B)253 749 y FC(j)285 736 y Fy(.)40 b(W)-7 b(e)29 b(ha)n(v)n(e:)949 932 y Fu(L)1011 890 y FC(n)1050 898 y Fl(0)1080 890 y Fo(+)p FC(p)1011 954 y Fo(0)1165 932 y Fq(1)p Fy(\()p Fv(z)t Fy(\))76 b(=)1609 847 y Fs(X)1548 1038 y FC(F)1597 1018 y Fn(n)1632 1031 y Fl(0)1666 1038 y FC(z)1699 1021 y Fm(0)1719 1038 y Fo(=)p FC(z)1815 932 y Fv(g)1857 945 y FC(n)1896 953 y Fl(0)1930 932 y Fy(\()p Fv(z)2009 895 y FB(0)2030 932 y Fy(\))p Fu(L)2126 890 y FC(p)2126 954 y Fo(0)2162 932 y Fq(1)p Fy(\()p Fv(z)2291 895 y FB(0)2312 932 y Fy(\))1404 1147 y Fu(\025)e Fv(g)1590 1160 y FC(n)1629 1168 y Fl(0)1663 1147 y Fy(\()p Fv(z)1738 1160 y Fo(0)1774 1147 y Fy(\))p Fv(\013)25 b Fy(:=)h Fv(\013)2064 1110 y FB(0)0 1325 y Fy(Using)35 b(\(4.4\))j(and)e(\(4.5\),)k(w)n(e)c(get)h Fu(L)1246 1293 y FC(n)1246 1347 y Fo(0)1289 1325 y Fq(1)p Fy(\()p Fv(x)p Fy(\))h Fu(\025)g Fv(\013)1660 1293 y FB(0)1680 1325 y Fv(=K)43 b Fy(for)37 b(an)n(y)f Fv(n)i Fu(2)g Fr(N)7 b Fy(,)45 b Fv(x)38 b Fu(2)f Fv(B)2725 1338 y FC(q)2759 1325 y Fy(.)62 b(This)36 b(pro)n(v)n(es)0 1434 y(that)44 b(either)f(sup)600 1455 y FC(B)648 1464 y Fn(j)696 1434 y Fu(L)758 1401 y FC(n)758 1456 y Fo(0)800 1434 y Fq(1)50 b Fu(!)f Fy(0)43 b(for)h(all)e Fv(j)55 b Fy(=)50 b(1)p Fv(;)15 b(:)g(:)g(:)h(;)f(p)44 b Fy(or)g(there)f(exists)g Fv(K)2646 1401 y FB(0)2717 1434 y Fv(>)49 b Fy(0)44 b(suc)n(h)e(that)0 1542 y(inf)106 1555 y FC(j)s Fo(=1)p FC(;:::)o(;p)355 1542 y Fy(inf)462 1555 y FC(B)510 1564 y Fn(j)557 1542 y Fu(L)619 1509 y FC(n)619 1564 y Fo(0)662 1542 y Fq(1)25 b Fu(\025)g Fv(K)913 1509 y FB(0)963 1542 y Fy(for)k(an)n(y)g Fv(n)c Fu(2)g Fr(N)7 b Fy(.)0 1650 y(Assume)38 b(sup)467 1671 y FC(B)515 1680 y Fn(j)563 1650 y Fu(L)625 1618 y FC(n)625 1672 y Fo(0)667 1650 y Fq(1)43 b Fu(!)e Fy(0)e(for)g(all)f Fv(j)47 b Fy(=)42 b(1)p Fv(;)15 b(:)g(:)g(:)h(;)f(p)p Fy(,)43 b(let)c Fv(")g Fy(b)r(e)g(\014xed,)j(let)d Fv(n)2578 1663 y Fo(0)2653 1650 y Fy(b)r(e)g(suc)n(h)f(that)h(for)0 1759 y Fv(n)34 b Fu(\025)f Fv(n)241 1772 y Fo(0)277 1759 y Fy(,)j Fu(L)399 1726 y FC(n)399 1781 y Fo(0)441 1759 y Fq(1)p Fy(\()p Fv(x)p Fy(\))e Fu(\024)g Fv(")g Fy(for)g(all)f Fv(x)h Fu(2)f Fy(\001)1338 1772 y Fo(0)1407 1759 y Fy(=)1509 1691 y Fs(S)1584 1785 y FC(j)s Fo(=1)p FC(;:::)o(;p)1833 1759 y Fv(B)1900 1772 y FC(j)1933 1759 y Fy(,)j(then)e Fu(L)2260 1726 y FC(n)2260 1781 y Fo(0)2302 1759 y Fq(1)p Fy(\()p Fv(x)p Fy(\))g Fv(<)g(")g Fy(for)g(an)n(y)g Fv(x)g Fu(2)f Fy(\001)3241 1772 y FC(`)3271 1759 y Fy(,)0 1867 y(pro)n(vided)28 b Fv(n)20 b Fu(\000)g Fv(`)25 b Fu(\025)f Fv(n)731 1880 y Fo(0)767 1867 y Fy(,)30 b(c)n(ho)r(ose)e Fv(n)e Fu(\025)e Fv(n)1323 1880 y Fo(0)1389 1867 y Fy(so)k(that)35 b(^)-49 b Fv(\027)6 b Fy(\()1770 1800 y Fs(S)1844 1893 y FC(`>n)p FB(\000)p FC(n)2046 1901 y Fl(0)2096 1867 y Fy(\001)2169 1880 y FC(`)2199 1867 y Fy(\))25 b Fv(<)g(")p Fy(,)30 b(w)n(e)f(ha)n(v)n(e:)624 1978 y Fs(Z)674 2181 y Fo(\001)745 2101 y Fu(L)807 2063 y FC(n)807 2123 y Fo(0)849 2101 y Fq(1)p Fv(d)5 b Fy(^)-49 b Fv(\027)82 b Fy(=)1214 2015 y Fs(X)1265 2206 y FC(`)1358 1978 y Fs(Z)1408 2181 y Fo(\001)1461 2192 y Fn(`)1507 2101 y Fu(L)1569 2063 y FC(n)1569 2123 y Fo(0)1611 2101 y Fq(1)p Fv(d)5 b Fy(^)-49 b Fv(\027)1070 2367 y Fy(=)1265 2282 y Fs(X)1214 2473 y FC(`)p FB(\024)p FC(n)p FB(\000)p FC(n)1416 2481 y Fl(0)1462 2245 y Fs(Z)1511 2448 y Fo(\001)1564 2459 y Fn(`)1610 2367 y Fu(L)1672 2330 y FC(n)1672 2389 y Fo(0)1714 2367 y Fq(1)p Fv(d)5 b Fy(^)-49 b Fv(\027)55 b Fy(+)2078 2282 y Fs(X)2026 2473 y FC(`>n)p FB(\000)p FC(n)2228 2481 y Fl(0)2274 2245 y Fs(Z)2324 2448 y Fo(\001)2377 2459 y Fn(`)2422 2367 y Fu(L)2484 2330 y FC(n)2484 2389 y Fo(0)2526 2367 y Fq(1)p Fv(d)5 b Fy(^)-49 b Fv(\027)1069 2589 y Fu(\024)75 b Fv(")p Fy(\(1)20 b(+)g Fv(K)6 b Fy(\))p Fv(:)0 2767 y Fy(But,)30 b(since)j(^)-49 b Fv(\027)35 b Fy(is)28 b(conformal,)34 b(^)-49 b Fv(\027)6 b Fy(\()p Fu(L)1157 2735 y FC(n)1157 2789 y Fo(0)1199 2767 y Fq(1)p Fy(\))26 b(=)f(1,)30 b(w)n(e)f(are)g(lead)g(to)g(a)g(con)n(tradiction.) 554 b Fg(\004)27 2915 y Fy(Theorems)26 b(2.5)h(and)f(3.2)h(in)e([ADU])i (imply)f(that)h(\()p Fv(F)r(;)22 b Fy(^)-51 b Fv(\026)p Fy(\))27 b(is)e(then)i(exact)g(hence)f(mixing)f(and)h(that)7 3023 y(^)-51 b Fv(\026)29 b Fy(is)f(the)i(only)j(^)-49 b Fv(\027)6 b Fy(-a.c.i.m.)41 b(The)29 b(follo)n(wing)e(lemma)i(is)f(a) h(direct)g(consequence)f(of)h(mixing.)0 3212 y Fj(Lemma)34 b(4.5)43 b Fx(There)33 b(exists)g(p)r(ositiv)n(e)e(n)n(um)n(b)r(ers)h Fv(A)i Fx(and)e Fv(B)38 b Fx(suc)n(h)32 b(that)h(for)g(an)n(y)g Fv(f)10 b Fx(,)34 b Fv(g)h Fu(2)c Fv(L)3118 3180 y Fo(2)3154 3212 y Fy(\()5 b(^)-49 b Fv(\027)6 b Fy(\))p Fx(,)0 3321 y(with)36 b Fy(^)-51 b Fv(\026)p Fy(\()p Fv(f)10 b Fy(\))25 b Fv(>)g Fy(0)p Fx(,)36 b Fy(^)-50 b Fv(\026)o Fy(\()p Fv(g)s Fy(\))26 b Fv(>)f Fy(0)p Fx(,)30 b(there)f(exists)f Fv(n)1490 3334 y Fo(0)1556 3321 y Fx(suc)n(h)g(that)h(for)g Fv(n)c Fu(\025)g Fv(n)2302 3334 y Fo(0)2338 3321 y Fx(,)1198 3548 y Fv(B)30 b Fu(\024)1405 3487 y Fy(^)-51 b Fv(\026)p Fy(\()p Fv(f)30 b Fu(\016)20 b Fv(F)1692 3455 y FC(n)1754 3487 y Fu(\001)g Fv(g)s Fy(\))p 1398 3527 480 4 v 1475 3609 a(^)-51 b Fv(\026)p Fy(\()p Fv(f)10 b Fy(\))d(^)-51 b Fv(\026)p Fy(\()p Fv(g)s Fy(\))1912 3548 y Fu(\024)25 b Fv(A:)0 3837 y Fp(4.2)120 b(De\014nition)42 b(of)e(the)g(cones)0 4003 y Fy(Let)32 b Fu(L)g Fy(b)r(e)h(the)f(normalized)e(transfer)i(op)r (erator:)46 b Fu(L)p Fv(f)39 b Fy(=)1958 3968 y Fo(1)p 1955 3983 37 4 v 1955 4034 a FC(h)2001 4003 y Fu(L)2063 4016 y Fo(0)2098 4003 y Fy(\()p Fv(f)10 b(h)p Fy(\),)33 b(it)f(satis\014es:)43 b Fu(L)p Fq(1)30 b Fy(=)h Fq(1)p Fy(.)48 b(The)0 4111 y(Jacobian)28 b(of)h Fv(F)41 b Fy(with)29 b(resp)r(ect)h(to)36 b(^)-51 b Fv(\026)29 b Fy(is:)1450 4268 y Fv(J)8 b(F)32 b Fu(\001)20 b Fv(h)f Fu(\016)h Fv(F)p 1450 4308 395 4 v 1622 4390 a(h)0 4529 y Fy(>F)-7 b(rom)34 b(no)n(w)e(one,)j Fv(J)8 b(F)45 b Fy(will)31 b(b)r(e)j(the)f(Jacobian)e(of)i Fv(F)45 b Fy(with)33 b(resp)r(ect)g(to)h(the)f(in)n(v)-5 b(arian)n(t)31 b(measure)7 4638 y(^)-51 b Fv(\026)p Fy(.)39 b(Remark)30 b(that)f(the)g(pro)r(of)g (of)g(Lemma)h(4.1)f(and)g(Lemma)g(4.4)h(giv)n(es)e(for)h(some)g Fv(C)i(>)25 b Fy(0:)926 4851 y Fv(C)995 4814 y FB(\000)p Fo(1)1087 4851 y Fy(^)-51 b Fv(\026)p Fy(\()p Fv(C)1230 4864 y FC(k)1268 4851 y Fy(\()p Fv(x)p Fy(\)\))25 b Fu(\024)1668 4790 y Fy(1)p 1549 4830 283 4 v 1549 4912 a Fv(J)8 b(F)1675 4886 y FC(k)1713 4912 y Fy(\()p Fv(x)p Fy(\))1865 4851 y Fu(\024)25 b Fv(C)13 b Fy(^)-51 b Fv(\026)p Fy(\()p Fv(C)2179 4864 y FC(k)2217 4851 y Fy(\()p Fv(x)p Fy(\)\))746 b(\(4.6\))p eop %%Page: 15 15 15 14 bop 832 26 a Fx(Piecewise)28 b(in)n(v)n(ertible)f(maps)i(in)f (higher)g(dimensions)745 b Fy(15)85 302 y(F)-7 b(or)26 b(tec)n(hnical)d(reasons,)j(w)n(e)f(need)g(to)h(in)n(tro)r(duce)e Fv(\015)30 b(<)25 b(\015)1969 270 y FB(0)2014 302 y Fv(<)g Fy(1)h(\(recall)e(that)i Fv(\015)j Fy(is)c(giv)n(en)f(b)n(y)i (\(A.I\)\).)85 411 y(W)-7 b(e)29 b(shall)f(denote)g(b)n(y)36 b(^)-51 b Fv(\026)899 424 y FC(\015)968 411 y Fy(the)30 b(\(\014nite\))e(measure)h(de\014ned)g(b)n(y)35 b(^)-50 b Fv(\026)2232 424 y FC(\015)2271 411 y Fy(\()p Fv(A)p Fy(\))25 b(=)2524 343 y Fs(P)2618 437 y FC(`)2670 411 y Fy(^)-51 b Fv(\026)p Fy(\()p Fv(A)20 b Fu(\\)f Fy(\001)2988 424 y FC(`)3018 411 y Fy(\))h Fu(\001)g Fv(\015)3168 378 y FB(0\000)p FC(`)3264 411 y Fy(.)85 519 y(Let)36 b Fv(s)h Fy(and)f Fv(t)g Fy(b)r(e)h(t)n(w)n(o)e(in)n(tegers)g(whic)n(h) h(will)f(b)r(e)h(\014xed)g(later)g(\(see)h(the)f(pro)r(of)g(of)g(Prop)r (osisition)85 627 y(4.9\).)k(Let)29 b Fu(Q)c Fy(=)g Fu(f)p Fv(P)745 640 y FB(1)812 627 y Fu(g)20 b([)g(Q)1030 640 y Fo(1)1094 627 y Fy(b)r(e)30 b(the)f(\014nite)g(partition)e(of)i(\001) h(de\014ned)e(b)n(y:)216 811 y Fu(\017)44 b Fv(P)362 824 y FB(1)453 811 y Fy(=)25 b Fu(f)p Fy(\()p Fv(x;)15 b(`)p Fy(\))26 b Fu(2)e Fy(\001)30 b Fv(=)1090 788 y Fy(^)1071 811 y Fv(R)p Fy(\()p Fv(x)p Fy(\))25 b Fu(\025)g Fv(t)p Fu(g)216 995 y(\017)44 b(Q)378 1008 y Fo(1)443 995 y Fy(is)28 b(the)h(partition)f(of)h(\001)20 b Fu(n)g Fv(P)1363 1008 y FB(1)1459 995 y Fy(in)n(to)28 b Fv(s)p Fy(-cylinders.)85 1193 y Fq(Remark)33 b(4.6)45 b Fx(The)29 b Fy(^)-51 b Fv(\026)866 1206 y FC(\015)903 1190 y Fm(0)926 1193 y Fx(-measure)22 b(of)g Fv(P)1440 1206 y FB(1)1528 1193 y Fx(is)f(as)h(small)e(as)i(w)n(e)f(w)n(an)n(t,)i(pro)n(vided)e Fv(t)h Fx(is)f(large)g(enough.)85 1302 y(W)-7 b(e)31 b(ma)n(y)g(c)n(ho)r(ose)g Fv(s)g Fx(to)g(get)h Fv(R)1108 1315 y Fo(0)1143 1302 y Fy(\()p Fv(s)p Fy(\))d(:=)1401 1235 y Fs(P)1496 1328 y FC(k)r FB(\025)p FC(s)p Fo(+1)1709 1302 y Fv(!)1764 1315 y FC(k)1802 1302 y Fy(\()p Fv(g)s Fy(\))j Fx(as)f(small)e(as)i(w)n(e)g(w)n(an)n(t.)45 b Fv(R)2851 1315 y Fo(0)2886 1302 y Fy(\()p Fv(s)p Fy(\))32 b Fx(is)e(the)h Fv(d)3316 1315 y Fo(0)3351 1302 y Fx(-)85 1410 y(diameter)e(of)g(the)g(partition)f Fu(Q)1141 1423 y Fo(1)1177 1410 y Fx(.)85 1579 y Fy(The)g(follo)n(wing)d(lemma)i (directly)g(follo)n(ws)f(from)h(Lemmas)h(4.4)g(and)f(4.5)g(applied)f (to)i(the)f(\(\014nitely)85 1687 y(man)n(y\))i(functions)f Fq(1)787 1700 y FC(P)869 1687 y Fy(and)h Fq(1)1090 1700 y FC(P)1163 1687 y Fu(\001)20 b Fv( )32 b Fy(with)d Fv(P)37 b Fu(2)25 b(Q)p Fy(,)30 b Fv( )s Fy(\()p Fv(x)p Fy(\))25 b(=)g Fv(\015)2150 1655 y FB(0\000)p FC(`)2275 1687 y Fy(for)k Fv(x)c Fu(2)g Fy(\001)2642 1700 y FC(`)2672 1687 y Fy(.)85 1886 y Fj(Lemma)34 b(4.7)43 b Fx(There)35 b(exist)f Fv(A)h(>)g Fy(0)f Fx(and)h Fv(B)k(>)34 b Fy(0)h Fx(suc)n(h)f(that)h(for)f(all)g Fv(s)h Fx(and)g Fv(t)p Fx(,)h(there)f(exists)f Fv(k)3348 1853 y FB(0)3345 1908 y Fo(0)85 1994 y Fx(suc)n(h)28 b(that)h(for)g Fv(k)f Fu(\025)d Fv(k)825 1962 y FB(0)822 2016 y Fo(0)886 1994 y Fx(and)k Fv(P)r(;)15 b(P)1224 1962 y FB(0)1270 1994 y Fu(2)24 b(Q)p Fx(,)655 2249 y Fv(B)30 b Fu(\024)862 2188 y Fy(^)-51 b Fv(\026)p Fy(\()p Fv(F)1011 2155 y FB(\000)p FC(k)1099 2188 y Fv(P)32 b Fu(\\)19 b Fv(P)1336 2155 y FB(0)1357 2188 y Fy(\))p 855 2228 537 4 v 930 2310 a(^)-51 b Fv(\026)p Fy(\()p Fv(P)12 b Fy(\))7 b(^)-51 b Fv(\026)p Fy(\()p Fv(P)1269 2284 y FB(0)1289 2310 y Fy(\))1425 2249 y Fu(\024)25 b Fv(A)k Fx(and)g Fv(B)g Fu(\024)1992 2188 y Fy(^)-51 b Fv(\026)2038 2201 y FC(\015)2075 2184 y Fm(0)2098 2188 y Fy(\()p Fv(F)2201 2155 y FB(\000)p FC(k)2289 2188 y Fv(P)32 b Fu(\\)20 b Fv(P)2515 2201 y FB(1)2581 2188 y Fy(\))p 1985 2228 631 4 v 2060 2310 a(^)-51 b Fv(\026)2106 2323 y FC(\015)2143 2307 y Fm(0)2166 2310 y Fy(\()p Fv(P)2257 2323 y FB(1)2324 2310 y Fy(\))7 b(^)-51 b Fv(\026)p Fy(\()p Fv(P)12 b Fy(\))2649 2249 y Fu(\024)25 b Fv(A)217 2495 y Fy(Let)f(us)g(de\014ne)g(inductiv)n(ely) f(a)i(sequence)f Fv(d)1647 2508 y FC(j)1704 2495 y Fy(of)g(metrics)g (and)h(a)f(sequence)g Fv(k)2742 2508 y FC(j)2799 2495 y Fy(of)h(in)n(tegers.)36 b(Let)85 2603 y Fv(k)131 2616 y Fo(0)198 2603 y Fu(\025)d Fv(k)350 2571 y FB(0)347 2626 y Fo(0)415 2603 y Fy(and)g Fv(D)i(>)e Fy(1)g(to)h(b)r(e)f(c)n (hosen)g(later)g(\(see)g(the)h(pro)r(of)f(of)h(Prop)r(osition)e(4.9\).) 52 b(F)-7 b(or)34 b Fv(`)e Fu(\025)g Fy(1,)85 2712 y(de\014ne:)900 2838 y Fv(k)946 2851 y FC(`)1001 2838 y Fy(=)25 b(inf)5 b Fu(f)p Fv(k)27 b Fu(\025)e Fv(k)1459 2851 y Fo(0)1548 2838 y Fy(:)55 b Fv(R)1694 2851 y FC(`)p FB(\000)p Fo(1)1805 2838 y Fy(\()p Fv(s)20 b Fy(+)g Fv(k)s Fy(\))25 b Fu(\024)2215 2777 y Fy(1)p 2200 2817 76 4 v 2200 2900 a Fv(D)2284 2838 y(R)2351 2851 y Fo(0)2386 2838 y Fy(\()p Fv(s)p Fy(\))p Fu(g)p Fv(:)85 3018 y Fy(F)-7 b(or)29 b Fv(x;)45 b(y)28 b Fu(2)c Fy(\001,)30 b(let)861 3217 y Fv(d)907 3230 y FC(`)936 3217 y Fy(\()p Fv(x;)15 b(y)s Fy(\))26 b(=)f Fv(D)r Fy([)p Fv(R)1424 3230 y FC(`)p FB(\000)p Fo(1)1535 3217 y Fy(\()p Fv(k)1615 3230 y FC(`)1665 3217 y Fy(+)20 b Fv(s)p Fy(\()p Fv(x;)15 b(y)s Fy(\)\))21 b(+)f Fv(R)2207 3230 y Fo(0)2242 3217 y Fy(\()p Fv(s)p Fy(\()p Fv(x;)15 b(y)s Fy(\)\)])p Fv(:)85 3415 y Fy(De\014ne)29 b Fv(R)424 3428 y FC(`)453 3415 y Fy(\()p Fv(n)p Fy(\))d(=)f(sup)824 3436 y FC(s)p Fo(\()p FC(x;y)r Fo(\))p FB(\025)p FC(n)1097 3415 y Fv(d)1143 3428 y FC(`)1172 3415 y Fy(\()p Fv(x;)15 b(y)s Fy(\))30 b(for)f Fv(`)c Fu(\025)f Fy(0.)40 b(The)29 b Fv(d)2029 3428 y FC(`)2058 3415 y Fy(-diameter)h(of)e Fu(Q)2626 3428 y Fo(1)2691 3415 y Fy(is)h Fv(R)2847 3428 y FC(`)2876 3415 y Fy(\()p Fv(s)p Fy(\).)217 3523 y(The)e(follo)n(wing) e(lemma)i(follo)n(ws)e(directly)h(from)i(the)f(ab)r(o)n(v)n(e)g (de\014nitions.)36 b(It)27 b(pro)n(v)n(es)f(that)h(the)85 3632 y Fv(d)131 3645 y FC(`)160 3632 y Fy(-diameter)i(of)g(the)h (partition)d Fu(Q)1245 3645 y Fo(1)1310 3632 y Fy(is)i(uniformly)f(b)r (ounded)g(\(in)h Fv(`)p Fy(\).)85 3831 y Fj(Lemma)34 b(4.8)43 b Fx(F)-7 b(or)29 b(all)f Fv(`)d Fu(2)f Fr(N)7 b Fx(,)36 b Fv(R)1202 3844 y FC(`)1231 3831 y Fy(\()p Fv(s)p Fy(\))26 b Fu(\024)f Fv(R)1528 3844 y Fo(0)1563 3831 y Fy(\()p Fv(s)p Fy(\)\()p Fv(D)e Fy(+)d(1\))p Fx(.)217 3999 y Fy(W)-7 b(e)36 b(are)h(no)n(w)g(in)f(p)r(osition)f(to)i (de\014ne)f(our)h(cones.)61 b(They)37 b(will)f(dep)r(end)g(on)h(three)g (p)r(ositiv)n(e)85 4108 y(n)n(um)n(b)r(ers)28 b Fv(a)p Fy(,)i Fv(b)p Fy(,)g Fv(c)f Fy(whic)n(h)f(will)g(b)r(e)h(\014xed)h (later)e(\(see)i(the)f(pro)r(of)g(of)g(Prop)r(osition)e(4.9\).)217 4216 y(F)-7 b(or)29 b Fv(j)h Fu(2)25 b Fr(N)7 b Fy(,)35 b Fv(C)706 4229 y FC(j)739 4216 y Fy(\()p Fv(a;)15 b(b;)g(c)p Fy(\))30 b(is)f(the)g(cone)g(of)g(functions)e Fv(f)40 b Fy(that)29 b(satisfy:)192 4446 y(1.)45 b Fu(8)p Fv(P)35 b Fu(2)25 b(Q)p Fv(;)45 b Fy(0)24 b Fu(\024)920 4385 y Fy(1)p 847 4425 190 4 v 854 4507 a(^)-51 b Fv(\026)p Fy(\()p Fv(P)12 b Fy(\))1061 4324 y Fs(Z)1061 4579 y FC(P)1165 4446 y Fv(f)e(d)d Fy(^)-51 b Fv(\026)25 b Fu(\024)f Fv(a)1497 4324 y Fs(Z)1495 4579 y Fo(\001)1602 4446 y Fv(f)10 b(d)d Fy(^)-51 b Fv(\026)o Fy(.)192 4793 y(2.)45 b(for)29 b(all)f Fv(x;)44 b(y)32 b Fy(in)d(the)g(same)g Fv(B)1295 4806 y FC(q)r(;`)1372 4793 y Fy(,)h Fu(j)p Fv(f)10 b Fy(\()p Fv(x)p Fy(\))20 b Fu(\000)g Fv(f)10 b Fy(\()p Fv(y)s Fy(\))p Fu(j)25 b(\024)f Fv(d)2089 4806 y FC(j)2122 4793 y Fy(\()p Fv(x;)15 b(y)s Fy(\))20 b Fu(\001)g Fv(b)2472 4671 y Fs(Z)2470 4926 y Fo(\001)2576 4793 y Fv(f)10 b(d)d Fy(^)-51 b Fv(\026)o Fy(.)p eop %%Page: 16 16 16 15 bop 0 26 a Fy(16)745 b Fx(J)n(\023)-41 b(er^)d(ome)29 b(Buzzi)g(&)g(V)n(\023)-41 b(eronique)28 b(Maume-Desc)n(hamps)107 335 y Fy(3.)89 b(sup)219 408 y 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1722 y Fy(Before)27 b(pro)n(ving)e(Prop)r(osition)g(4.9,)j(let)e(us)g (state)g(t)n(w)n(o)h(prop)r(erties)f(of)g(the)h(cones)f Fv(C)2781 1735 y FC(j)2813 1722 y Fy(\()p Fv(a;)15 b(b;)g(c)p Fy(\).)40 b(The)0 1830 y(\014rst)27 b(one)g(is)g(a)g(b)r(ound)g(on)g (the)g(suprem)n(um)g(on)g(\001)1653 1843 y FC(`)1710 1830 y Fy(of)h(functions)d(in)i(the)g(cones.)38 b(The)28 b(second)f(one)0 1939 y(comes)i(from)g(a)g(\\Lasota-Y)-7 b(ork)n(e)29 b(t)n(yp)r(e)g(inequalit)n(y".)0 2131 y Fj(Lemma)34 b(4.10)42 b Fx(F)-7 b(or)33 b(an)n(y)g Fv(j)j Fu(2)31 b Fr(N)7 b Fx(,)40 b(for)32 b(an)n(y)h Fv(f)41 b Fu(2)31 b Fv(C)1717 2144 y FC(j)1749 2131 y Fy(\()p Fv(a;)15 b(b;)g(c)p Fy(\))p Fx(,)36 b(for)c(an)n(y)h Fv(`)e Fu(2)f Fr(N)7 b Fx(,)40 b(for)33 b(an)n(y)g Fv(P)42 b Fu(2)31 b(Q)3235 2144 y Fo(1)3271 2131 y Fx(,)0 2239 y(w)n(e)e(ha)n(v)n(e:)822 2369 y Fy(inf)812 2425 y FC(x)p FB(2)p FC(P)954 2369 y Fu(j)p Fv(f)10 b Fy(\()p Fv(x)p Fy(\))p Fu(j)25 b(\025)f(\000)p Fv(b)15 b(R)1483 2382 y FC(j)1516 2369 y 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1011 y FC(F)709 994 y Fn(r)740 1011 y Fo(\()p FC(x)801 994 y Fm(0)820 1011 y Fo(\)=)p FC(x)945 902 y Fu(j)p Fv(f)g Fy(\()p Fv(y)1103 865 y FB(0)1124 902 y Fy(\))p Fu(j)p Fv(J)e(F)1309 865 y FC(r)1343 902 y Fy(\()p Fv(y)1423 865 y FB(0)1444 902 y Fy(\))1478 865 y FB(\000)p Fo(1)1578 772 y Fs(\014)1578 826 y(\014)1578 880 y(\014)1578 933 y(\014)1619 841 y Fv(J)g(F)1745 809 y FC(r)1779 841 y Fy(\()p Fv(y)1859 809 y FB(0)1880 841 y Fy(\))p 1617 881 300 4 v 1617 963 a Fv(J)g(F)1743 938 y FC(r)1777 963 y Fy(\()p Fv(x)1861 938 y FB(0)1882 963 y Fy(\))1945 902 y Fu(\000)20 b Fy(1)2079 772 y Fs(\014)2079 826 y(\014)2079 880 y(\014)2079 933 y(\014)515 1187 y Fu(\024)75 b Fv(bR)764 1200 y FC(j)s FB(\000)p Fo(1)878 1187 y Fy(\()p Fv(r)22 b Fy(+)e Fv(s)p Fy(\()p Fv(x;)15 b(y)s Fy(\)\))g(sup)g Fu(L)1565 1150 y FC(r)1599 1187 y Fq(1)1679 1065 y Fs(Z)1678 1320 y Fo(\001)1784 1187 y Fv(f)10 b(d)d Fy(^)-51 b Fv(\026)516 1569 y Fy(+)76 b Fv(R)727 1582 y Fo(0)762 1569 y Fy(\()p Fv(s)p Fy(\()p Fv(x;)15 b(y)s Fy(\)\))1089 1362 y Fs(2)1089 1520 y(6)1089 1578 y(4)1192 1484 y(X)1159 1672 y Fn(F)1201 1659 y(r)1232 1672 y(y)1262 1659 y Fm(0)1282 1672 y Fl(=)p Fn(y)1166 1726 y(y)1196 1713 y Fm(0)1216 1726 y(2)p Fn(P)1288 1733 y Fm(1)1378 1569 y Fu(j)p Fv(f)10 b Fy(\()p Fv(y)1536 1532 y FB(0)1557 1569 y Fy(\))p Fu(j)p Fv(J)e(F)1742 1532 y FC(r)1776 1569 y Fy(\()p Fv(y)1856 1532 y FB(0)1877 1569 y Fy(\))1911 1532 y FB(\000)p Fo(1)2016 1569 y Fy(+)2146 1484 y Fs(X)2113 1672 y Fn(F)2155 1659 y(r)2186 1672 y(y)2216 1659 y Fm(0)2236 1672 y Fl(=)p Fn(y)2120 1726 y(y)2150 1713 y Fm(0)2170 1726 y(62)p Fn(P)2242 1733 y Fm(1)2332 1569 y Fu(j)p Fv(f)i Fy(\()p Fv(y)2490 1532 y FB(0)2511 1569 y Fy(\))p Fu(j)p Fv(J)e(F)2696 1532 y FC(r)2730 1569 y Fy(\()p Fv(y)2810 1532 y FB(0)2831 1569 y Fy(\))2865 1532 y FB(\000)p Fo(1)2950 1362 y Fs(3)2950 1520 y(7)2950 1578 y(5)515 1896 y Fu(\024)75 b Fv(bR)764 1909 y FC(j)s FB(\000)p Fo(1)878 1896 y Fy(\()p Fv(r)22 b Fy(+)e Fv(s)p Fy(\()p Fv(x;)15 b(y)s Fy(\)\))1371 1774 y Fs(Z)1369 2029 y Fo(\001)1475 1896 y Fv(f)10 b(d)d Fy(^)-51 b Fv(\026)516 2190 y Fy(+)76 b Fv(R)727 2203 y Fo(0)762 2190 y Fy(\()p Fv(s)p Fy(\()p Fv(x;)15 b(y)s Fy(\)\))1089 2068 y Fs(Z)1087 2323 y Fo(\001)1195 2190 y Fv(f)10 b(d)d Fy(^)-51 b Fv(\026)o Fy([)p Fv(cC)13 b Fy(^)-51 b Fv(\026)1530 2203 y FC(\015)1567 2186 y Fm(0)1590 2190 y Fy(\()p Fv(P)1681 2203 y FB(1)1748 2190 y Fy(\))20 b(+)g Fv(bR)1994 2203 y FC(j)s FB(\000)p Fo(1)2109 2190 y Fy(\()p Fv(s)p Fy(\))g(+)g Fv(a)p Fy(])p Fv(;)85 2505 y Fy(\(w)n(e)43 b(used)e(\(4.6\))i(and)f(\(4.8\))h(to)g(b)r(ound) 1551 2420 y Fs(X)1519 2607 y Fn(F)1561 2594 y(r)1592 2607 y(y)1622 2594 y Fm(0)1642 2607 y Fl(=)p Fn(y)1526 2661 y(y)1556 2648 y Fm(0)1576 2661 y(2)p Fn(P)1648 2668 y Fm(1)1738 2505 y Fu(j)p Fv(f)10 b Fy(\()p Fv(y)1896 2468 y FB(0)1917 2505 y Fy(\))p Fu(j)p Fv(J)e(F)2102 2468 y FC(r)2136 2505 y Fy(\()p Fv(y)2216 2468 y FB(0)2237 2505 y Fy(\))2271 2468 y FB(\000)p Fo(1)2398 2505 y Fy(and)42 b(then)g(\(4.7\))h(to)g(b)r(ound)127 2696 y Fs(X)94 2883 y Fn(F)136 2870 y(r)167 2883 y(y)197 2870 y Fm(0)217 2883 y Fl(=)p Fn(y)101 2937 y(y)131 2924 y Fm(0)151 2937 y(62)p Fn(P)223 2944 y Fm(1)313 2781 y Fu(j)p Fv(f)10 b Fy(\()p Fv(y)471 2744 y FB(0)492 2781 y Fy(\))p Fu(j)p Fv(J)e(F)677 2744 y FC(r)711 2781 y Fy(\()p Fv(y)791 2744 y FB(0)812 2781 y Fy(\))846 2744 y FB(\000)p Fo(1)931 2781 y Fy(.\))40 b(This)27 b(pro)n(v)n(es)h(the)h(result)e(for)i Fv(x)p Fy(,)g Fv(y)j Fy(in)c(\001)2380 2794 y Fo(0)2444 2781 y Fy(\(tak)n(e)h Fv(r)e Fy(=)e Fv(k)2874 2794 y FC(j)2906 2781 y Fy(\).)40 b(F)-7 b(or)28 b Fv(x)h Fy(and)85 3037 y Fv(y)36 b Fy(in)31 b Fv(B)336 3050 y FC(q)r(;`)446 3037 y Fy(if)h Fv(x)580 3004 y FB(0)633 3037 y Fy(and)g Fv(y)853 3004 y FB(0)907 3037 y Fy(are)g(the)h(unique)e(preimage)h(of)g Fv(x)g Fy(and)g Fv(y)k Fy(b)n(y)c Fv(F)2510 3004 y FC(`)2572 3037 y Fy(then)g Fv(x)2825 3004 y FB(0)2846 3037 y Fv(;)48 b(y)2964 3004 y FB(0)3015 3037 y Fu(2)30 b Fv(B)3172 3050 y FC(q)3239 3037 y Fy(and)85 3145 y Fv(R)152 3158 y FC(j)s FB(\000)p Fo(1)266 3145 y Fy(\()p Fv(k)346 3158 y FC(j)398 3145 y Fu(\000)20 b Fv(`)g Fy(+)g Fv(s)p Fy(\()p Fv(x)758 3112 y FB(0)779 3145 y Fv(;)15 b(y)864 3112 y FB(0)886 3145 y Fy(\)\))25 b(=)g Fv(R)1139 3158 y FC(j)s FB(\000)p Fo(1)1253 3145 y Fy(\()p Fv(k)1333 3158 y FC(j)1386 3145 y Fy(+)20 b Fv(s)p Fy(\()p Fv(x;)15 b(y)s Fy(\)\),)31 b Fv(R)1874 3158 y Fo(0)1909 3145 y Fy(\()p Fv(s)p Fy(\()p Fv(x)2068 3112 y FB(0)2089 3145 y Fv(;)15 b(y)2174 3112 y FB(0)2196 3145 y Fy(\)\))25 b Fu(\024)f Fv(R)2450 3158 y Fo(0)2486 3145 y Fy(\()p Fv(s)p Fy(\()p Fv(x;)15 b(y)s Fy(\)\).)40 b(So,)129 3325 y Fu(jL)216 3288 y FC(k)249 3297 y Fn(j)281 3325 y Fv(f)10 b Fy(\()p Fv(x)p Fy(\))20 b Fu(\000)g(L)624 3288 y FC(k)657 3297 y Fn(j)690 3325 y Fv(f)10 b Fy(\()p Fv(y)s Fy(\))p Fu(j)25 b Fy(=)g Fu(jL)1087 3288 y FC(k)1120 3297 y Fn(j)1149 3288 y FB(\000)p FC(`)1228 3325 y Fv(f)10 b Fy(\()p Fv(x)1365 3288 y FB(0)1386 3325 y Fy(\))20 b Fu(\000)g(L)1592 3288 y FC(k)1625 3297 y Fn(j)1654 3288 y FB(\000)p FC(`)1733 3325 y Fv(f)10 b Fy(\()p Fv(y)1866 3288 y FB(0)1887 3325 y Fy(\))p Fu(j)278 3499 y(\024)25 b Fv(b)425 3377 y Fs(Z)423 3632 y Fo(\001)530 3499 y Fv(f)10 b(d)d Fy(^)-51 b Fv(\026)o Fy([)p Fv(R)772 3512 y FC(j)s FB(\000)p Fo(1)886 3499 y Fy(\()p Fv(k)966 3512 y FC(j)1019 3499 y Fu(\000)19 b Fv(`)h Fy(+)g Fv(s)p Fy(\()p Fv(x)1378 3462 y FB(0)1400 3499 y Fv(;)15 b(y)1485 3462 y FB(0)1506 3499 y Fy(\)\))20 b(+)g Fv(b)1719 3462 y FB(\000)p Fo(1)1805 3499 y Fv(R)1872 3512 y Fo(0)1907 3499 y Fy(\()p Fv(s)p Fy(\()p Fv(x)2066 3462 y FB(0)2087 3499 y Fv(;)15 b(y)2172 3462 y FB(0)2194 3499 y Fy(\)\)\()p Fv(cC)e Fy(^)-51 b Fv(\026)2456 3462 y FB(0)2456 3522 y FC(\015)2496 3499 y Fy(\()p Fv(P)2587 3512 y FB(1)2654 3499 y Fy(\))20 b(+)g Fv(bR)2900 3512 y FC(j)s FB(\000)p Fo(1)3014 3499 y Fy(\()p Fv(s)p Fy(\))h(+)f Fv(a)p Fy(\)])278 3799 y Fu(\024)25 b Fv(b)440 3677 y Fs(Z)438 3932 y Fo(\001)544 3799 y Fv(f)10 b(d)d Fy(^)-51 b Fv(\026)p Fy([)p Fv(R)787 3812 y FC(j)s FB(\000)p Fo(1)901 3799 y Fy(\()p Fv(k)981 3812 y FC(j)1034 3799 y Fy(+)20 b Fv(s)p Fy(\()p Fv(x;)15 b(y)s Fy(\)\))21 b(+)1518 3738 y Fv(cC)13 b Fy(^)-51 b Fv(\026)1678 3751 y FC(\015)1715 3735 y Fm(0)1738 3738 y Fy(\()p Fv(P)1829 3751 y FB(1)1896 3738 y Fy(\))20 b(+)g Fv(bR)2142 3751 y FC(j)s FB(\000)p Fo(1)2256 3738 y Fy(\()p Fv(s)p Fy(\))h(+)f Fv(a)p 1518 3778 1003 4 v 2000 3861 a(b)2529 3799 y(R)2596 3812 y Fo(0)2631 3799 y Fy(\()p Fv(s)p Fy(\()p Fv(x;)15 b(y)s Fy(\)\)])p Fv(:)3311 4081 y Fg(\004)85 4337 y Ft(Pr)l(o)l(of)35 b(of)h(Pr)l(op)l(osition)g (4.9)p Fy(:)99 b(Let)34 b Fv(f)45 b Fy(b)r(elong)33 b(to)i Fv(C)1856 4350 y FC(j)s FB(\000)p Fo(1)1970 4337 y Fy(\()p Fv(a;)15 b(b;)g(c)p Fy(\).)56 b(W)-7 b(e)35 b(are)f(going)g(to)g(c)n (hec)n(k)g(that)85 4446 y(eac)n(h)24 b(of)h(the)g(three)g(conditions)d (in)n(v)n(olv)n(ed)h(in)h(the)h(de\014nition)e(of)h(the)h(cones)f(is)g (con)n(tracted)g(b)n(y)h Fu(L)3290 4413 y FC(k)3323 4422 y Fn(j)3356 4446 y Fy(.)85 4554 y(Condition)j(1.)p 85 4568 478 4 v 29 w(Let)i Fv(P)36 b Fu(2)25 b(Q)p Fy(.)311 4712 y(1)p 238 4752 190 4 v 245 4835 a(^)-51 b Fv(\026)p Fy(\()p Fv(P)12 b Fy(\))452 4651 y Fs(Z)452 4906 y FC(P)557 4773 y Fu(L)619 4736 y FC(k)652 4745 y Fn(j)684 4773 y Fv(f)e(d)d Fy(^)-51 b Fv(\026)25 b Fy(=)971 4688 y Fs(X)954 4879 y FC(P)1003 4862 y Fm(0)1023 4879 y FB(2Q)1214 4712 y Fy(1)p 1141 4752 V 1148 4835 a(^)-51 b Fv(\026)p Fy(\()p Fv(P)12 b Fy(\))1468 4651 y Fs(Z)1335 4926 y FC(F)1384 4896 y Fm(\000)p Fn(k)1457 4910 y(j)1490 4926 y FC(P)d FB(\\)p FC(P)1630 4909 y Fm(0)1685 4773 y Fv(f)h(d)d Fy(^)-51 b Fv(\026)p eop %%Page: 18 18 18 17 bop 0 26 a Fy(18)745 b Fx(J)n(\023)-41 b(er^)d(ome)29 b(Buzzi)g(&)g(V)n(\023)-41 b(eronique)28 b(Maume-Desc)n(hamps)219 396 y Fu(\024)396 311 y Fs(X)363 503 y FC(P)412 486 y Fm(0)433 503 y FB(2Q)527 511 y Fl(1)589 336 y Fy(^)-51 b Fv(\026)p Fy(\()p Fv(P)738 303 y FB(0)778 336 y Fu(\\)20 b Fv(F)927 303 y FB(\000)p FC(k)1009 312 y Fn(j)1043 336 y Fv(P)12 b Fy(\))p 582 376 564 4 v 670 458 a(^)-51 b Fv(\026)p Fy(\()p Fv(P)819 432 y FB(0)840 458 y Fy(\))7 b(^)-51 b Fv(\026)p Fy(\()p Fv(P)12 b Fy(\))1169 216 y Fs(2)1169 378 y(4)1229 274 y(Z)1220 531 y FC(P)1269 514 y Fm(0)1334 396 y Fv(f)e(d)d Fy(^)-51 b Fv(\026)48 b Fy(+)i Fv(bR)1756 409 y FC(j)s FB(\000)p Fo(1)1870 396 y Fy(\()p Fv(s)p Fy(\))1994 274 y Fs(Z)1992 529 y Fo(\001)2099 396 y Fv(f)10 b(d)d Fy(^)-51 b Fv(\026)36 b Fy(^)-51 b Fv(\026)p Fy(\()p Fv(P)2436 359 y FB(0)2456 396 y Fy(\))2490 216 y Fs(3)2490 378 y(5)2741 396 y Fy(using)28 b(\(4.7\))220 704 y(+)75 b Fv(c)416 582 y Fs(Z)414 837 y Fo(\001)521 704 y Fv(f)10 b(d)d Fy(^)-51 b Fv(\026)775 619 y Fs(X)793 800 y Fn(`)p Fm(2)p Fk(N)696 850 y Fl(\001)741 866 y Fn(`)768 850 y Fm(\\)p Fn(P)840 857 y Fm(1)899 850 y(6)p Fl(=)p Fk(?)1023 643 y Fy(^)h Fv(\026)o Fy(\()p Fv(P)1160 656 y FB(1)1247 643 y Fu(\\)20 b Fv(F)1396 611 y FB(\000)p FC(k)1478 620 y Fn(j)1511 643 y Fv(P)32 b Fu(\\)20 b Fy(\001)1753 656 y FC(`)1783 643 y Fy(\))g Fu(\001)27 b Fy(^)-51 b Fv(\026)1935 656 y FC(\015)1972 640 y Fm(0)1995 643 y Fy(\()p Fv(P)2086 656 y FB(1)2153 643 y Fy(\))p 1017 684 1171 4 v 1248 766 a(^)g Fv(\026)p Fy(\()p Fv(P)12 b Fy(\))20 b Fu(\001)26 b Fy(^)-51 b Fv(\026)1548 779 y FC(\015)1585 762 y Fm(0)1608 766 y Fy(\()p Fv(P)1699 779 y FB(1)1766 766 y Fy(\))20 b Fu(\001)g Fv(\015)1916 740 y FB(0)p FC(`)2371 704 y Fy(using)28 b(\(4.8\))219 1027 y Fu(\024)74 b Fv(B)450 905 y Fs(Z)448 1160 y Fo(\001)554 1027 y Fv(f)10 b(d)d Fy(^)-51 b Fv(\026)14 b Fy([)q(1)20 b(+)g Fv(bR)1001 1040 y FC(j)s FB(\000)p Fo(1)1115 1027 y Fy(\()p Fv(s)p Fy(\))h(+)f Fv(c)7 b Fy(^)-51 b Fv(\026)1424 1040 y FC(\015)1461 1023 y Fm(0)1483 1027 y Fy(\()p Fv(P)1574 1040 y FB(1)1641 1027 y Fy(\)])191 b(using)28 b(Lemma)i(4.7)p Fv(:)545 b Fy(\(4.9\))0 1288 y(Similarly)-7 b(,)638 1374 y(1)p 565 1414 190 4 v 572 1496 a(^)-51 b Fv(\026)p Fy(\()p Fv(P)12 b Fy(\))779 1313 y Fs(Z)779 1568 y FC(P)883 1435 y Fu(L)945 1398 y FC(k)978 1407 y Fn(j)1011 1435 y Fv(f)e(d)d Fy(^)-51 b Fv(\026)24 b Fu(\025)1282 1313 y Fs(Z)1281 1568 y Fo(\001)1387 1435 y Fv(f)10 b(d)d Fy(^)-51 b Fv(\026)14 b Fy([)p Fv(A)20 b Fu(\000)g Fv(B)5 b Fy(\()p Fv(bR)1963 1448 y FC(j)s FB(\000)p Fo(1)2076 1435 y Fy(\()p Fv(s)p Fy(\))21 b(+)f(2)p Fv(c)7 b Fy(^)-51 b Fv(\026)2429 1448 y FC(\015)2466 1431 y Fm(0)2489 1435 y Fy(\()p Fv(P)2580 1448 y FB(1)2647 1435 y Fy(\)\)])332 b(\(4.10\))0 1678 y(Condition)28 b(2.)p 0 1692 478 4 v 29 w(Lemma)i(4.11)f(giv)n(es:)419 1875 y Fu(jL)506 1838 y FC(k)539 1847 y Fn(j)572 1875 y Fv(f)10 b Fy(\()p Fv(x)p Fy(\))20 b Fu(\000)g(L)915 1838 y FC(k)948 1847 y Fn(j)980 1875 y Fv(f)10 b Fy(\()p Fv(y)s Fy(\))p Fu(j)75 b(\024)g Fv(b)1459 1753 y Fs(Z)1457 2008 y Fo(\001)1564 1875 y Fv(f)10 b(d)d Fy(^)-51 b Fv(\026)o Fy([)p Fv(R)1806 1888 y FC(j)s FB(\000)p Fo(1)1920 1875 y Fy(\()p Fv(k)2000 1888 y FC(j)2053 1875 y Fy(+)20 b Fv(s)p Fy(\()p Fv(x;)15 b(y)s Fy(\)\))1248 2174 y(+)1401 2114 y Fv(cC)e Fy(^)-51 b Fv(\026)1561 2127 y FC(\015)1598 2110 y Fm(0)1621 2114 y Fy(\()p Fv(P)1712 2127 y FB(1)1779 2114 y Fy(\))20 b(+)g Fv(bR)2025 2127 y FC(j)s FB(\000)p Fo(1)2139 2114 y Fy(\()p Fv(s)p Fy(\))h(+)f Fv(a)p 1401 2154 1003 4 v 1883 2236 a(b)2412 2174 y(R)2479 2187 y Fo(0)2514 2174 y Fy(\()p Fv(s)p Fy(\()p Fv(x;)15 b(y)s Fy(\)\)])p Fv(:)197 b Fy(\(4.11\))0 2358 y(Condition)28 b(3.)p 0 2372 478 4 v 31 w(Let)i Fv(x)h Fy(b)r(elongs)e(to)i(\001)1246 2371 y FC(`)1297 2358 y Fu(\\)20 b Fv(P)1434 2371 y FB(1)1501 2358 y Fy(.)44 b(If)30 b Fv(`)d Fu(\025)g Fv(k)1865 2371 y FC(j)1928 2358 y Fy(then)j Fv(x)h Fy(has)f(only)g(one)g(preimage)g Fv(x)3151 2326 y FB(0)3203 2358 y Fy(b)n(y)0 2467 y Fv(F)69 2434 y FC(k)102 2443 y Fn(j)135 2467 y Fy(,)g(the)f(de\014nition)e(of)i Fv(P)887 2480 y FB(1)983 2467 y Fy(implies)e(that)i Fv(x)1521 2434 y FB(0)1572 2467 y Fy(b)r(elong)f(also)g(to)h Fv(P)2192 2480 y FB(1)2259 2467 y Fy(,)h(so)911 2663 y Fu(jL)998 2626 y FC(k)1031 2635 y Fn(j)1064 2663 y Fv(f)10 b Fy(\()p Fv(x)p Fy(\))p Fu(j)24 b Fy(=)i Fu(j)p Fv(f)10 b Fy(\()p Fv(x)1540 2626 y FB(0)1560 2663 y Fy(\))p Fu(j)75 b(\024)1933 2602 y Fv(c)p 1847 2642 210 4 v 1847 2724 a(\015)1898 2697 y FB(0)p FC(`)p FB(\000)p FC(k)2023 2706 y Fn(j)2080 2541 y Fs(Z)2079 2796 y Fo(\001)2185 2663 y Fv(f)10 b(d)d Fy(^)-51 b Fv(\026)1694 2957 y Fu(\024)1877 2896 y Fv(c)p 1847 2936 98 4 v 1847 3018 a(\015)1898 2992 y FB(0)p FC(`)1954 2957 y Fv(\015)2005 2920 y FB(0)p FC(k)2055 2928 y Fl(0)2104 2835 y Fs(Z)2103 3090 y Fo(\001)2209 2957 y Fv(f)10 b(d)d Fy(^)-51 b Fv(\026)o(:)0 3230 y Fy(If)31 b Fv(`)f(<)f(k)300 3243 y FC(j)365 3230 y Fy(then)j Fu(jL)655 3197 y FC(k)688 3206 y Fn(j)720 3230 y Fv(f)10 b Fy(\()p Fv(x)p Fy(\))p Fu(j)30 b Fy(=)g Fu(jL)1131 3197 y FC(k)1164 3206 y Fn(j)1193 3197 y FB(\000)p FC(`)1272 3230 y Fv(f)10 b Fy(\()p Fv(x)1409 3197 y FB(0)1430 3230 y Fy(\))p Fu(j)32 b Fy(where)g Fv(x)1827 3197 y FB(0)1877 3230 y Fu(2)d Fy(\001)2039 3243 y Fo(0)2107 3230 y Fy(is)i(the)h (unique)f(preimage)g(of)h Fv(x)g Fy(b)n(y)0 3338 y Fv(F)69 3305 y FC(`)99 3338 y Fy(.)39 b(So,)29 b(it)g(su\016ces)g(to)g (consider)f Fv(x)d Fu(2)f Fy(\001)1376 3351 y Fo(0)1441 3338 y Fy(and)29 b Fv(p)c Fu(\024)g Fv(k)1822 3351 y FC(j)1854 3338 y Fy(.)439 3548 y Fu(jL)526 3511 y FC(p)562 3548 y Fv(f)10 b Fy(\()p Fv(x)p Fy(\))p Fu(j)74 b(\024)1020 3462 y Fs(X)986 3650 y Fn(F)1028 3636 y(p)1060 3650 y(x)1091 3637 y Fm(0)1112 3650 y Fl(=)p Fn(x)994 3700 y(x)1025 3687 y Fm(0)1045 3700 y(2)p Fn(P)1117 3707 y Fm(1)1247 3487 y Fu(j)p Fv(f)10 b Fy(\()p Fv(x)1409 3454 y FB(0)1430 3487 y Fy(\))p Fu(j)p 1218 3527 301 4 v 1218 3609 a Fv(J)e(F)1344 3583 y FC(p)1379 3609 y Fy(\()p Fv(x)1463 3583 y FB(0)1484 3609 y Fy(\))1577 3548 y(+)1738 3462 y Fs(X)1703 3650 y Fn(F)1745 3636 y(p)1778 3650 y(x)1809 3637 y Fm(0)1829 3650 y Fl(=)p Fn(x)1711 3700 y(x)1742 3687 y Fm(0)1763 3700 y(62)p Fn(P)1835 3707 y Fm(1)1964 3487 y Fu(j)p Fv(f)i Fy(\()p Fv(x)2126 3454 y FB(0)2147 3487 y Fy(\))p Fu(j)p 1935 3527 V 1935 3609 a Fv(J)e(F)2061 3583 y FC(p)2097 3609 y Fy(\()p Fv(x)2181 3583 y FB(0)2202 3609 y Fy(\))832 3870 y Fu(\024)75 b Fv(c)1030 3748 y Fs(Z)1028 4003 y Fo(\001)1134 3870 y Fv(f)10 b(d)d Fy(^)-51 b Fv(\026)19 b Fu(\001)h Fv(C)13 b Fy(^)-51 b Fv(\026)1472 3883 y FC(\015)1509 3867 y Fm(0)1532 3870 y Fy(\()p Fv(P)1623 3883 y FB(1)1690 3870 y Fy(\))20 b(+)g Fv(bR)1936 3883 y FC(j)s FB(\000)p Fo(1)2051 3870 y Fy(\()p Fv(s)p Fy(\))2175 3748 y Fs(Z)2173 4003 y Fo(\001)2280 3870 y Fv(f)10 b(d)d Fy(^)-51 b Fv(\026)19 b Fy(+)h Fv(a)2600 3748 y Fs(Z)2598 4003 y Fo(\001)2705 3870 y Fv(f)10 b(d)d Fy(^)-51 b Fv(\026)832 4164 y Fu(\024)977 4042 y Fs(Z)975 4297 y Fo(\001)1081 4164 y Fv(f)10 b(d)d Fy(^)-51 b Fv(\026)14 b Fy([)q Fv(cC)f Fy(^)-51 b Fv(\026)1432 4177 y FC(\015)1469 4160 y Fm(0)1492 4164 y Fy(\()p Fv(P)1583 4177 y FB(1)1650 4164 y Fy(\))20 b(+)g Fv(bR)1896 4177 y FC(j)s FB(\000)p Fo(1)2010 4164 y Fy(\()p Fv(s)p Fy(\))h(+)f Fv(a)p Fy(])15 b Fv(:)0 4425 y Fy(So,)29 b(for)g(an)n(y)g Fv(`)c(<)g(k)647 4438 y FC(j)680 4425 y Fy(,)k Fv(x)c Fu(2)g Fy(\001)966 4438 y FC(`)1016 4425 y Fu(\\)19 b Fv(P)1152 4438 y FB(1)1219 4425 y Fy(,)733 4629 y Fu(jL)820 4592 y FC(k)853 4601 y Fn(j)886 4629 y Fv(f)10 b Fy(\()p Fv(x)p Fy(\))p Fu(j)25 b(\024)1201 4507 y Fs(Z)1200 4762 y Fo(\001)1306 4629 y Fv(f)10 b(d)d Fy(^)-51 b Fv(\026)14 b Fy([)p Fv(cC)f Fy(^)-51 b Fv(\026)1656 4642 y FC(\015)1693 4625 y Fm(0)1716 4629 y Fy(\()p Fv(P)1807 4642 y FB(1)1874 4629 y Fy(\))20 b(+)h Fv(bR)2121 4642 y FC(j)s FB(\000)p Fo(1)2235 4629 y Fy(\()p Fv(s)p Fy(\))f(+)h Fv(a)p Fy(])15 b Fv(:)509 b Fy(\(4.12\))132 4890 y(No)n(w)29 b(c)n(ho)r(ose)f(the)h(parameters)g Fv(a)p Fy(,)h Fv(b)p Fy(,)g Fv(c)p Fy(,)g Fv(D)r Fy(,)g Fv(k)1673 4903 y Fo(0)1708 4890 y Fy(,)f Fv(s)h Fy(and)f Fv(t)g Fy(as)f(follo)n(ws:)p eop %%Page: 19 19 19 18 bop 832 26 a Fx(Piecewise)28 b(in)n(v)n(ertible)f(maps)i(in)f (higher)g(dimensions)745 b Fy(19)216 355 y Fu(\017)44 b Fv(D)27 b(>)e Fy(1)k(suc)n(h)f(that)969 294 y Fv(D)22 b Fy(+)f(1)p 968 334 229 4 v 968 416 a Fv(D)i Fu(\000)c Fy(1)1231 355 y Fu(\025)1335 289 y Fy(4)p Fv(B)5 b Fy([1)19 b(+)1653 254 y FC(A)p 1635 269 83 4 v 1635 320 a Fo(2)p FC(B)1727 289 y Fy(])p 1335 334 417 4 v 1510 416 a Fv(A)1760 355 y Fy(,)216 604 y Fu(\017)44 b Fv(a)29 b Fy(suc)n(h)f(that)h Fv(a)c Fu(\025)g Fv(D)r Fy([)p Fv(B)g Fy(+)1222 568 y FC(A)p 1222 583 48 4 v 1230 634 a Fo(2)1278 604 y Fy(],)216 808 y Fu(\017)44 b Fv(b)29 b Fy(suc)n(h)f(that)h Fv(b)d Fu(\025)e Fy(\(2)c(+)g Fv(a)p Fy(\),)216 1053 y Fu(\017)44 b Fv(\033)358 1020 y FB(0)404 1053 y Fv(<)25 b(D)572 1020 y FB(\000)p Fo(1)657 1053 y Fy(,)30 b Fv(c)f Fy(suc)n(h)f(that)h Fv(c)c(>)1404 992 y Fy(1)19 b(+)i Fv(a)p 1331 1032 344 4 v 1331 1114 a(D)1406 1088 y FB(\000)p Fo(1)1511 1114 y Fu(\000)f Fv(\033)1654 1088 y FB(0)1684 1053 y Fy(,)216 1301 y Fu(\017)44 b Fv(s)29 b Fy(suc)n(h)f(that)h Fv(R)830 1314 y FC(j)s FB(\000)p Fo(1)969 1301 y Fu(\024)c Fv(R)1131 1314 y Fo(0)1166 1301 y Fy(\()p Fv(s)p Fy(\)\()p Fv(D)e Fy(+)d(1\))25 b Fu(\024)1700 1266 y Fo(1)p 1700 1281 32 4 v 1702 1332 a FC(b)1770 1301 y Fy(and)k Fv(R)2008 1314 y FC(j)s FB(\000)p Fo(1)2146 1301 y Fu(\024)c Fv(R)2308 1314 y Fo(0)2343 1301 y Fy(\()p Fv(s)p Fy(\)\()p Fv(D)e Fy(+)e(1\))k Fu(\024)2908 1266 y FC(A)p 2877 1281 110 4 v 2877 1332 a Fo(4)p FC(B)s(b)216 1517 y Fu(\017)44 b Fv(t)28 b Fy(suc)n(h)h(that)36 b(^)-51 b Fv(\026)807 1530 y FC(\015)844 1514 y Fm(0)867 1517 y Fy(\()p Fv(P)958 1530 y FB(1)1025 1517 y Fy(\))25 b Fu(\024)1187 1482 y FC(\033)1225 1461 y Fm(0)p 1187 1497 59 4 v 1192 1548 a FC(C)1255 1517 y Fy(,)36 b(^)-50 b Fv(\026)1362 1530 y FC(\015)1399 1514 y Fm(0)1421 1517 y Fy(\()p Fv(P)1512 1530 y FB(1)1579 1517 y Fy(\))25 b Fu(\024)1765 1482 y Fo(1)p 1742 1497 77 4 v 1742 1548 a FC(cC)1857 1517 y Fy(and)36 b(^)-51 b Fv(\026)2081 1530 y FC(\015)2118 1514 y Fm(0)2141 1517 y Fy(\()p Fv(P)2232 1530 y FB(1)2299 1517 y Fy(\))25 b Fu(\024)2518 1482 y FC(A)p 2462 1497 160 4 v 2462 1548 a Fo(4)p FC(cB)s(C)2630 1517 y Fy(,)216 1722 y Fu(\017)44 b Fv(k)351 1735 y Fo(0)425 1722 y Fu(\025)c Fv(k)584 1689 y FB(0)581 1744 y Fo(0)654 1722 y Fy(suc)n(h)d(that)h Fv(D)k Fu(\024)d Fv(\015)1335 1689 y FB(0\000)p FC(k)1434 1697 y Fl(0)1507 1722 y Fy(\(recall)e(that)h Fv(k)2032 1689 y FB(0)2029 1744 y Fo(0)2102 1722 y Fy(is)f(giv)n(en)g(b)n(y)h (Lemma)g(4.7)g(and)f(that)305 1830 y Fv(\015)356 1798 y FB(0)401 1830 y Fv(<)25 b Fy(1\).)85 2035 y(With)39 b(these)g(c)n(hoices,)h(using)e(\(4.9,)k(4.11,)g(4.12\),)g(w)n(e)e(sho) n(w)e(that)h Fu(L)2448 2002 y FC(k)2481 2011 y Fn(j)2514 2035 y Fv(f)49 b Fy(b)r(elongs)38 b(to)h(the)h(cone)85 2143 y Fv(C)148 2156 y FC(j)181 2143 y Fy(\()p Fv(D)290 2111 y FB(\000)p Fo(1)375 2143 y Fv(a;)15 b(D)535 2111 y FB(\000)p Fo(1)621 2143 y Fv(b;)g(D)772 2111 y FB(\000)p Fo(1)858 2143 y Fv(c)p Fy(\).)39 b(Moreo)n(v)n(er,)30 b(\(4.10\))f(leads)f(to:)40 b Fu(8)p Fv(P)35 b Fu(2)25 b(Q)p Fy(,)1376 2340 y(1)p 1303 2380 190 4 v 1310 2462 a(^)-51 b Fv(\026)o Fy(\()p Fv(P)12 b Fy(\))1516 2278 y Fs(Z)1517 2533 y FC(P)1621 2400 y Fu(L)1683 2363 y FC(k)1716 2372 y Fn(j)1749 2400 y Fv(f)e(d)d Fy(^)-51 b Fv(\026)24 b Fu(\025)2029 2340 y Fy(1)p 2029 2380 44 4 v 2029 2462 a(4)2082 2400 y Fv(A:)984 b Fy(\(4.13\))85 2726 y(It)36 b(remains)f(to)h(pro)n(v)n(e)g(that)g(the)g(pro)5 b(jectiv)n(e)35 b(diameter)h(of)f Fu(L)2189 2694 y FC(k)2222 2703 y Fn(j)2255 2726 y Fv(C)2318 2739 y FC(j)s FB(\000)p Fo(1)2432 2726 y Fy(\()p Fv(a;)15 b(b;)g(c)p Fy(\))37 b(in)n(to)e Fv(C)2984 2739 y FC(j)3017 2726 y Fy(\()p Fv(a;)15 b(b;)g(c)p Fy(\))37 b(is)85 2835 y(b)r(ounded)31 b(b)n(y)h(\000)d(:=)h(2)15 b(log)973 2799 y FC(D)r Fo(+1)p 973 2814 136 4 v 973 2866 a FC(D)r FB(\000)p Fo(1)1117 2835 y Fy(.)48 b(A)32 b(standard)f(computation)f(\(see)i([KMS)q(],)h ([M)q(]\))g(giv)n(es)d(that)i(the)85 2943 y(pro)5 b(jectiv)n(e)28 b(diameter)h(is)f(b)r(ounded)h(b)n(y)596 3369 y(2)15 b(log)g(max)961 3082 y Fs(2)961 3239 y(6)961 3293 y(6)961 3347 y(6)961 3401 y(6)961 3458 y(4)1031 3309 y Fv(D)22 b Fy(+)e(1)p 1030 3349 229 4 v 1030 3431 a Fv(D)i Fu(\000)e Fy(1)1268 3369 y Fv(;)1449 3193 y Fy(sup)1316 3270 y FC(f)7 b FB(2)p FC(C)1441 3279 y Fn(j)s Fm(\000)p Fl(1)1541 3270 y Fo(\()p FC(a;b;c)p Fo(\))1734 3193 y Fy(sup)1729 3266 y FC(P)i FB(2Q)1894 3193 y Fy(^)-51 b Fv(\026)p Fy(\()p Fv(P)12 b Fy(\))2077 3155 y FB(\000)p Fo(1)2177 3070 y Fs(Z)2177 3326 y FC(P)2281 3193 y Fv(f)e(d)d Fy(^)-51 b Fv(\026)p 1316 3349 1117 4 v 1462 3482 a Fy(inf)1316 3541 y FC(f)7 b FB(2)p FC(C)1441 3550 y Fn(j)s Fm(\000)p Fl(1)1541 3541 y Fo(\()p FC(a;b;c)p Fo(\))1747 3482 y Fy(inf)1729 3537 y FC(P)i FB(2Q)1894 3482 y Fy(^)-51 b Fv(\026)p Fy(\()p Fv(P)12 b Fy(\))2077 3445 y FB(\000)p Fo(1)2177 3360 y Fs(Z)2177 3615 y FC(P)2281 3482 y Fv(f)e(d)d Fy(^)-51 b Fv(\026)2441 3082 y Fs(3)2441 3239 y(7)2441 3293 y(7)2441 3347 y(7)2441 3401 y(7)2441 3458 y(5)670 3810 y Fu(\024)75 b Fy(2)15 b(log)g(max)1180 3657 y Fs(")1242 3750 y Fv(D)23 b Fy(+)d(1)p 1242 3790 229 4 v 1242 3872 a Fv(D)i Fu(\000)e Fy(1)1479 3810 y Fv(;)1557 3745 y Fy(4)p Fv(B)5 b Fy([1)19 b(+)1875 3709 y FC(A)p 1858 3724 83 4 v 1858 3775 a Fo(2)p FC(B)1949 3745 y Fy(])p 1557 3790 417 4 v 1732 3872 a Fv(A)1982 3657 y Fs(#)2225 3810 y Fy(using)28 b(\(4.9,)i(4.13\))p Fv(:)85 4095 y Fy(The)g(c)n(hoice)d(of)i Fv(D)j Fy(concludes)c(the)h(pro)r(of.)3311 4207 y Fg(\004)85 4607 y Fp(4.4)120 b(Conclusion)85 4782 y Fy(T)-7 b(o)35 b(conclude)e(the)i(pro)r(of)g(of)f(Theorem)h(2.1,)i(w) n(e)e(need)f(to)h(deriv)n(e)f(from)h(the)g(pro)5 b(jectiv)n(e)33 b(metric)85 4890 y(b)r(ound)20 b(obtained)g(ab)r(o)n(v)n(e,)i(b)r (ounds)e(on)g(the)h(correlations.)34 b(The)21 b(follo)n(wing)d(lemma)j (ma)n(y)f(b)r(e)h(pro)n(v)n(ed)p eop %%Page: 20 20 20 19 bop 0 26 a Fy(20)745 b Fx(J)n(\023)-41 b(er^)d(ome)29 b(Buzzi)g(&)g(V)n(\023)-41 b(eronique)28 b(Maume-Desc)n(hamps)0 302 y Fy(in)h(a)h(standard)f(w)n(a)n(y)g(\(see)h([KMS)q(])g(page)f (687,)h([M)q(])g(Lemmas)g(3.9-3.10\).)41 b(F)-7 b(or)30 b Fv(d)c(>)g Fy(0,)k(let)f Fu(k)h(k)3139 315 y FC(d)3205 302 y Fy(b)r(e)0 411 y(the)f(norm)g(on)g(b)r(ounded)g(functions)e (de\014ned)i(b)n(y:)564 710 y Fu(k)p Fv(f)10 b Fu(k)707 723 y FC(d)768 710 y Fy(=)25 b(max)1039 530 y Fs(2)1039 691 y(4)1099 710 y Fv(d)1160 580 y Fs(\014)1160 634 y(\014)1160 688 y(\014)1160 741 y(\014)1189 588 y(Z)1239 791 y Fo(\001)1311 710 y Fv(f)10 b(d)d Fy(^)-51 b Fv(\026)1462 580 y Fs(\014)1462 634 y(\014)1462 688 y(\014)1462 741 y(\014)1507 710 y Fv(;)50 b Fy(sup)1575 783 y FC(P)9 b FB(2Q)1741 710 y Fy(^)-51 b Fv(\026)p Fy(\()p Fv(P)12 b Fy(\))1924 673 y FB(\000)p Fo(1)2023 526 y Fs(\014)2023 580 y(\014)2023 634 y(\014)2023 688 y(\014)2023 741 y(\014)2023 795 y(\014)2053 588 y(Z)2054 843 y FC(P)2158 710 y Fv(f)e(d)d Fy(^)-51 b Fv(\026)2309 526 y Fs(\014)2309 580 y(\014)2309 634 y(\014)2309 688 y(\014)2309 741 y(\014)2309 795 y(\014)2354 710 y Fv(;)45 b Fu(k)p Fv(f)10 b Fu(k)2566 723 y FB(1)2632 530 y Fs(3)2632 691 y(5)2707 710 y Fv(:)0 1018 y Fj(Lemma)34 b(4.12)42 b Fx(The)27 b(norm)g Fu(k)g(k)1077 1031 y FC(d)1140 1018 y Fx(and)g(the)f(homogeneous)g(form)h Fv(f)35 b Fu(7!)c Fy(^)-51 b Fv(\026)p Fy(\()p Fv(f)10 b Fy(\))27 b Fx(are)g(adapted)f(to)h(the)0 1126 y(cones)i Fv(C)298 1139 y FC(j)330 1126 y Fy(\()p Fv(a;)15 b(b;)g(c)p Fy(\))p Fx(,)31 b(pro)n(vided)d Fv(d)d Fu(\025)f Fy(max)q([)p Fv(c;)44 b(R)1539 1139 y FC(j)1572 1126 y Fv(b)p Fy(])p Fv(:)0 1234 y Fx(F)-7 b(or)29 b(all)f Fv(p)d Fu(2)g Fr(N)7 b Fx(,)35 b(the)29 b(function)f Fu(L)1108 1202 y FC(p)1144 1234 y Fq(1)i Fx(b)r(elongs)e(to)h(all)f(the)h(cones)g Fv(C)2217 1247 y FC(j)2249 1234 y Fy(\()p Fv(a;)15 b(b;)g(c)p Fy(\))p Fx(.)0 1343 y(F)-7 b(or)31 b(an)n(y)g Fv(f)38 b Fu(2)28 b Fq(L)p Fy(\(\001\))p Fx(,)k(there)f(exists)g Fv(R)q Fy(\()p Fv(f)10 b Fy(\))28 b Fv(>)g Fy(0)j Fx(suc)n(h)f(that)h Fv(f)g Fy(+)22 b Fv(R)q Fy(\()p Fv(f)10 b Fy(\))p Fq(1)28 b Fu(2)g Fv(C)2587 1356 y Fo(0)2622 1343 y Fy(\()p Fv(a;)15 b(b;)g(c)p Fy(\))p Fx(,)34 b(moreo)n(v)n(er)0 1451 y Fv(R)q Fy(\()p Fv(f)10 b Fy(\))25 b Fu(\024)f Fv(C)6 b Fu(k)p Fv(f)k Fu(k)520 1465 y Ff(L)p Fo(\(\001\))670 1451 y Fx(.)0 1620 y Fy(Let)20 b Fv(f)34 b Fu(2)25 b Fv(C)373 1633 y Fo(0)408 1620 y Fy(\()p Fv(a;)15 b(b;)g(c)p Fy(\),)23 b(b)n(y)c(Prop)r(osition)f(4.9)i(\(remark)g(that)g Fq(1)25 b Fu(2)g Fv(C)2157 1633 y Fo(0)2192 1620 y Fy(\()p Fv(a;)15 b(b;)g(c)p Fy(\)\),)23 b(for)c(an)n(y)g Fv(`)p Fy(,)j Fu(L)2965 1588 y FC(k)2998 1596 y Fl(1)3029 1588 y Fo(+)p FB(\001\001\001)p Fo(+)p FC(k)3214 1599 y Fn(`)3245 1620 y Fv(f)0 1729 y Fy(and)29 b Fu(L)233 1696 y FC(k)266 1704 y Fl(1)297 1696 y Fo(+)p FB(\001\001\001)o Fo(+)p FC(k)481 1707 y Fn(`)512 1729 y Fq(1)d Fy(=)f Fq(1)30 b Fy(b)r(elong)e(to)h Fv(C)1211 1742 y FC(`)1241 1729 y Fy(.)39 b(Applying)28 b Fv(`)19 b Fu(\000)h Fy(1)29 b(times)g(Theorem)g(3.1,)h(w)n(e)g(get:)464 1966 y Fv(\016)503 1979 y FC(C)548 1990 y Fn(`)579 1966 y Fy(\()p Fu(L)675 1929 y FC(k)708 1937 y Fl(1)740 1929 y Fo(+)p FB(\001\001\001)o Fo(+)p FC(k)924 1940 y Fn(`)955 1966 y Fv(f)5 b(;)15 b Fq(1)p Fy(\))26 b Fu(\024)e Fy([tanh)1470 1906 y(\000)p 1470 1946 55 4 v 1475 2028 a(4)1534 1966 y(])1558 1929 y FC(`)p FB(\000)p Fo(1)1689 1966 y Fu(\001)c Fv(\016)1773 1979 y Fo(1)1809 1966 y Fy(\()p Fu(L)1905 1929 y FC(k)1938 1937 y Fl(1)1972 1966 y Fv(f)5 b(;)15 b Fq(1)p Fy(\))26 b Fu(\024)f Fy([tanh)2488 1906 y(\000)p 2488 1946 V 2493 2028 a(4)2552 1966 y(])2576 1929 y FC(`)p FB(\000)p Fo(1)2707 1966 y Fu(\001)20 b Fy(\000)p Fv(;)0 2202 y Fy(where)29 b(\000)g(is)f(giv)n(en)g(b)n(y)g(Prop)r(osition)f(4.9.)39 b(Remark)29 b(that)g(tanh)2142 2167 y Fo(\000)p 2142 2182 40 4 v 2146 2233 a(4)2215 2202 y Fy(=)c Fv(D)2383 2169 y FB(\000)p Fo(1)2469 2202 y Fy(.)39 b(Since)28 b(the)h(norm)f Fu(k)h(k)3259 2215 y FC(d)0 2310 y Fy(is)e(adapted)h(to) g(the)g(cones)f Fv(C)975 2323 y FC(j)1008 2310 y Fy(\()p Fv(a;)15 b(b;)g(c)p Fy(\))29 b(and)f(is)f(equiv)-5 b(alen)n(t)27 b(to)h(the)g(uniform)f(norm,)i(Theorem)f(3.2)0 2419 y(giv)n(es)g(for)h Fv(f)39 b Fy(in)29 b Fv(C)597 2432 y Fo(0)632 2419 y Fy(\()p Fv(a;)15 b(b;)g(c)p Fy(\))30 b(with)36 b(^)-51 b Fv(\026)p Fy(\()p Fv(f)10 b Fy(\))25 b(=)h(1,)1067 2618 y Fu(kL)1174 2581 y FC(k)1207 2589 y Fl(1)1238 2581 y Fo(+)p FB(\001\001\001)o FC(k)1373 2592 y Fn(`)1404 2618 y Fv(f)k Fu(\000)20 b Fq(1)p Fu(k)1662 2631 y FB(1)1754 2618 y Fu(\024)25 b Fy(const)o Fv(D)2124 2581 y FB(\000)p FC(`)2204 2618 y Fv(:)0 2817 y Fy(F)-7 b(or)29 b Fv(n)d Fu(2)e Fr(N)7 b Fy(,)36 b(let)29 b Fv(`)p Fy(\()p Fv(n)p Fy(\))g(b)r(e)h(de\014ned)e(b)n(y:)922 3016 y Fv(n)d Fy(=)h Fv(k)1139 3029 y Fo(1)1194 3016 y Fy(+)20 b Fu(\001)15 b(\001)g(\001)g Fv(k)1448 3030 y FC(`)p Fo(\()p FC(n)p Fo(\))1585 3016 y Fy(+)20 b Fv(r)31 b Fy(with)e Fv(r)e(<)e(k)2150 3030 y FC(`)p Fo(\()p FC(n)p Fo(\)+1)2349 3016 y Fv(;)0 3215 y Fy(w)n(e)k(ha)n(v)n(e)536 3324 y Fu(kL)643 3287 y FC(n)685 3324 y Fv(f)g Fu(\000)20 b Fq(1)p Fu(k)942 3337 y FB(1)1034 3324 y Fu(\024)25 b(kL)1236 3287 y FC(r)1270 3324 y Fq(1)p Fu(k)1365 3337 y FB(1)1462 3324 y Fu(kL)1569 3287 y FC(k)1602 3295 y Fl(1)1633 3287 y Fo(+)p FB(\001\001\001)o Fo(+)p FC(k)1817 3298 y Fn(`)1848 3324 y Fv(f)30 b Fu(\000)19 b Fq(1)p Fu(k)2105 3337 y FB(1)2198 3324 y Fu(\024)25 b Fy(const)o Fv(D)2568 3287 y FB(\000)p FC(`)p Fo(\()p FC(n)p Fo(\))2735 3324 y Fv(:)0 3491 y Fy(F)-7 b(or)29 b(an)n(y)g(function)f Fv(f)35 b Fu(2)24 b Fq(L)p Fy(\(\001\),)30 b(applying)e(the)h(ab)r(o)n(v)n(e)g(inequalit)n(y)e(to)2393 3448 y FC(f)7 b Fo(+)p FC(R)p Fo(\()p FC(f)g Fo(\))p Ff(1)p 2367 3470 309 4 v 2372 3522 a Fo(^)-37 b FC(\026)p Fo(\()p FC(f)7 b Fo(\)+)p FC(R)p Fo(\()p FC(f)g Fo(\))2714 3491 y Fy(giv)n(es)877 3717 y Fu(kL)984 3680 y FC(n)1026 3717 y Fv(f)30 b Fu(\000)c Fy(^)-51 b Fv(\026)p Fy(\()p Fv(f)10 b Fy(\))p Fu(k)1407 3730 y FB(1)1499 3717 y Fu(\024)25 b Fy(const)19 b Fu(\001)h Fv(D)1934 3680 y FB(\000)p FC(`)p Fo(\()p FC(n)p Fo(\))2101 3717 y Fu(k)p Fv(f)10 b Fu(k)2244 3731 y Ff(L)p Fo(\(\001\))2394 3717 y Fv(:)0 3916 y Fy(The)29 b(deca)n(y)g(of)g(correlations)f(follo)n(ws:)37 b(for)29 b Fv(f)35 b Fu(2)25 b Fv(L)k Fy(and)g Fv(g)f Fu(2)c Fv(L)2091 3883 y Fo(1)2127 3916 y Fy(\()7 b(^)-51 b Fv(\026)p Fy(\),)415 4036 y Fs(\014)415 4090 y(\014)415 4144 y(\014)415 4198 y(\014)415 4252 y(\014)415 4305 y(\014)445 4098 y(Z)444 4353 y Fo(\001)550 4220 y Fv(g)23 b Fu(\016)d Fv(F)749 4183 y FC(n)811 4220 y Fu(\001)g Fv(f)10 b(d)d Fy(^)-51 b Fv(\026)19 b Fu(\000)1117 4098 y Fs(Z)1115 4353 y Fo(\001)1221 4220 y Fv(f)10 b(d)d Fy(^)-51 b Fv(\026)1388 4098 y Fs(Z)1386 4353 y Fo(\001)1492 4220 y Fv(g)s(d)7 b Fy(^)-51 b Fv(\026)1636 4036 y Fs(\014)1636 4090 y(\014)1636 4144 y(\014)1636 4198 y(\014)1636 4252 y(\014)1636 4305 y(\014)1741 4220 y Fy(=)1885 4036 y Fs(\014)1885 4090 y(\014)1885 4144 y(\014)1885 4198 y(\014)1885 4252 y(\014)1885 4305 y(\014)1915 4098 y(Z)1913 4353 y Fo(\001)2019 4220 y Fv(g)s Fy([)p Fu(L)2150 4183 y FC(n)2193 4220 y Fy(\()p Fv(f)10 b Fy(\))20 b Fu(\000)27 b Fy(^)-51 b Fv(\026)p Fy(\()p Fv(f)10 b Fy(\)])p Fv(d)d Fy(^)-51 b Fv(\026)2721 4036 y Fs(\014)2721 4090 y(\014)2721 4144 y(\014)2721 4198 y(\014)2721 4252 y(\014)2721 4305 y(\014)1741 4474 y Fu(\024)74 b Fy(const)19 b Fu(\001)h Fv(D)2225 4437 y FB(\000)p FC(`)p Fo(\()p FC(n)p Fo(\))2393 4474 y Fu(k)p Fv(f)10 b Fu(k)2536 4488 y Ff(L)p Fo(\(\001\))2685 4474 y Fu(k)p Fv(g)s Fu(k)2820 4487 y Fo(1)2855 4474 y Fv(:)0 4673 y Fy(Th)n(us)28 b(w)n(e)i(can)f(tak)n(e)g Fv(u)753 4686 y FC(n)820 4673 y Fy(=)c Fv(D)988 4640 y FB(\000)p FC(`)p Fo(\()p FC(n)p Fo(\))1156 4673 y Fy(.)132 4782 y(W)-7 b(e)32 b(no)n(w)g(pro)n(v)n(e)g(that)h Fv(u)956 4795 y FC(n)1031 4782 y Fy(has)e(the)i(announced)e(b)r(eha)n(vior)h (for)g(exp)r(onen)n(tial)f(or)i(p)r(olynomial)0 4890 y(sequences)28 b Fv(!)453 4903 y FC(n)495 4890 y Fy(\()p Fv(g)s Fy(\).)p eop %%Page: 21 21 21 20 bop 832 26 a Fx(Piecewise)28 b(in)n(v)n(ertible)f(maps)i(in)f (higher)g(dimensions)745 b Fy(21)85 302 y Fq(Estimates)32 b(on)i Fv(u)740 315 y FC(n)85 469 y Fy(T)-7 b(o)30 b(estimate)g(the)g (rate)g(of)g(mixing)f Fv(u)1343 482 y FC(n)1386 469 y Fy(,)h(w)n(e)h(ha)n(v)n(e)e(to)h(analyse)f(the)h(asymptotic)g(b)r(eha)n (vior)f(of)h(the)85 577 y(sequence)f Fv(k)495 590 y FC(j)527 577 y Fy(.)40 b(Recall)27 b(that:)979 822 y Fv(k)1025 835 y FC(`)p Fo(+1)1161 822 y Fy(=)e(inf)5 b Fu(f)p Fv(k)28 b Fu(\025)c Fv(k)1619 835 y Fo(0)1679 822 y Fy(:)i Fv(R)1796 835 y FC(`)1825 822 y Fy(\()p Fv(s)21 b Fy(+)f Fv(k)s Fy(\))25 b Fu(\024)2220 762 y Fv(R)2287 775 y Fo(0)2322 762 y Fy(\()p Fv(s)p Fy(\))p 2220 802 212 4 v 2288 884 a Fv(D)2441 822 y Fu(g)85 1038 y Fy(and)k(observ)n(e)f(that)i(an)f (easy)g(induction)e(using)h(the)h(de\014nitions)e(giv)n(es:)401 1311 y Fv(R)468 1324 y FC(`)498 1311 y Fy(\()p Fv(n)p Fy(\))f(=)f Fv(D)812 1273 y FC(`)842 1311 y Fv(R)909 1324 y Fo(0)944 1311 y Fy(\()p Fv(k)1024 1324 y Fo(1)1079 1311 y Fy(+)20 b Fu(\001)15 b(\001)g(\001)20 b Fy(+)g Fv(k)1426 1324 y FC(`)1475 1311 y Fy(+)h Fv(n)p Fy(\))f(+)1810 1198 y FC(`)1758 1225 y Fs(X)1772 1414 y FC(i)p Fo(=1)1903 1311 y Fv(D)1978 1273 y FC(`)p Fo(+1)p FB(\000)p FC(i)2160 1311 y Fv(R)2227 1324 y Fo(0)2262 1311 y Fy(\()p Fv(k)2342 1324 y FC(i)p Fo(+1)2469 1311 y Fy(+)g Fu(\001)15 b(\001)g(\001)k Fy(+)i Fv(k)2816 1324 y FC(`)2865 1311 y Fy(+)f Fv(n)p Fy(\))p Fv(:)85 1583 y Fy(where)30 b Fv(R)406 1596 y Fo(0)441 1583 y Fy(\()p Fv(m)p Fy(\))25 b(=)704 1516 y Fs(P)799 1609 y FC(k)r(>m)958 1583 y Fv(!)1013 1596 y FC(k)1051 1583 y Fy(\()p Fv(g)s Fy(\).)85 1691 y(>F)-7 b(rom)30 b(these)f(remarks)g(w)n(e)h(get)f(the)g(follo)n(wing)e (results.)216 1870 y Fu(\017)44 b Fy(If)29 b(there)g(exists)g(0)c Fv(<)h(\032)f(<)h Fy(1)j(suc)n(h)f(that)i Fv(!)1704 1883 y FC(n)1771 1870 y Fy(=)c Fv(O)r Fy(\()p Fv(\032)2013 1837 y FC(n)2056 1870 y Fy(\))k(then)f Fv(R)2387 1883 y Fo(0)2422 1870 y Fy(\()p Fv(n)p Fy(\))d(=)g Fv(O)r Fy(\()p Fv(\032)2810 1837 y FC(n)2853 1870 y Fy(\))j(and)g(one)h(can) 305 1978 y(tak)n(e)f Fv(k)541 1991 y FC(j)598 1978 y Fy(=)c Fv(p)g Fy(=)h(log)15 b Fv(D)r(=)p Fu(j)g Fy(log)g Fv(\032)p Fu(j)20 b Fy(+)h(10.)39 b(This)28 b(implies)f Fv(u)2147 1991 y FC(n)2214 1978 y Fy(=)f Fv(D)2383 1946 y FB(\000)p FC(`)p Fo(\()p FC(n)p Fo(\))2575 1978 y Fy(=)f Fv(D)2743 1946 y FB(\000)p FC(n=p)2899 1978 y Fy(.)216 2160 y Fu(\017)44 b Fy(If)f(there)h(exists)f Fv(\013)50 b(>)g Fy(1)44 b(suc)n(h)e(that)i Fv(!)1687 2173 y FC(n)1779 2160 y Fy(=)50 b Fv(O)r Fy(\()p Fv(n)2052 2128 y FB(\000)p FC(\013)2147 2160 y Fy(\))44 b(then)f Fv(R)2506 2173 y Fo(0)2541 2160 y Fy(\()p Fv(n)p Fy(\))51 b(=)f Fv(O)r Fy(\()p Fv(n)2985 2128 y FB(\000)p FC(\013)p Fo(+1)3161 2160 y Fy(\))44 b(and)305 2268 y Fv(R)372 2281 y FC(`)401 2268 y Fy(\()p Fv(s)20 b Fy(+)h Fv(k)631 2281 y FC(`)p Fo(+1)741 2268 y Fy(\))k(=)g Fv(O)978 2196 y Fs(\000)1019 2268 y Fv(R)1086 2281 y Fo(0)1121 2268 y Fy(\()p Fv(s)c Fy(+)f Fv(k)1351 2281 y FC(`)p Fo(+1)1461 2268 y Fy(\))g Fu(\001)g Fv(D)1635 2236 y FC(`)1665 2196 y Fs(\001)1736 2268 y Fy(so,)29 b(the)g Fv(k)2065 2281 y FC(`)2094 2268 y Fy('s)h(ma)n(y)f(b)r(e)g(tak)n(en)g(to)h(b)r(e)305 2402 y Fu(\030)24 b Fy(const)p Fv(D)726 2338 y Fn(`)p 684 2349 107 3 v 684 2386 a(\013)p Fm(\000)p Fl(1)833 2402 y Fy(so)k Fv(`)p Fy(\()p Fv(n)p Fy(\))e Fu(\030)e Fy(\()p Fv(\013)d Fu(\000)f Fy(1\))1512 2363 y Fo(log)11 b FC(n)p 1505 2382 147 4 v 1505 2433 a Fo(log)f FC(D)1680 2402 y Fy(+)20 b(const)29 b(and)g(th)n(us)1031 2619 y Fv(u)1081 2632 y FC(n)1148 2619 y Fy(=)c Fv(D)1316 2582 y FB(\000)p FC(`)p Fo(\()p FC(n)p Fo(\))1508 2619 y Fy(=)h Fv(O)r Fy(\()p Fv(e)1746 2582 y FB(\000)p Fo(\()p FC(\013)p FB(\000)p Fo(1\))10 b(log)h FC(n)2112 2619 y Fy(\))25 b(=)g Fv(O)r Fy(\()p Fv(n)2419 2582 y FB(\000)p FC(\013)p Fo(+1)2596 2619 y Fy(\))p Fv(:)85 2918 y Fh(References)85 3115 y Fy([ADU])44 b(J.)29 b(Aaronson,)f(M.)h(Denk)n(er)f(&)h(M.)g (Urbanski)e Ft(Er)l(go)l(dic)k(the)l(ory)f(for)g(Markov)g(\014b)l(er)l (e)l(d)h(sys-)376 3223 y(tems)39 b(and)f(p)l(ar)l(ab)l(olic)f(r)l (ational)h(maps.)e Fy(T)-7 b(rans.)36 b(Amer.)h(Math.)h(So)r(c.)f (\(1993\),)i Fq(337)f Fy(\(2\),)376 3331 y(495-548.)85 3513 y([Bi])156 b(G.)31 b(Birk)n(oh\013)e Ft(Extensions)34 b(of)e(Jentzch's)h(the)l(or)l(em.)c Fy(T)-7 b(rans.)30 b(Amer.)h(Math.)f(So)r(c.)h(\(1957\),)376 3621 y Fq(85)p Fy(,)g(219-227.)85 3803 y([Bo])137 b(R.)23 b(Bo)n(w)n(en)g Ft(Equilibrium)k(states)f(and)f(the)h(er)l(go)l(dic)g(the)l(ory)e(of)i (A)n(nosov)g(di\013e)l(omorphisms.)376 3911 y Fy(Lecture)k(Notes)e(in)h (Mathematics,)g(V)-7 b(ol.)28 b Fq(470)p Fy(.)j(Springer-V)-7 b(erlag,)28 b(Berlin,)h(1975.)85 4093 y([Bu1])88 b(J.)32 b(Buzzi)e Ft(A)n(bsolutely)j(c)l(ontinuous)g(invariant)g(me)l(asur)l (es)f(for)g(generic)j(pie)l(c)l(ewise)d(a\016ne)376 4201 y(and)g(exp)l(anding)g(maps)c Fy(In)n(t.)h(J.)h(Chaos)f(&)g(Bif.)g (\(1999\),)h Fq(9)f Fy(\(9\),)h(1743-1750.)85 4383 y([Bu2])88 b(J.)48 b(Buzzi)e Ft(A.c.i.m.'s)h(as)g(e)l(quilibrium)h(states)g(for)f (pie)l(c)l(ewise)h(invertible)h(maps)d Fy(pr)n(\023)-41 b(e-)376 4491 y(publication)27 b(I.M.L.)j Fq(98-23)h Fy(\(1998\))e(\(see)g Fd(http://iml.univ-mrs.fr/)q Fy(~)t Fd(buzzi)p Fy(\))85 4673 y([BPS])72 b(J.)43 b(Buzzi,)h(F.)f(P)n (accaut,)h(B.)f(Sc)n(hmitt)e Ft(Conformal)h(me)l(asur)l(es)g(for)h (multidimensional)376 4782 y(exp)l(anding)d(maps.)e Fy(Preprin)n(t)e (Lab)r(oratoire)i(de)g(T)-7 b(op)r(ologie)36 b(#196)i(\(Dijon,)i (1999\))e(\(see)376 4890 y Fd(http://iml.univ-mrs.fr/)q Fy(~)5 b Fd(buzzi)p Fy(\))p eop %%Page: 22 22 22 21 bop 0 26 a Fy(22)745 b Fx(J)n(\023)-41 b(er^)d(ome)29 b(Buzzi)g(&)g(V)n(\023)-41 b(eronique)28 b(Maume-Desc)n(hamps)0 302 y Fy([Co])136 b(W.)23 b(Co)n(wieson,)f Ft(Pie)l(c)l(ewise)k(smo)l (oth)d(exp)l(anding)i(maps)f(in)g Fr(R)2292 270 y FC(d)2335 302 y Ft(,)f Fy(Ph.D.)f(Thesis,)g(U.C.L.A.,)291 411 y(Berk)n(eley)30 b(\(see)f Fd(http://math.usc.edu/)q Fy(~)t Fd(cowieson)p Fy(\).)0 607 y([DKS])59 b(M.)36 b(Denk)n(er,)h(C.)e(Grillen)n(b)r (erger,)g(K.)h(Sygm)n(und)e Ft(Er)l(go)l(dic)i(the)l(ory)g(on)h(c)l (omp)l(act)e(sp)l(ac)l(es.)291 715 y Fy(Lecture)30 b(notes)e(in)h (mathematics)f Fq(527)p Fy(,)j(Springer,)e(Berlin,)f(1976.)0 911 y([FS])137 b(P)-7 b(.)22 b(F)-7 b(errero,)23 b(B.)f(Sc)n(hmitt)f Ft(R)n(uel)t(le)26 b(Perr)l(on)e(F)-7 b(r)l(ob)l(enius)24 b(the)l(or)l(ems)g(and)g(pr)l(oje)l(ctive)g(metrics.)291 1020 y Fy(Collo)r(que)29 b(Math.)h(So)r(c.)g(J.)g(Boly)n(ai)e(Random)g (Fields.)g(Estergom)h(\(Hungary\))g(\(1979\).)0 1216 y([Go])130 b(P)-7 b(.)66 b(Gora)g Ft(Pr)l(op)l(erties)f(of)g(invariant) h(me)l(asur)l(es)e(for)h(pie)l(c)l(ewise)h(exp)l(anding)f(one-)291 1324 y(dimensional)28 b(tr)l(ansformations)e(with)i(summable)f(oscil)t (lations)i(of)e(derivative.)f Fy(Erg.)f(Th.)291 1432 y(and)k(Dyn.)h(Syst.)f(\(1994\),)h Fq(14)p Fy(,)g(475-492.)0 1629 y([Ho])133 b(F.)44 b(Hofbauer)e Ft(Pie)l(c)l(ewise)j(invertible)h (dynamic)l(al)d(systems.)g Fy(Probab.)f(Theory)i(Relat.)291 1737 y(Fields)28 b Fq(72)i Fy(\(1986\),)g(no.)f(3,)h(359{386.)0 1933 y([HK])109 b(F.)35 b(Hofbauer,)f(G.)g(Keller)f Ft(Er)l(go)l(dic)i (pr)l(op)l(erties)g(of)g(invariant)h(me)l(asur)l(es)f(for)g(pie)l(c)l (ewise)291 2041 y(monotonic)d(tr)l(ansformations)c Fy(Math.)i(Z.)f Fq(180)p Fy(,)i(\(1982\),)f(119{140.)0 2237 y([Ke])136 b(G.)26 b(Keller)f Ft(Markov)i(extensions,)j(zeta)e(functions,)h(and)f (F)-7 b(r)l(e)l(dholm)27 b(the)l(ory)h(for)f(pie)l(c)l(ewise)291 2346 y(invertible)k(dynamic)l(al)e(systems.)d Fy(T)-7 b(rans.)27 b(Amer.)g(Math.)g(So)r(c.)h Fq(314)g Fy(\(1989\),)f(no.)g (2,)h(433{)291 2454 y(497.)0 2650 y([KMS])46 b(A.)55 b(Kondah,)60 b(V.)54 b(Maume)h(&)f(B.)h(Sc)n(hmitt)e Ft(Vitesse)i(de)f(c)l(onver)l(genc)l(e)i(vers)e(l')n(\023)-43 b(etat)291 2759 y(d')n(\023)g(equilibr)l(e)45 b(p)l(our)f(des)g (dynamiques)g(markoviennes)i(non)e(h\177)-45 b(old)n(\023)i(eriennes.) 45 b Fy(Ann.)e(Inst.)291 2867 y(P)n(oincar)n(\023)-41 b(e)27 b(Sec.)j(Prob.)f(Stat.)g(\(1997\))g Fq(33)i Fy(\(6\))e(675-695.) 0 3063 y([Li])163 b(C.)30 b(Liv)n(erani)e Ft(De)l(c)l(ay)j(of)g(c)l (orr)l(elations.)e Fy(Ann.)g(of)g(Math.)h(\(1995\),)f Fq(142)i Fy(\(2\),)f(239-301)0 3259 y([LSV])73 b(C.)25 b(Liv)n(erani,)e(B.)h(Saussol,)f(S.)h(V)-7 b(aien)n(ti)21 b Ft(Conformal)k(me)l(asur)l(e)h(and)g(de)l(c)l(ay)f(of)g(c)l(orr)l (elations)291 3368 y(for)j(c)l(overing)h(weighte)l(d)g(systems)c Fy(Ergo)r(dic)g(th.)h(&)h(dynam.)f(syst.)g Fq(18)p Fy(,)h(1998,)g (1399{1420.)0 3564 y([M])163 b(V.)48 b(Maume-Desc)n(hamps)e Ft(Pr)l(oje)l(ctive)j(metrics)f(and)g(mixing)g(pr)l(op)l(erties)f(on)h (towers.)291 3672 y Fy(Preprin)n(t)28 b(\(see)h Fd (http://www.unige.ch/math/biblio/preprint/pp9)q(9.html)p Fy(\).)0 3868 y([R])178 b(M.)23 b(Ryc)n(hlik)d Ft(Bounde)l(d)k (variation)g(and)h(invariant)g(me)l(asur)l(es)c Fy(Studia)f(Math.)j Fq(76)p Fy(,)h(\(1983\),)291 3976 y(69{80.)0 4173 y([Sc])155 b(B.)39 b(Sc)n(hmitt)f Ft(Existenc)l(e)j(of)f(acip)g(for)f(exp)l (anding)i(maps)e(of)g(the)i(interval.)e Fy(Dynamical)291 4281 y(Systems)29 b(and)g(Ergo)r(dic)g(Theory)g(Banac)n(h)g(Cen)n(ter)g (Publications.)e(W)-7 b(arsza)n(w)n(a)28 b(\(1989\).)0 4477 y([Y1])133 b(L.-S.)28 b(Y)-7 b(oung)25 b Ft(R)l(eturn)k(times)g (and)g(r)l(ates)g(of)g(mixing)e Fy(Israel)e(J.)i(Math)g Fq(110)p Fy(,)i(\(1999\),)f(153{)291 4585 y(188.)0 4782 y([Y2])133 b(L.-S.)32 b(Y)-7 b(oung)31 b Ft(De)l(c)l(ay)h(of)h(Corr)l 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