This is a multi-part message in MIME format. ---------------9910130249372 Content-Type: text/plain; name="99-383.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-383.comments" The present note is aimed at proving an existence result of SRB measures for a class of topological Markov chains with infinitely many states, extending to saddle type sets previous results which generalizes several known versions of the Folklore Theorem to higher dimensional Markov transformations with infinitely many expanding branches. To the best of my knowledge these type of problems were considered for the first time by Newhouse and Jacobson. Here, as for Markov piecewise expanding endomorphisms, there is an essential difference between a finite and an infinite number of hyperbolic branches since derivatives grows to infinite with the number of branches so relations between the first and the second derivative become crucial. ---------------9910130249372 Content-Type: text/plain; name="99-383.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-383.keywords" SRB measures, invariant measures, Markov transformations, ergodic ---------------9910130249372 Content-Type: application/postscript; name="folklorethmtwo.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="folklorethmtwo.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.76 (MiKTeX 1.11) Copyright 1997 Radical Eye Software (www.radicaleye.com) %%Title: folklorethmtwo.dvi %%CreationDate: Wed Oct 13 00:25:10 1999 %%Pages: 20 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: 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Fp(z)t Fq(\))56 b Fm(8)14 b Fp(\036)22 b Fm(2)i Fp(C)2878 3358 y Fo(0)2915 3392 y Fq(\()p Fp(M)9 b Fq(\))p Fm(g)23 b Fp(>)g Fq(0)515 3654 y(By)k(this)h(reason)e(they)i(are)e(sometimes)i(called)f Fn(er)l(go)l(dic)k(attr)l(actors)p Fq(.)37 b(Cf.)g([5])27 b(and)h([10)o(].)639 3754 y(As)34 b(far)g(as)f(I)h(kno)n(w,)h(h)n(yp)r (erb)r(olic)e(dynamical)g(systems)h(are)f(the)h(unique)g(where)g(ex-) 515 3853 y(istence)d(and)h(prop)r(erties)f(of)g(SRB)h(measures)e(are)h (w)n(ell)h(understo)r(o)r(d.)48 b(Ma)5 b(jor)31 b(attempts)515 3953 y(to)j(pro)n(v)n(e)g(existence)g(of)h(SRB)g(measures)e(in)i(non)g (h)n(yp)r(erb)r(olic)f(systems)g(w)n(ere)g(made)h(in)515 4053 y(the)d(nineties)g(at)g(lo)n(w)n(er)e(dimensional)i(settings,)g (for)g(in)n(terv)-5 b(al)31 b(transformations)f(as)h(w)n(ell)515 4152 y(as)40 b(H)n(\023)-39 b(enon)40 b(maps.)76 b(See)41 b([3)o(,)k(Chapter)40 b(V],)h([13)o(])g(and)g(references)f(therein.)76 b(Recen)n(tly)515 4252 y(some)24 b(new)h(higher)f(dimensional)h (examples,)f(including)h(partially)f(h)n(yp)r(erb)r(olic)h(ones,)g(ex-) 515 4351 y(hibiting)k(SRB)g(measures)e(with)i(large)f(unstable)g (dimension)h(w)n(ere)f(in)n(tro)r(duced.)40 b(Cf.)g([1])515 4451 y(and)33 b([14)o(].)54 b(Nev)n(ertheless,)33 b(to)h(giv)n(e)e (e\013ectiv)n(ely)h(v)n(eri\014ed)f(conditions)h(for)g(the)g(existence) 515 4551 y(of)f(SRB)h(mesures)f(in)h(systems)f(whic)n(h)g(are)g(h)n(yp) r(erb)r(olic)g(in)h(large)e(parts)h(of)g(their)h(phase)515 4650 y(space)27 b(without)h(b)r(eing)f(uniformly)h(h)n(yp)r(erb)r(olic) f(remains)g(b)n(y)g(large)f(an)h(op)r(en)h(question.)639 4750 y(The)22 b(presen)n(t)f(note)g(is)h(aimed)f(at)h(pro)n(ving)e(an)h (existence)h(result)f(of)h(SRB)f(measures)g(for)515 4850 y(a)32 b(class)f(of)h(top)r(ological)f(Mark)n(o)n(v)f(c)n(hains)h(with) i(in\014nitely)g(man)n(y)f(states,)h(extending)f(to)515 4949 y(saddle)23 b(t)n(yp)r(e)g(sets)g(previous)f(results)h(whic)n(h)g (generalizes)f(sev)n(eral)f(kno)n(wn)i(v)n(ersions)e(of)j(the)1926 5255 y(1)p eop %%Page: 2 2 2 1 bop 515 523 a Fq(F)-7 b(olklore)23 b(Theorem)g(to)h(higher)g (dimensional)f(Mark)n(o)n(v)f(transformations)h(with)h(in\014nitely)515 623 y(man)n(y)j(expanding)g(branc)n(hes.)639 722 y(Recen)n(tly)-7 b(,)24 b(L.S.)f(Y)-7 b(oung)22 b([14)o(])h(prop)r(osed)e(a)i(general)e (abstract)h(mo)r(del)g(somewhat)g(simi-)515 822 y(lar)c(to)h(that)h(in) n(tro)r(duced)f(here,)i(where)e(su\016cien)n(t)g(conditions)g(are)f (giv)n(en)h(for)g(the)g(existence)515 922 y(of)k(SRB)h(measures)f(in)h (certain)f(non)g(h)n(yp)r(erb)r(olic)h(systems.)35 b(It)24 b(is)g(w)n(orth)e(to)i(men)n(tion)g(that)515 1021 y(generalized)19 b(horsesho)r(es)f(presen)n(ted)i(here)f(app)r(ear)h(naturally)f(when)h (inducing)h(h)n(yp)r(erb)r(ol-)515 1121 y(icit)n(y)k(in)g(non-uniform)f (h)n(yp)r(erb)r(olic)h(systems.)35 b(Namely)-7 b(,)26 b(w)n(e)e(can)h(sho)n(w)f(that,)i(giv)n(en)e(a)g Fp(C)3341 1091 y Fo(2)515 1220 y Fq(di\013eomorphism)h Fp(f)34 b Fq(of)25 b(a)g(riemannian)g(manifold)g(preserving)f(a)h(h)n(yp)r(erb) r(olic)g(non-atomic)515 1320 y(measure)31 b Fp(\026)p Fq(,)j(and)e(giv)n(en)g(a)g(p)r(oin)n(t)g Fp(x)g Fm(2)f Fp(supp)h(\026)p Fq(,)h(there)f(is)h(a)f(neigh)n(b)r(orho)r(o)r(d)f Fp(R)h Fq(=)e Fp(R)q Fq(\()p Fp(x)p Fq(\))515 1420 y(con)n(taining)h(a) h(horsesho)r(e)e(with)j(in\014nitely)g(man)n(y)e(branc)n(hes)g(\012)i (obtained)e(from)h(a)g(suit-)515 1519 y(able)20 b(sequence)h(of)g (returns)f(de\014ning)h(coun)n(tably)f(man)n(y)g(\(in)i(general)e (in\014nite\))i(h)n(yp)r(erb)r(olic)515 1619 y(branc)n(hes)27 b Fp(f)910 1589 y Fl(n)951 1597 y Fi(i)1006 1619 y Fq(:)e Fp(S)1105 1631 y Fl(i)1157 1619 y Fm(\000)-15 b(!)25 b Fp(U)1372 1631 y Fl(i)1400 1619 y Fq(.)39 b(\012)29 b(has)f(the)h(same)f(top)r(ological)f(picture)i(and)f(regular)f(mea-) 515 1719 y(sure)38 b(theoretical)g(prop)r(erties)g(of)h(generalized)e (horsesho)r(es)g(in)n(tro)r(duced)i(here.)70 b(These)515 1818 y(results)29 b(strongly)g(suggests)g(that)i(the)f(presence)g(of)g (horsersho)r(es)e(with)j(in\014nitely)g(man)n(y)515 1918 y(branc)n(hes)g(ma)n(y)h(b)r(e)h(indeed)h(a)e(necessary)f(and)i (su\016cien)n(t)f(condition)h(for)f(the)h(existence)515 2017 y(of)f(SRB)g(measures)f(in)h(certain)g(non-uniformly)f(h)n(yp)r (erb)r(olic)h(systems.)50 b(P)n(esin's)31 b(theory)515 2117 y(tec)n(hniques)f(and)h(ideas)g(from)f([4])h(are)f(b)r(een)h(used) g(to)g(k)n(eep)f(trac)n(k)g(this)h(t)n(yp)r(e)h(of)f(results.)515 2217 y(The)c(follo)n(wing)g(is)g(our)g(main)h(result.)515 2416 y Fh(F)-8 b(olklore)36 b(Theorem)64 b Fn(L)l(et)34 b Fg(R)c Fq(=)h Fp(B)1720 2386 y Fl(u)1785 2416 y Fm(\002)22 b Fp(B)1939 2386 y Fl(s)2008 2416 y Fn(b)l(e)35 b(a)f(r)l(e)l(ctangle)h (in)f Ff(R)2703 2386 y Fl(m)2806 2416 y Fn(and)h Fp(T)42 b Fq(:)31 b(\012)g Fe(\011)j Fn(b)l(e)515 2516 y Fp(C)580 2485 y Fo(2)652 2516 y Fn(horsesho)l(e)i(c)l(onstructe)l(d)d Fg(R)i Fn(de\014ne)l(d)f(by)h(in\014nitely)g(many)f Fp(C)2606 2485 y Fo(2)2678 2516 y Fn(hyp)l(erb)l(olic)j(br)l(anches)515 2615 y Fp(T)564 2627 y Fl(i)614 2615 y Fq(:)23 b Fp(S)711 2627 y Fl(i)762 2615 y Fm(\000)-15 b(!)24 b Fp(U)976 2627 y Fl(i)1032 2615 y Fn(such)30 b(that)f Fm(f)p Fp(S)1483 2627 y Fl(i)1510 2615 y Fm(g)g Fn(c)l(overs)h Fg(R)f Fn(up)g(to)g(a)h(subset)f(of)h(zer)l(o)f(L)l(eb)l(esgue)h(me)l(asur)l (e.)515 2715 y(Supp)l(ose)h(in)h(addition)h(that)e(the)g(non-line)l(ar) h(exp)l(ansion)f(along)h(invariant)g(manifolds)i(is)515 2814 y(b)l(ounde)l(d)27 b(fr)l(om)f(b)l(elow)i(by)f(some)f Fp(\025)e(>)f Fq(1)i Fn(and)i(that)g(the)f(fol)t(lowing)j Fh(b)s(ounded)e(distortion)515 2914 y(condition)i Fn(holds:)836 3134 y Fq(\()p Fh(BD)p Fq(\))253 b(sup)1302 3202 y Fl(i>)p Fo(0)1690 3073 y Fq(sup)1815 3093 y Fl(z)r Fk(2)p Fl(S)1935 3101 y Fi(i)1979 3073 y Fm(k)p Fp(D)2092 3043 y Fo(2)2090 3096 y Fl(k)q(l)2151 3073 y Fp(T)2200 3085 y Fl(i)2227 3073 y Fq(\()p Fp(z)t Fq(\))p Fm(k)p 1456 3115 1154 4 v 1456 3191 a Fq(inf)1557 3203 y Fl(z)r(;w)r Fk(2)p Fl(S)1747 3211 y Fi(i)1790 3191 y Fm(k)p Fp(D)1901 3203 y Fl(j)1936 3191 y Fp(T)1985 3203 y Fl(i)2012 3191 y Fq(\()p Fp(z)t Fq(\))p Fm(k)18 b(\001)g(k)p Fp(D)2331 3203 y Fl(j)2365 3191 y Fp(T)2414 3203 y Fl(i)2441 3191 y Fq(\()p Fp(w)r Fq(\))p Fm(k)2642 3134 y(\024)23 b Fq(\001)g Fp(<)g Fq(+)p Fm(1)215 b Fq(\(1\))515 3364 y Fn(for)33 b Fp(j;)28 b(k)s(;)g(l)h Fq(=)f(1)p Fp(;)14 b Fm(\001)g(\001)g(\001)27 b Fp(;)14 b(m)32 b Fn(and)h(for)h(every)f Fp(i)28 b(>)g Fq(0)p Fn(.)46 b(Then)34 b Fp(T)43 b Fn(pr)l(eserves)33 b(a)g(unique)g(er)l (go)l(dic)515 3464 y(SRB)c(me)l(asur)l(e)h Fp(\026)23 b Fq(=)g Fp(\026)1243 3476 y Fo(\012)1325 3464 y Fn(supp)l(orte)l(d)30 b(on)g Fq(\012)p Fn(.)39 b(F)-6 b(urther,)30 b(the)g(er)l(go)l(dic)i(b) l(asin)e(of)h Fp(\026)f Fn(c)l(overs)g Fg(R)515 3563 y Fn(up)f(to)h(a)g(L)l(eb)l(esgue)g(me)l(asur)l(e)f(zer)l(o)h(set.)515 3763 y Fq(W)-7 b(e)32 b(refer)e(to)i(next)f(Section)h(for)f (de\014nitions)g(and)h(pro)r(ofs.)48 b(T)-7 b(o)31 b(the)h(b)r(est)f (of)h(m)n(y)f(kno)n(wl-)515 3862 y(edge)24 b(these)g(t)n(yp)r(e)g(of)h (problems)e(w)n(ere)h(considered)f(for)h(the)g(\014rst)g(time)h(b)n(y)f (Newhouse)g(and)515 3962 y(Jacobson.)45 b(See)30 b([7])h(and)g (references)e(therein.)47 b(Nev)n(ertheless,)31 b(our)f(approac)n(h)f (not)h(only)515 4061 y(pro)n(vides)d(higher)i(dimensional)f (generalizations)f(but)j(it)f(also)f(giv)n(es)g(a)h(new)g(pro)r(of)f (of)h([7,)515 4161 y(Theorem)k(1].)58 b(Here,)36 b(as)e(for)g(Mark)n(o) n(v)e(piecewise)i(expanding)g(endomorphisms,)h(there)515 4261 y(is)30 b(an)f(essen)n(tial)g(di\013erence)h(b)r(et)n(w)n(een)g(a) f(\014nite)i(and)f(an)f(in\014nite)i(n)n(um)n(b)r(er)f(of)g(h)n(yp)r (erb)r(olic)515 4360 y(branc)n(hes)d(since)i(deriv)-5 b(ativ)n(es)27 b(gro)n(ws)g(with)i Fp(i)g Fq(and)f(relations)g(b)r(et)n (w)n(een)h(the)g(\014rst)f(and)h(the)515 4460 y(second)c(deriv)-5 b(ativ)n(e)26 b(b)r(ecome)g(crucial.)35 b(In)27 b(this)f(w)n(ork)f(w)n (e)g(p)r(ermit)i(to)f(the)h(stable)e(cylinder)515 4560 y Fp(S)566 4572 y Fl(i)621 4560 y Fq(to)i(accum)n(ulate)g(to)n(w)n(ard) f(admissible)h(manifolds)h(\(see)f(b)r(elo)n(w\).)639 4659 y(Condition)i(\(1\))g(ab)r(o)n(v)n(e)f(is)h(little)h(bit)f (stronger)e(that)j(b)r(ounded)f(distortion)f(condition)515 4759 y(\(D1\))j(in)g([7].)46 b(Ho)n(w)n(ev)n(er,)30 b(it)h(allo)n(w)f (us)h(to)g(go)f(b)r(ey)n(ond)g(dimension)h(t)n(w)n(o)f(and)g(to)h (consider)515 4858 y(generalized)c(horsesho)r(es)g(with)i(unstable)f (Can)n(tor)f(set)i(with)g(p)r(ositiv)n(e)f(measure)f(as)h(w)n(ell,)515 4958 y(impro)n(ving)e(Jacobson-Newhouse's)f(results,)i(ev)n(en)g(in)h (dimension)f(t)n(w)n(o.)1926 5255 y(2)p eop %%Page: 3 3 3 2 bop 639 523 a Fq(Roughly)33 b(sp)r(eaking)f(what)i(w)n(e)f(do)g(is) g(to)g(pro)5 b(ject)32 b(the)i(dynamics)f(along)f(the)i(lea)n(v)n(es) 515 623 y(of)g(an)h(stable)f(lamination)g(on)h(lo)r(cal)f(unstable)h (manifolds)f Fp(W)47 b Fq(=)34 b Fp(W)2771 593 y Fl(u)2759 646 y(loc)2848 623 y Fq(.)58 b(Then)35 b(w)n(e)f(in-)515 722 y(tro)r(duce)28 b(certain)g(dynamically)g(de\014ned)g(Carath)n (\023)-39 b(eo)r(dory's)25 b(structure)j(\(or)g(shortly)-7 b(,)28 b(a)g(C-)515 822 y(structure\))i([8,)i(Chapter)e(1])g(on)h(the)g (unstable)f(Can)n(tor)g(sets)g(of)h(\012,)h(pro)n(ving)d(that)i(these) 515 922 y(pro)5 b(jections)22 b(are)h(in)h(fact)f(measure-theoretical)f (expanding)h(transformations)e([6,)k(Chap-)515 1021 y(ter)31 b(I)r(I)r(I].)g(Then)g(w)n(e)g(use)g(Ruelle's)g(theorem)f(to)h(see)g (pro)n(v)n(e)e(that)i(this)h(pro)5 b(jection)30 b(lea)n(v)n(es)515 1121 y(in)n(v)-5 b(arian)n(t)36 b(a)i(unique)g(ergo)r(dic)e(asymptotic) i(measure,)h(sa)n(y)e Fp(\026)2532 1133 y Fl(W)2607 1121 y Fq(,)k(whic)n(h)c(is)h(absolutely)515 1220 y(con)n(tin)n(uous)30 b(wrt)g(the)i(C-structure.)46 b(Let)31 b(us)g(stress)f(that)h(w)n(e)g (don't)g(require)f(Jacobians)515 1320 y(to)25 b(satisfy)h(a)f(H\177)-42 b(older)25 b(condition,)h(so)f(w)n(e)h(ha)n(v)n(e)e(to)i(use)g(a)f(pro) r(of)g(of)h(Ruelle's)g(theorem)f(due)515 1420 y(to)34 b(W)-7 b(alters)35 b([12)o(,)h(Theorem)e(8].)58 b(This)35 b(put)g(ourselv)n(es)e(in)i(a)g(setting)f(similar)g(to)h(previ-)515 1519 y(ous)30 b([11)o(].)47 b(Actually)-7 b(,)32 b(w)n(e)e(pro)n(v)n(e) f(that)i Fp(\026)1807 1531 y Fl(W)1914 1519 y Fq(is)f(absolutely)g(con) n(tin)n(uous)g(wrt)h(riemannian)515 1619 y(v)n(olume)26 b(of)h Fp(W)39 b Fq(i\013)28 b(the)f(Carath)n(\023)-39 b(eo)r(dory)24 b(dimensional)i(c)n(haracteristic)f(de\014ned)i(b)n(y)g (our)f(C-)515 1719 y(structure)31 b(attains)g(its)h(maxim)n(um)f(on)h (unstable)f(Can)n(tor)g(sets)g(and)g(this)h(o)r(ccurs)f(if)h(and)515 1818 y(only)d(if)h(unstable)f(Can)n(tor)g(sets)g(has)g(p)r(ositiv)n(e)g (Leb)r(esgue)g(measure.)42 b(Then)29 b(w)n(e)g(use)h(that)515 1918 y(this)k(dynamically)f(de\014ned)h(C-structure)e(is)i(in)n(v)-5 b(arian)n(t)32 b(b)n(y)i(the)g(holonom)n(y)e(of)i(the)g(sta-)515 2017 y(ble)d(lamination)g(to)g("in)n(tegrate")e(wrt)i(the)g(stable)g (lamination)g(getting)g(an)g(asymptotic)515 2117 y(SRB)f(measure.)43 b(Let)30 b(us)g(notice)f(that)i(this)f(approac)n(h)e(pro)n(vides)g (absolutely)i(con)n(tin)n(uit)n(y)515 2217 y(of)c(the)h(stable)g (lamination)f(wrt)g(Leb)r(esgue)g(measure)g(without)h(additional)f (assumptions)515 2316 y(as)h(condition)g(\(G4\))h(used)g(in)f([7].)515 2516 y Fh(Ac)m(kno)m(wledgmen)m(ts:)49 b Fq(I)35 b(w)n(ould)f(lik)n(e)h (to)f(express)g(m)n(y)g(gratitude)g(to)h(CIMA)-7 b(T)35 b(for)f(its)515 2615 y(hospitalit)n(y)i(and)h(to)g(prof.)66 b(Gonzalo)36 b(Con)n(treras)f(for)h(useful)i(discussions)e(during)h (the)515 2715 y(preparation)26 b(of)h(this)h(man)n(uscript.)515 3083 y Fr(2)134 b(Pro)t(of)45 b(of)h(the)f(F)-11 b(olklore)46 b(Theorem)515 3265 y Fq(W)-7 b(e)40 b(recall)e(\014rst)i(some)f (de\014nitions)h(and)f(terminology)g(to)g(state)h(prop)r(erly)e(our)h (main)515 3364 y(results.)h(Let)29 b Fg(R)c Fq(=)g Fp(B)1201 3334 y Fl(s)1256 3364 y Fm(\002)19 b Fp(B)1407 3334 y Fl(u)1476 3364 y Fm(\022)24 b Ff(R)1619 3334 y Fl(m)1688 3364 y Fq(,)30 b(where)e Fp(B)2049 3334 y Fl(s)2110 3364 y Fm(\022)d Ff(R)2254 3334 y Fl(s)2324 3364 y Fq(and)k Fp(B)2554 3334 y Fl(u)2623 3364 y Fm(\022)c Ff(R)2767 3334 y Fl(u)2845 3364 y Fq(are)j(unit)i(closed)515 3464 y(balls.)35 b(W)-7 b(e)26 b(shall)f(supp)r(ose)g(that)h Fg(R)f Fq(is)g(endo)n(w)n(ed)g(with)h(t)n(w)n(o)f(cone)f(\014elds)i Fg(K)2853 3434 y Fl(u)2919 3464 y Fq(=)d Fg(K)3065 3434 y Fl(u)3108 3464 y Fq(\()p Fp(x)p Fq(\))j(and)515 3564 y Fg(K)573 3534 y Fl(s)634 3564 y Fq(=)f Fg(K)782 3534 y Fl(s)818 3564 y Fq(\()p Fp(x)p Fq(\))30 b(de\014ned)g(at)f(ev)n(ery)f (p)r(oin)n(t)i(in)f Fp(x)e Fm(2)f Fg(R)p Fq(.)42 b(W)-7 b(e)30 b(also)e(supp)r(ose)h(that)g(these)h(cones)515 3672 y(families)g(extend)h(con)n(tin)n(uously)e(to)i(a)f(sligh)n(tly)g (larger)e(neigh)n(b)r(orho)r(o)r(d)2818 3650 y(^)2812 3672 y Fg(R)i Fq(con)n(taining)f Fg(R)p Fq(.)515 3772 y(W)-7 b(e)27 b(asso)r(ciate)e(with)i(these)g(cone)f(\014elds)g Fn(admissible)31 b(stable)e(and)h(unstable)e(submanifolds)515 3872 y Fq(\000)567 3841 y Fl(s)602 3872 y Fq(\()p Fg(R)p Fq(\))i(=)g Fm(f)p Fp(\015)935 3841 y Fl(s)970 3872 y Fm(g)h Fq(and)h(\000)1261 3841 y Fl(u)1304 3872 y Fq(\()p Fg(R)p Fq(\))f(=)e Fm(f)p Fp(\015)1637 3841 y Fl(u)1680 3872 y Fm(g)p Fq(.)49 b(Eac)n(h)31 b Fp(\015)2053 3841 y Fl(s)2118 3872 y Fm(2)f Fq(\000)2255 3841 y Fl(s)2291 3872 y Fq(\()p Fg(R)p Fq(\))i(is)g(the)g(graph)e(of)i(a)g Fp(C)3153 3841 y Fo(1)3222 3872 y Fq(map)515 3971 y Fp(\036)23 b Fq(:)g Fp(B)700 3941 y Fl(s)759 3971 y Fm(\000)-14 b(!)23 b Fp(B)983 3941 y Fl(u)1049 3971 y Fq(suc)n(h)g(that)g Fp(T)1456 3983 y Fl(x)1497 3971 y Fp(W)35 b Fm(\022)23 b Fg(K)1756 3941 y Fl(s)1791 3971 y Fq(\()p Fp(x)p Fq(\))h(for)e(ev)n (ery)g Fp(x)h Fm(2)h Fp(\015)2460 3941 y Fl(s)2495 3971 y Fq(.)35 b(Lik)n(ewise)22 b(for)g(admissible)515 4071 y(unstable)35 b(submanifolds.)58 b(Also)35 b Fn(stable)i(and)g (unstable)f(cylinders)h Fq(can)d(b)r(e)h(de\014ned.)60 b(A)515 4171 y(compact)34 b(connected)g(subset)h(subset)f Fp(S)39 b Fm(\022)34 b Fg(R)h Fq(is)f(an)g(admissible)g(stable)h (cylinder)f(if)h(it)515 4270 y(admits)25 b(a)f(foliation)h(b)n(y)f (admissible)h(stable)f(submanifolds)h(and)f(if)i(its)f(unstable)g (sections)515 4370 y Fp(S)40 b Fm(\\)d Fp(\015)746 4340 y Fl(u)822 4370 y Fq(are)32 b(con)n(v)n(ex)f(sets)i(for)f(ev)n(ery)g Fp(\015)1816 4340 y Fl(u)1891 4370 y Fm(2)h Fq(\000)2031 4340 y Fl(u)2074 4370 y Fq(\()p Fg(R)p Fq(\).)53 b(Unstable)33 b(cylinders)g(are)f(de\014ned)515 4469 y(similarly)-7 b(.)639 4569 y(A)28 b Fp(C)794 4539 y Fo(2)859 4569 y Fq(horsesho)r(e)e(\012)h(with)h(in\014nitely)g(man)n(y)f(branc)n(hes)f (or)h(a)g(generalized)f(horsesho)r(e)515 4669 y(will)i(b)r(e)g (de\014ned)g(b)n(y)f(a)g(coun)n(table)g(collection)g(of)g(maps)h Fm(f)p Fp(T)2403 4681 y Fl(i)2452 4669 y Fq(:)c Fp(S)2550 4681 y Fl(i)2600 4669 y Fm(\000)-14 b(!)23 b Fp(U)2814 4681 y Fl(i)2841 4669 y Fm(g)k Fq(where:)616 4807 y(1.)41 b Fm(f)p Fp(S)815 4819 y Fl(i)842 4807 y Fm(g)25 b Fq(\(resp.)p Fm(f)p Fp(U)1212 4819 y Fl(i)1239 4807 y Fm(g)p Fq(\))g(are)f(coun)n (table)g(collections)g(of)h(stable)g(\(resp.)36 b(unstable\))25 b(cylin-)722 4907 y(ders)k(with)g Fn(non)i(overlapping)j(interiors)p Fq(,)c(that)f(is,)h Fp(int)14 b(S)2527 4919 y Fl(i)2573 4907 y Fm(\\)20 b Fp(int)14 b(S)2822 4919 y Fl(j)2881 4907 y Fq(=)25 b Fm(;)k Fq(whenev)n(er)722 5006 y Fp(i)23 b Fm(6)p Fq(=)g Fp(j)5 b Fq(;)1926 5255 y(3)p eop %%Page: 4 4 4 3 bop 616 523 a Fq(2.)41 b(eac)n(h)27 b Fp(T)958 535 y Fl(i)1008 523 y Fq(:)d Fp(S)1106 535 y Fl(i)1156 523 y Fm(\000)-14 b(!)23 b Fp(U)1370 535 y Fl(i)1425 523 y Fq(has)k(a)h Fp(C)1708 493 y Fo(2)1773 523 y Fq(extension)2156 502 y(^)2140 523 y Fp(T)2189 535 y Fl(i)2239 523 y Fq(:)2300 502 y(^)2286 523 y Fp(S)2337 535 y Fl(i)2387 523 y Fm(\000)-14 b(!)2559 502 y Fq(^)2544 523 y Fp(U)2601 535 y Fl(i)2656 523 y Fq(to)28 b(neigh)n(b)r(orho)r(o)r(ds)e(of)722 623 y Fp(S)773 635 y Fl(i)829 623 y Fq(and)h Fp(U)1047 635 y Fl(i)1102 623 y Fq(whic)n(h)g(are)g(stable)g(and)h(unstable)f (cylinders)g(in)2664 601 y(^)2658 623 y Fg(R)p Fq(;)616 782 y(3.)738 761 y(^)722 782 y Fp(T)771 794 y Fl(i)836 782 y Fq(maps)37 b(h)n(yp)r(erb)r(olically)1624 761 y(^)1610 782 y Fp(S)1661 794 y Fl(i)1726 782 y Fq(on)n(to)1937 761 y(^)1922 782 y Fp(U)1979 794 y Fl(i)2007 782 y Fq(,)i(that)f(is,)h Fn(pr)l(eserving)h(strictly)f(the)g(c)l(one)722 882 y(families)i Fg(K)1099 851 y Fl(s)1173 882 y Fn(and)f Fg(K)1402 851 y Fl(u)1445 882 y Fq(:)57 b Fg(K)1583 851 y Fl(s)1618 882 y Fq(\()p Fp(T)1699 894 y Fl(i)1727 882 y Fq(\()p Fp(x)p Fq(\)\))41 b Fm(\022)f Fq(in)n(t)14 b Fp(D)r(T)2249 894 y Fl(i)2276 882 y Fq(\()p Fp(x)p Fq(\))p Fg(K)2445 851 y Fl(s)2481 882 y Fq(\()p Fp(x)p Fq(\))39 b(and)e Fp(D)r(T)2922 894 y Fl(i)2949 882 y Fq(\()p Fp(x)p Fq(\))p Fg(K)3118 851 y Fl(u)3162 882 y Fq(\()p Fp(x)p Fq(\))k Fm(\022)722 981 y Fq(in)n(t)14 b Fg(K)893 951 y Fl(u)936 981 y Fq(\()p Fp(T)1017 993 y Fl(i)1045 981 y Fq(\()p Fp(x)p Fq(\)\).)639 1134 y(W)-7 b(e)25 b(shall)f(supp)r(ose)g(that)h (non)f(linear)g(expansion)g(along)f(unstable)h(admissible)g(mani-)515 1233 y(folds)e(is)g(b)r(ounded)h(from)e(b)r(elo)n(w,)i(namely:)34 b Fp(d)1923 1250 y Fl(\015)1962 1234 y Fi(k)2002 1233 y Fq(\()p Fp(T)2095 1203 y Fl(k)2136 1233 y Fq(\()p Fp(x)p Fq(\))p Fp(;)14 b(T)2345 1203 y Fl(k)2385 1233 y Fq(\()p Fp(y)s Fq(\)\))24 b Fm(\025)f Fp(\025)2685 1203 y Fl(n)p Fk(\000)p Fl(k)q Fo(+1)2903 1233 y Fp(d)2946 1248 y Fl(\015)2985 1232 y Fd(0)3021 1233 y Fq(\()p Fp(x;)14 b(y)s Fq(\))23 b(uni-)515 1333 y(formly)d(in)h Fp(x)g Fq(and)f Fp(y)j Fq(in)e Fp(\015)1285 1303 y Fo(0)1345 1333 y Fq(=)h Fp(\015)1480 1303 y Fl(u)1542 1333 y Fm(\\)c Fp(P)12 b Fq(,)22 b Fp(k)k Fq(=)c(0)p Fp(;)14 b Fm(\001)g(\001)g(\001)g Fp(n)20 b Fq(and)g(for)g(ev)n(ery)f Fp(\015)2675 1303 y Fl(u)2742 1333 y Fm(2)k Fq(\000)2872 1303 y Fl(u)2915 1333 y Fq(,)f Fp(P)35 b Fm(2)24 b Fp(})3180 1345 y Fl(n)3245 1333 y Fq(and)515 1433 y Fp(n)f Fm(\025)f Fq(0.)35 b(Here)23 b Fp(W)1057 1403 y Fl(k)1121 1433 y Fq(=)g Fp(T)1270 1403 y Fl(k)1310 1433 y Fq(\()p Fp(\015)1390 1403 y Fo(0)1427 1433 y Fq(\).)36 b(W)-7 b(e)24 b(also)e(supp)r(ose)h(that)h(a)f (similar)g(statemen)n(t)g(holds)g(true)515 1532 y(for)28 b(the)i(in)n(v)n(erse)d Fp(T)1125 1502 y Fk(\000)p Fo(1)1214 1532 y Fq(.)41 b(Due)30 b(to)f(the)g(b)r(ounded)h(geometry)e(of)h (admissible)f(submanifolds)515 1632 y(\(see)18 b(b)r(elo)n(w\))g(these) h(conditions)f(are)f(equiv)-5 b(alen)n(t)18 b(to)g Fm(k)p Fp(T)2231 1602 y Fl(k)2271 1632 y Fq(\()p Fp(x)p Fq(\))p Fm(\000)p Fp(T)2508 1602 y Fl(k)2548 1632 y Fq(\()p Fp(y)s Fq(\))p Fm(k)23 b(\025)g Fp(C)6 b(\025)2922 1602 y Fl(n)p Fk(\000)p Fl(k)q Fo(+1)3140 1632 y Fm(k)p Fp(x)p Fm(\000)p Fp(y)s Fm(k)515 1732 y Fq(for)21 b(ev)n(ery)g Fp(x)14 b(;)28 b(y)e Fm(2)d Fp(\015)1155 1701 y Fo(0)1215 1732 y Fq(=)g Fp(\015)1351 1701 y Fl(u)1415 1732 y Fm(\\)f Fp(P)33 b Fq(and)22 b Fp(k)k Fq(=)d(0)p Fp(;)14 b Fm(\001)g(\001)g (\001)f Fp(n)p Fq(,)23 b Fp(\015)2224 1701 y Fl(u)2290 1732 y Fm(2)h Fq(\000)2421 1701 y Fl(u)2486 1732 y Fq(and)e(ev)n(ery)f (stable)g(cylinder)515 1831 y(of)32 b(lev)n(el)f Fp(n)p Fq(,)j Fp(P)42 b Fm(2)31 b Fp(})1148 1843 y Fl(n)1193 1831 y Fq(.)51 b(Admissible)32 b(manifolds)g(ha)n(v)n(e)f(the)h(follo)n (wing)f(useful)i(prop)r(erties)515 1931 y(whic)n(h)27 b(shall)g(b)r(e)h(used)g(throughout:)515 2130 y(i\))76 b(let)39 b Fp(W)52 b Fm(2)41 b Fq(\000)1055 2100 y Fl(u)1137 2130 y Fq(and)d Fp(W)1399 2100 y Fk(0)1462 2130 y Fm(2)j Fq(\000)1610 2100 y Fl(s)1684 2130 y Fq(the)d(W)h(in)n(tersects)e (transv)n(ersally)e(W')k(at)f(a)g(single)515 2230 y(p)r(oin)n(t)28 b(x)f(and)g(the)h(angle)f Fe(\\)p Fq(\()p Fp(T)1465 2242 y Fl(x)1507 2230 y Fp(W)n(;)14 b(T)1669 2242 y Fl(x)1710 2230 y Fp(W)1800 2200 y Fk(0)1823 2230 y Fq(\))28 b(is)g(uniformly)f(b) r(ounded)h(b)r(elo)n(w;)515 2429 y(ii\))52 b Fn(the)29 b(ge)l(ometry)g(of)g(admissible)i(manifolds)f(is)f(b)l(ounde)l(d)p Fq(,)e(that)f(is,)g(there)g(is)g(a)g(constan)n(t)515 2529 y Fp(C)j Fq(=)23 b Fp(C)6 b Fq(\(\000\))24 b Fp(>)e Fq(1)d(only)h(dep)r(ending)g(on)f Fp(\015)24 b Fq(and)c(the)g(diameter) f(of)h Fg(R)f Fq(suc)n(h)g(that)h(V)-7 b(ol)28 b(\()p Fp(\015)3150 2498 y Fl(u)3193 2529 y Fq(\))20 b(and)515 2628 y(diam)13 b(\()p Fp(\015)788 2598 y Fl(u)832 2628 y Fq(\))27 b(are)e(b)r(ounded)i(in)g([)14 b Fp(C)1564 2598 y Fk(\000)p Fo(1)1653 2628 y Fp(;)g(C)20 b Fq(].)37 b(F)-7 b(urther,)26 b Fp(C)2240 2598 y Fk(\000)p Fo(1)2353 2628 y Fm(\024)c Fq(dist)2588 2640 y Fl(\015)2627 2624 y Fi(u)2670 2628 y Fq(\()p Fp(x;)14 b(y)s Fq(\))g Fp(=)g Fm(k)p Fp(x)i Fm(\000)g Fp(y)s Fm(k)22 b(\024)h Fp(C)515 2728 y Fq(for)k(ev)n(ery)f Fp(x;)14 b(y)31 b Fq(in)d(an)f(unstable)g (admissible)h(submanifold)f Fp(\015)2480 2698 y Fl(u)2546 2728 y Fm(2)d Fq(\000)2677 2698 y Fl(u)2748 2728 y Fq(where)j(dist)3136 2740 y Fl(\015)3175 2724 y Fi(u)3245 2728 y Fq(and)515 2827 y(diam)708 2839 y Fl(\015)747 2823 y Fi(u)812 2827 y Fq(are)20 b(giv)n(en)g(b)n(y)h(the)h(in)n(trinsic)f(riemannian)f(v)n (olume,)i(as)f(w)n(ell)g(as)f(V)-7 b(ol)2933 2839 y Fl(\015)2972 2823 y Fi(u)3015 2827 y Fq(.)35 b(Lik)n(ewise)515 2927 y(for)27 b(for)g(the)h(admissible)f(s-submanifolds;)515 3126 y(iii\))42 b(The)21 b(set)g(\000)997 3096 y Fl(u)1062 3126 y Fq(\(resp.)34 b(\000)1352 3096 y Fl(s)1387 3126 y Fq(\))22 b(is)f(endo)n(w)n(ed)f(with)h(an)g(structure)f(of)h(Banac)n (h)f(space)g(giv)n(en)g(b)n(y)515 3226 y(the)33 b(iden)n(ti\014cation)f (with)h Fp(C)1427 3196 y Fo(1)1465 3226 y Fq(\()p Fp(B)1564 3196 y Fl(u)1608 3226 y Fp(;)14 b(B)1712 3196 y Fl(s)1747 3226 y Fq(\))33 b(equipp)r(ed)h(with)f(the)g Fp(C)2580 3196 y Fo(1)2650 3226 y Fq(norm.)52 b(In)32 b(particular)515 3326 y(it)k(is)f(a)g(complete)h(metric)f(space.)60 b(Moreo)n(v)n(er,)35 b(there)h(is)f(a)g(w)n(ell)h(de\014ned)g Fn(gr)l(aph)i(tr)l(ans-)515 3425 y(form)e Fq(\000)773 3437 y Fl(T)861 3425 y Fq(:)f(\000)971 3395 y Fl(u)1050 3425 y Fm(\000)-14 b(!)36 b Fq(\000)1272 3395 y Fl(u)1350 3425 y Fq(de\014ned)g(b)n(y)e(\000)1818 3437 y Fl(T)1871 3425 y Fq(\()p Fp(\015)1951 3395 y Fl(u)1994 3425 y Fq(\))i(=)f Fp(T)12 b Fq(\()p Fp(\015)2303 3395 y Fl(u)2345 3425 y Fq(\))24 b Fm(\\)g Fg(R)p Fq(.)59 b(The)35 b(follo)n(wing)f(result,)515 3525 y(due)c(to)g(Aleeksev)f(and) h(Moser,)g(will)g(b)r(e)g(used)g(elsewhere)f(without)i(further)e (commen)n(ts:)515 3624 y Fn(The)39 b(gr)l(aph)g(tr)l(ansform)g(is)g(a)f (c)l(ontr)l(action,)j(i.e.)66 b(ther)l(e)38 b(exists)g Fq(0)g Fp(<)g(\022)j(<)d Fq(1)g Fn(such)g(that)515 3724 y(dist)644 3739 y Fl(C)696 3722 y Fd(1)732 3724 y Fq(\(\000)816 3736 y Fl(T)869 3724 y Fq(\()p Fp(\015)949 3694 y Fl(u)944 3745 y Fo(1)992 3724 y Fq(\))p Fp(;)14 b Fq(\000)1113 3736 y Fl(T)1166 3724 y Fq(\()p Fp(\015)1246 3694 y Fl(u)1241 3745 y Fo(2)1289 3724 y Fq(\)\))28 b Fm(\024)f Fp(\022)16 b Fn(dist)1658 3739 y Fl(C)1710 3722 y Fd(1)1746 3724 y Fq(\()p Fp(\015)1826 3694 y Fl(u)1821 3745 y Fo(1)1869 3724 y Fp(;)e(\015)1954 3694 y Fl(u)1949 3745 y Fo(2)1997 3724 y Fq(\).)46 b(In)30 b(addition,)h(\()p Fp(\015)2638 3694 y Fl(u)2681 3724 y Fp(;)14 b(\015)2766 3694 y Fl(s)2801 3724 y Fq(\))28 b Fm(7\000)-14 b(!)28 b Fp(\015)3071 3694 y Fl(u)3134 3724 y Fm(\\)21 b Fp(\015)3258 3694 y Fl(s)3323 3724 y Fq(is)515 3824 y(a)27 b(Lipsc)n(hitz)g(map)h(from)f (\000)1371 3794 y Fl(u)1433 3824 y Fm(\002)18 b Fq(\000)1568 3794 y Fl(s)1631 3824 y Fq(to)27 b Fg(R)p Fq(.)639 3923 y(Giv)n(en)g(a)f(family)h(of)g(h)n(yp)r(erb)r(olic)g(branc)n(hes)e Fp(T)2092 3935 y Fl(i)2142 3923 y Fq(:)e Fp(S)2239 3935 y Fl(i)2290 3923 y Fm(\000)-14 b(!)23 b Fp(U)2504 3935 y Fl(i)2558 3923 y Fq(as)j(ab)r(o)n(v)n(e)g(w)n(e)g(can)h(de\014ne)515 4023 y(nested)c(sequences)g(of)h(stable)f(and)g(unstable)h(cylinders)f (con)n(v)n(erging)e(to)i(t)n(w)n(o)g(laminations)515 4123 y(of)32 b(stable)g(and)g(unstable)g(admissible)g(manifolds)g Fm(F)2218 4092 y Fl(s)2286 4123 y Fq(and)g Fm(F)2520 4092 y Fl(u)2563 4123 y Fq(,)i(resp)r(ectiv)n(ely)-7 b(.)49 b(In)33 b(fact,)515 4222 y(giv)n(en)27 b(a)g(sequence)g(\()p Fp(i)1206 4234 y Fo(0)1243 4222 y Fp(;)14 b Fm(\001)g(\001)g(\001)27 b Fp(;)14 b(i)1470 4234 y Fl(n)1515 4222 y Fp(;)g Fm(\001)g(\001)g (\001)g Fq(\))23 b Fm(2)g Ff(N)1850 4192 y Fc(N)1930 4222 y Fq(w)n(e)k(set)h(de\014ne)616 4467 y Fp(U)673 4479 y Fl(i)696 4487 y Fd(0)729 4479 y Fk(\001\001\001)o Fl(i)811 4487 y Fi(n)p Fb(\000)p Fd(1)953 4467 y Fq(=)1040 4363 y Fl(n)p Fk(\000)p Fo(1)1057 4388 y Fj(\\)1043 4567 y Fl(k)q Fo(=0)1180 4467 y Fp(T)1229 4479 y Fl(i)1252 4487 y Fi(n)p Fb(\000)p Fd(1)1403 4467 y Fm(\016)18 b(\001)c(\001)g (\001)k(\016)32 b Fp(T)1701 4479 y Fl(i)1724 4487 y Fd(0)1760 4467 y Fp(U)1817 4479 y Fl(i)1840 4487 y Fd(0)1988 4467 y Fp(S)2039 4479 y Fl(i)2062 4487 y Fd(0)2094 4479 y Fk(\001\001\001)p Fl(i)2177 4487 y Fi(n)p Fb(\000)p Fd(1)2318 4467 y Fq(=)2406 4363 y Fl(n)p Fk(\000)p Fo(1)2423 4388 y Fj(\\)2409 4567 y Fl(k)q Fo(=0)2546 4467 y Fp(T)2607 4432 y Fk(\000)p Fo(1)2595 4490 y Fl(i)2618 4498 y Fi(n)p Fb(\000)p Fd(1)2768 4467 y Fm(\016)18 b(\001)c(\001)g(\001)k(\016)32 b Fp(T)3078 4432 y Fk(\000)p Fo(1)3066 4490 y Fl(i)3089 4498 y Fd(0)3166 4467 y Fp(S)3217 4479 y Fl(i)3240 4487 y Fd(0)515 4707 y Fq(where)f(w)n(e)h(b)n(y)g(abuse)g(of)g(language)f(w) n(e)h(omit)g(the)g(domain)g(of)g(comp)r(ositions.)50 b(The)33 b(se-)515 4807 y(quence)39 b Fm(f)p Fp(S)894 4819 y Fl(i)917 4827 y Fd(0)949 4819 y Fk(\001\001\001)p Fl(i)1032 4827 y Fi(n)p Fb(\000)p Fd(1)1150 4807 y Fm(g)1192 4819 y Fl(n)p Fk(\025)p Fo(0)1361 4807 y Fq(is)h(a)f(nested)g(sequence) g(of)h(stable)f(cylinders.)72 b(Similarly)39 b(so)515 4907 y Fm(f)p Fp(U)614 4919 y Fl(i)637 4927 y Fd(0)669 4919 y Fk(\001\001\001)o Fl(i)751 4927 y Fi(n)p Fb(\000)p Fd(1)870 4907 y Fm(g)912 4919 y Fl(n)p Fk(\025)p Fo(0)1041 4907 y Fq(.)69 b(In)39 b(fact,)i(graph)c(transform)g(con)n(traction)g (prop)r(erties)g(implies)i(that)515 5006 y(there)c(is)g(a)g(unique)h (admissible)f(manifold)h Fp(\015)1991 4976 y Fl(s)2062 5006 y Fq(=)g Fp(\015)2211 4976 y Fl(s)2246 5006 y Fq(\()p Fh(i)p Fq(\))g(suc)n(h)f(that)g Fp(d)2798 5021 y Fl(C)2850 5004 y Fd(1)2887 5006 y Fq(\()p Fp(S)2970 5018 y Fl(i)2993 5026 y Fd(0)3026 5018 y Fk(\001\001\001)o Fl(i)3108 5026 y Fi(n)p Fb(\000)p Fd(1)3227 5006 y Fp(;)14 b(\015)3312 4976 y Fl(s)3347 5006 y Fq(\))1926 5255 y(4)p eop %%Page: 5 5 5 4 bop 515 523 a Fq(con)n(v)n(erges)26 b(to)i(zero)g(as)g Fp(n)c Fm(\000)-14 b(!)25 b Fq(+)p Fm(1)j Fq(and)g(lik)n(ewise)g(for)g (the)h(unstable)g(cylinders.)39 b(Hence,)515 623 y(w)n(e)27 b(de\014ne)h Fm(F)945 593 y Fl(s)1003 623 y Fq(=)23 b Fm(f)p Fp(\015)1181 593 y Fl(s)1215 623 y Fq(\()p Fh(i)p Fq(\))h(:)f Fh(i)f Fm(2)i Ff(N)1558 593 y Fc(N)1610 623 y Fm(g)j Fq(\(resp.)37 b Fm(F)1988 593 y Fl(u)2054 623 y Fq(=)23 b Fm(f)p Fp(\015)2232 593 y Fl(u)2274 623 y Fq(\()p Fh(i)p Fq(\))g(:)g Fh(i)g Fm(2)g Ff(N)2616 593 y Fc(N)2668 623 y Fm(g)p Fq(\).)639 722 y(W)-7 b(e)26 b(set)f(\012)e(=)1077 660 y Fj(S)1174 722 y Fm(F)1242 692 y Fl(s)1305 722 y Fm(\\)1387 660 y Fj(S)1484 722 y Fm(F)1552 692 y Fl(u)1596 722 y Fq(.)36 b(\012)25 b(is)g(a)f (compact,)i(p)r(erfect,)g(subset)f(of)g Ff(R)2937 692 y Fl(m)3031 722 y Fq(con)n(tained)515 822 y(in)34 b(the)h(cub)r(e)g Fg(R)p Fq(.)56 b(T)-7 b(op)r(ologically)32 b(it)j(migh)n(t)f(b)r(e)h(a) f(Can)n(tor)f(set)h(times)g(an)g(in)n(terv)-5 b(al)34 b(or)f(a)515 922 y(pro)r(duct)27 b(of)h(t)n(w)n(o)f(Can)n(tor)f(sets)h (or)g(ev)n(en)g(it)h(migh)n(t)g(\014lls)f Fg(R)h Fq(up)g(to)f(a)g (measure)g(zero)f(set.)639 1021 y(W)-7 b(e)26 b(can)e(use)h(the)h(h)n (yp)r(erb)r(olic)f(branc)n(hes)e Fp(T)2005 1033 y Fl(i)2058 1021 y Fq(to)i(de\014ne)g(a)g(piecewise)g(h)n(yp)r(erb)r(olic)f(map)515 1121 y Fp(T)56 b Fq(:)689 1059 y Fj(S)758 1146 y Fl(i)813 1121 y Fp(int)14 b(S)987 1133 y Fl(i)1059 1121 y Fm(\000)-14 b(!)1239 1059 y Fj(S)1308 1146 y Fl(i)1363 1121 y Fp(int)14 b(U)1543 1133 y Fl(i)1570 1121 y Fq(,)44 b(b)n(y)d(setting)g Fp(T)56 b Fm(j)k Fp(int)14 b(S)2414 1133 y Fl(i)2486 1121 y Fq(=)45 b Fp(T)2645 1133 y Fl(i)2713 1121 y Fq(and)c(extends)g Fp(T)52 b Fq(to)515 1233 y(some)27 b(w)n(ell)h(de\014ned)g(measurable)f (transformation)2196 1212 y(^)2179 1233 y Fp(T)35 b Fq(:)2310 1170 y Fj(S)2379 1258 y Fl(i)2435 1233 y Fp(S)2486 1245 y Fl(i)2537 1233 y Fm(\000)-14 b(!)2694 1170 y Fj(S)2764 1258 y Fl(i)2819 1233 y Fp(U)2876 1245 y Fl(i)2903 1233 y Fq(,)29 b(whic)n(h)f(is)f Fp(C)3341 1203 y Fo(2)515 1332 y Fq(smo)r(oth)f(in)g(a)g(dense)g(subset)g(of)g(its)h(domain)f (and)g(preserv)n(es)e(the)i(admissible)g(manifolds.)515 1432 y(\012)33 b(is)h(the)g(maximal)f(in)n(v)-5 b(arian)n(t)33 b(subset)h(of)1924 1411 y(^)1908 1432 y Fp(T)45 b Fq(and)34 b Fm(S)39 b Fq(=)33 b Fm(f)p Fp(x)g Fm(2)2568 1370 y Fj(S)2637 1457 y Fl(i)2679 1432 y Fp(@)18 b(S)2792 1444 y Fl(i)2853 1432 y Fq(:)101 b Fp(x)34 b Fm(2)f Fp(@)19 b(S)3260 1444 y Fl(i)3324 1432 y Fm(\\)515 1544 y Fp(@)f(S)628 1556 y Fl(j)677 1544 y Fp(;)90 b Fq(for)28 b(some)75 b Fp(i;)14 b(j)28 b(>)23 b Fq(0)p Fm(g)h Fq(is)h(its)h(singular)e(set.) 36 b(Extensions)2600 1523 y(^)2584 1544 y Fp(T)g Fq(are)25 b(rather)f(arbitrary)515 1643 y(and)e(they)g(are)f(not)h(unique.)36 b(Ho)n(w)n(ev)n(er,)21 b(as)h(long)f(as)h(their)g(singular)f(singular)g (is)h(negligible)515 1743 y(for)28 b(all)g(the)i(purp)r(oses)d(of)i (the)g(ergo)r(dic)f(theory)g(w)n(e)g(can)h(omit)g(this)g (arbitrariness,)d(so)i(w)n(e)515 1843 y(will)g(con)n(tin)n(ue)f (denoting)g(extensions)1761 1822 y(^)1745 1843 y Fp(T)39 b Fq(b)n(y)27 b Fp(T)39 b Fq(to)27 b(a)n(v)n(oid)f(messy)h(notations.) 639 1942 y(Giv)n(en)j(one)f(suc)n(h)g(extension)g(it)h(is)f(easy)g(to)g (see)g(that)h Fm(F)2438 1912 y Fl(s)2503 1942 y Fq(and)f Fm(F)2734 1912 y Fl(u)2807 1942 y Fq(are)g(T-in)n(v)-5 b(arian)n(t.)515 2042 y(Moreo)n(v)n(er,)40 b(due)g(to)f(no-linear)g (expansion)g(prop)r(erties)f(along)h(admissible)g(manifolds,)515 2142 y(w)n(e)31 b(conclude)h(that)g Fp(W)1261 2111 y Fl(s)1297 2142 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\))31 b(=)f Fp(\015)1673 2111 y Fl(s)1708 2142 y Fq(\()p Fh(i)p Fq(\))i(and)g Fp(W)2087 2111 y Fl(u)2130 2142 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\))32 b(=)e Fp(\015)2507 2111 y Fl(u)2550 2142 y Fq(\()p Fh(j)p Fq(\))i(are)f(the)i(lo)r(cal)e(stable)515 2241 y(manifold)25 b(of)h Fp(x)g Fq(where)e Fm(f)p Fp(x)p Fm(g)f Fq(=)g Fp(\015)1549 2211 y Fl(s)1584 2241 y Fq(\()p Fh(i)p Fq(\))14 b Fm(\\)g Fp(\015)1806 2211 y Fl(u)1849 2241 y Fq(\()p Fh(j)p Fq(\).)37 b(Indeed,)26 b Fp(W)2383 2211 y Fl(s)2419 2241 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\))23 b(=)g Fm(f)p Fp(y)i Fm(2)f Fg(R)e Fq(:)i Fm(k)p Fp(T)3145 2211 y Fl(n)3188 2241 y Fq(\()p Fp(x)p Fq(\))15 b Fm(\000)515 2341 y Fp(T)576 2311 y Fl(n)620 2341 y Fq(\()p Fp(y)s Fq(\))p Fm(k)27 b(\024)f Fp(C)6 b(\025)1001 2311 y Fk(\000)p Fl(n)1159 2341 y Fm(8)29 b Fp(n)e Fm(\025)f Fq(0)p Fm(g)j Fq(and)h(similarly)f(so)g(for)g(the)i(unstable)e(lo)r(cal)h(manifold)g (for)515 2440 y(bac)n(kw)n(ard)24 b(orbits.)36 b(Therefore)25 b(\012)k Fn(is)g(endowe)l(d)h(with)g(a)f(hyp)l(erb)l(olic)i(pr)l(o)l (duct)e(structur)l(e)c Fq(in)515 2540 y(the)i(sense)g(that)g(ev)n(ery)f (p)r(oin)n(t)i(is)f(the)g(in)n(tersection)f(of)i(lo)r(cal)e(in)n(v)-5 b(arian)n(t)26 b(manifolds)h(whic)n(h)515 2640 y(in)n(tersects)g (transv)n(ersally)e(at)i(angle)g(b)r(ounded)h(from)f(b)r(elo)n(w.)515 2806 y Fh(De\014nition)k(2.1.)40 b Fq(A)29 b(set)h(\012)g(with)g(a)f(h) n(yp)r(erb)r(olic)g(pro)r(duct)g(structure)g(and)g(a)h(dynamics)515 2905 y Fp(T)k Fq(:)23 b(\012)g Fe(\011)f Fq(giv)n(en)f(b)n(y)h(a)f (collection)g(of)h Fp(C)1715 2875 y Fo(2)1775 2905 y Fq(h)n(yp)r(erb)r(olic)g(branc)n(hes)e Fp(T)2563 2917 y Fl(i)2613 2905 y Fq(:)k Fp(S)2711 2917 y Fl(i)2761 2905 y Fm(\000)-14 b(!)23 b Fp(U)2975 2917 y Fl(i)3024 2905 y Fq(as)f(de\014ned)515 3005 y(ab)r(o)n(v)n(e)32 b(shall)i(b)r(e)g(called)f(a)h Fn(horsesho)l(e)j(with)f(in\014nitely)g (many)f(br)l(anches)i(or,)g(shortly,)i(a)515 3105 y(gener)l(alize)l(d) 31 b(horsesho)l(e)p Fq(.)639 3271 y(W)-7 b(e)44 b(in)n(tro)r(duce)e (for)h(further)g(use)g(the)g(follo)n(wing)f(notation:)67 b Fp(})2751 3283 y Fl(n)2845 3271 y Fq(=)48 b Fm(f)p Fp(S)3051 3283 y Fl(i)3074 3291 y Fd(0)3106 3283 y Fk(\001\001\001)p Fl(i)3189 3291 y Fi(n)p Fb(\000)p Fd(1)3356 3271 y Fq(:)515 3370 y Fh(i)e Fm(2)i Ff(N)745 3340 y Fc(N)797 3370 y Fm(g)41 b Fq(for)h Fp(n)k(>)h Fq(0)42 b(and)f Fp(})1542 3382 y Fo(0)1626 3370 y Fq(=)47 b Fm(f)p Fp(S)1831 3382 y Fl(i)1858 3370 y Fm(g)1900 3382 y Fl(i>)p Fo(0)2012 3370 y Fq(.)80 b(Notice)42 b(that)g Fp(})2637 3382 y Fl(n)2729 3370 y Fq(=)47 b Fp(T)2902 3340 y Fk(\000)p Fl(n)2998 3370 y Fp(})3051 3382 y Fo(0)3088 3370 y Fq(.)80 b Fp(})3244 3382 y Fo(0)3323 3370 y Fq(is)515 3470 y(a)41 b(Mark)n(o)n(v)e(for)i(\012)h(as)f(it)h(is)g(easy)e(to)i(v)n(erify)-7 b(.)78 b(F)-7 b(urther,)45 b(for)d(ev)n(ery)e Fp(x)47 b Fm(2)g Fq(\012)41 b(it)h(holds)515 3569 y Fm(f)p Fp(x)p Fm(g)29 b Fq(=)770 3507 y Fj(T)839 3594 y Fl(n)p Fk(2)p Fc(Z)982 3569 y Fp(T)1043 3539 y Fk(\000)p Fl(n)1139 3569 y Fp(})1192 3581 y Fo(0)1229 3569 y Fq(\()p Fp(T)1322 3539 y Fl(n)1366 3569 y Fq(\()p Fp(x)p Fq(\)\).)51 b(So)1703 3507 y Fj(W)1772 3594 y Fl(n)p Fk(2)p Fc(Z)1914 3569 y Fp(T)1975 3539 y Fk(\000)p Fl(n)2071 3569 y Fp(})2124 3581 y Fo(0)2193 3569 y Fq(generates)31 b(the)h Fp(\033)s Fq(-algebra)e(of)i(Bore-)515 3669 y(lian)h(sets)g(of)g(\012.)53 b(Also,)1304 3607 y Fj(W)1373 3694 y Fl(n)p Fk(\025)p Fo(0)1517 3669 y Fp(T)1578 3639 y Fk(\000)p Fl(n)1674 3669 y Fp(})1727 3681 y Fo(0)1797 3669 y Fq(generates)32 b(the)i Fp(\033)s Fq(-algebra)d(stable)i(saturated)f(of)515 3769 y(Borel)c(subsets,)j(that)f(is)g(those)f(Borelians)f(whic)n(h)i (are)f(the)h(reunion)f(of)h(stable)g(lea)n(v)n(es)e(of)515 3868 y Fm(F)583 3838 y Fl(s)618 3868 y Fq(.)37 b(W)-7 b(e)28 b(shall)f(denote)h Fp(})f Fq(the)h(reunion)f(of)h(all)f (cylinders)g(in)h Fp(})2515 3880 y Fl(n)2588 3868 y Fq(for)f Fp(n)c Fm(\025)f Fq(0.)639 3968 y(No)n(w,)h(let)g Fp(T)34 b Fq(:)23 b(\012)g Fe(\011)f Fq(b)r(e)h(a)f Fp(C)1498 3938 y Fo(2)1558 3968 y Fq(generalized)f(horsesho)r(e)g(with)i(b)r (ounded)g(distortion)e(and)515 4068 y(expansion)28 b(co)r(e\016cien)n (t)g(b)r(ounded)i(from)e(b)r(elo)n(w.)41 b(W)-7 b(e)29 b(in)n(tro)r(duce,)g(as)f(w)n(e)h(did)g(previously)515 4167 y(for)j(Mark)n(o)n(v)e(transformations,)i(a)g(dynamically)g (de\014ned)g(Charath)n(\023)-39 b(eo)r(dory's)30 b(structure)515 4267 y(to)38 b(study)h(some)f(regularit)n(y)f(prop)r(erties)h(of)h (unstable)g(Can)n(tor)e(sets)h(of)h(a)f(generalized)515 4366 y(horsesho)r(e.)d(F)-7 b(or)25 b(this)h(w)n(e)g(let)g Fp(\015)1518 4336 y Fl(u)1584 4366 y Fm(2)e Fq(\000)1715 4336 y Fl(u)1784 4366 y Fq(and)i(de\014ne)g(a)g(generating)e(net)j(of)f (subsets)f(on)h Fp(\015)3336 4336 y Fl(u)515 4466 y Fq(taking)h(the)h (in)n(tersections)e(of)i Fp(\015)1535 4436 y Fl(u)1605 4466 y Fq(with)h(cylinders)d(in)i Fp(})p Fq(:)719 4632 y Fp(})p Fq(\()p Fp(\015)852 4602 y Fl(u)895 4632 y Fq(\))23 b(=)g Fm(f)p Fp(P)44 b Fm(\\)32 b Fp(\015)1312 4602 y Fl(u)1378 4632 y Fq(:)23 b Fp(P)35 b Fm(2)24 b Fp(})1644 4644 y Fl(n)1716 4632 y Fq(is)k(an)f(stable)h(n-cylinder)e(for)i(some) 54 b Fp(n)23 b Fm(\025)g Fq(0)p Fm(g)p Fq(.)515 4798 y(Then)30 b(w)n(e)h(de\014ne)f(an)h(outer)f(measure)f(in)i Fp(\015)1915 4768 y Fl(u)1958 4798 y Fq(,)g(using)g(the)g(Carath)n (\023)-39 b(eo)r(dory)27 b(construction,)515 4898 y(b)n(y)k(taking)f (co)n(v)n(erings)f(b)n(y)i Fp(})p Fq(\()p Fp(\015)1507 4868 y Fl(u)1551 4898 y Fq(\)-sets)g(and)g(using)g(a)g(riemannian)f(v)n (olume)h(V)-7 b(ol)3097 4868 y Fl(u)3097 4918 y(\015)3171 4898 y Fq(as)31 b(set)1926 5255 y(5)p eop %%Page: 6 6 6 5 bop 515 523 a Fq(function.)67 b(More)37 b(precisely)-7 b(,)39 b(giv)n(en)e Fp(a)j(>)f Fq(0)e(w)n(e)g(de\014ne)h(an)f(outer)g (measure)g(for)g(these)515 623 y(unstable)27 b(Can)n(tor)g(sets)g(as)g (follo)n(ws:)1055 856 y Fm(D)1119 868 y Fl(a;};\015)1276 852 y Fi(u)1332 856 y Fq(\()p Fp(X)7 b Fq(\))23 b(=)54 b(lim)1583 914 y Fl(\016)r Fk(!)p Fo(0)1714 897 y Fd(+)1775 856 y Fq(inf)1802 910 y Fk(U)1891 752 y Fo(+)p Fk(1)1890 777 y Fj(X)1896 954 y Fl(i)p Fo(=1)2024 856 y Fq(V)-7 b(ol)2157 868 y Fl(\015)2196 852 y Fi(u)2239 856 y Fq(\()p Fp(U)2328 868 y Fl(i)2356 856 y Fq(\))2388 822 y Fl(a)2539 856 y Fp(X)29 b Fm(\022)23 b Fp(\015)2773 822 y Fl(u)2816 856 y Fp(;)515 1110 y Fq(taking)29 b(in\014m)n(um)i(o)n(v)n(er)e(all)h (the)h Fp(\016)s Fq(-co)n(v)n(ering)d(b)n(y)i Fp(})p Fq(\()p Fp(\015)2193 1080 y Fl(u)2236 1110 y Fq(\)-sets.)45 b(A)30 b(coun)n(table)g(family)h Fm(U)k Fq(=)515 1209 y Fm(f)p Fp(U)614 1221 y Fl(i)641 1209 y Fm(g)683 1221 y Fl(i>)p Fo(0)831 1209 y Fm(U)45 b(\022)37 b Fp(})f Fq(with)g(diam)14 b(\()p Fp(U)1599 1221 y Fl(i)1626 1209 y Fq(\))38 b Fp(<)e(\016)s Fq(,)j(will)d(b)r(e)g(called)g Fp(\016)s Fn(-c)l(overings)i(by)g Fp(})p Fq(\()p Fp(\015)3117 1179 y Fl(u)3160 1209 y Fq(\))p Fn(-sets)p Fq(.)515 1309 y(\(Throughout)c(this)h(w)n(ork)e Fp(V)19 b(ol)1527 1321 y Fl(\015)1566 1305 y Fi(u)1644 1309 y Fq(will)35 b(alw)n(a)n(ys)e (denote)i(the)g(v)n(olume)f(on)h Fp(\015)2970 1279 y Fl(u)3048 1309 y Fm(2)g Fq(\000)3190 1279 y Fl(u)3268 1309 y Fq(de-)515 1409 y(\014ned)25 b(b)n(y)f(its)h(in)n(trinsic)f (riemannian)g(metric)h(and)f(V)-7 b(ol)2237 1421 y Fl(\015)2276 1404 y Fi(u)2319 1409 y Fq(\()p Fp(X)7 b Fq(\))23 b(=)g(V)-7 b(ol)13 b(\()p Fp(X)20 b Fm(\\)13 b Fp(\015)2940 1378 y Fl(u)2983 1409 y Fq(\),)25 b(for)f(ev)n(ery)515 1508 y(Borel)k(subset)h Fp(X)j Fm(\022)25 b Ff(R)1238 1478 y Fl(m)1307 1508 y Fq(.\))42 b(This)29 b(construction)f(yields)h(a)g(w) n(ell)g(de\014ned)g(and)g(non-trivial)515 1608 y(measure)j(class)g(and) h(a)f(Carath)n(\023)-39 b(eo)r(dory's)30 b(dimensional)i(c)n (haracteristic)f(whic)n(h)i(shall)g(b)r(e)515 1707 y(called)d(the)i Fn(dynamic)l(al)j(me)l(asur)l(e)d(and)i(the)f(dynamic)l(al)i(dimension) f(of)g Fq(\012)e Fn(asso)l(ciate)l(d)j(to)515 1807 y(the)27 b(gener)l(ating)g(family)i Fp(})p Fq(.)35 b(Compare)24 b([8)o(,)h(Chapter)f(1])h(and)f([8,)h(Chapter)f(4].)35 b(W)-7 b(e)25 b(denote)515 1907 y(these)38 b(b)n(y)g Fm(D)928 1919 y Fl(\013;\015)1030 1903 y Fi(u)1111 1907 y Fq(and)g(dim)1421 1919 y Fl(\015)1460 1903 y Fi(u)1499 1919 y Fl(;})1565 1907 y Fq(,)j(resp)r(ectiv)n(ely)-7 b(.)68 b(Changing)37 b(the)h(riemannian)g(v)n(olume)515 2006 y(giv)n(es)26 b(equiv)-5 b(alen)n(t)27 b(measures.)35 b(Also)27 b(w)n(e)g(can)f(enlarge)g(the)i(co)n(v)n(ering)d(family)i Fp(})g Fq(using)g(the)515 2106 y(b)r(ounded)32 b(distortion)g(prop)r (ert)n(y)-7 b(.)50 b(But)32 b(these)g(pro)r(cedures)f(at)h(most)g(c)n (hanges)f(the)i(mea-)515 2206 y(sure)21 b(b)n(y)h(b)r(ounded)h(constan) n(ts)e(so)g(this)i(holds)e(inside)i(a)e(w)n(ell)h(de\014ned)h(measure)e (class.)34 b(The)515 2305 y(follo)n(wing)26 b(is)i(our)f(\014rst)g (main)g(result)515 2504 y Fh(Theorem)d(A)44 b Fn(L)l(et)24 b Fp(T)34 b Fq(:)23 b(\012)g Fe(\011)h Fn(b)l(e)g(a)h Fp(C)1697 2474 y Fo(2)1759 2504 y Fn(gener)l(alize)l(d)h(horsesho)l(e)g (with)f(b)l(ounde)l(d)g(distortion)515 2604 y(and)e(exp)l(ansion)f(c)l (o)l(e\016cient)h(b)l(ounde)l(d)g(fr)l(om)g(b)l(elow.)37 b(Then)23 b(ther)l(e)f(exists)g(a)h(unique)f(er)l(go)l(dic)515 2704 y(me)l(asur)l(e)29 b Fp(\026)23 b Fq(=)g Fp(\026)1047 2716 y Fo(\012)1128 2704 y Fn(such)30 b(that)g(for)g(every)h(lo)l(c)l (al)g(unstable)e(manifold)j Fp(W)j Fq(=)22 b Fp(W)2982 2674 y Fl(u)3026 2704 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\))613 2865 y Fn(1.)42 b Fq(dim)861 2877 y Fk(D)r Fl(;W)1009 2865 y Fq(\(\012\))24 b(=)f Fp(h)1293 2877 y Fl(\026)1337 2865 y Fq(\()p Fp(T)12 b Fq(\))i Fp(=)1545 2798 y Fj(R)1628 2865 y Fq(log)28 b Fp(J)1817 2835 y Fl(u)1860 2865 y Fp(T)12 b Fq(\()p Fp(x)p Fq(\))p Fp(d\026)p Fq(\()p Fp(x)p Fq(\))p Fn(;)613 3029 y(2.)42 b Fp(\026)772 3041 y Fl(W)848 3029 y Fn(,)34 b(the)f(pr)l(oje)l(ction)i(of)e Fp(\026)1587 3041 y Fo(\012)1672 3029 y Fn(along)h(the)f(stable)h(lamination)g(onto) f Fp(W)12 b Fn(,)34 b(is)g(e)l(quiva-)722 3128 y(lent)i(to)g Fm(D)1062 3140 y Fl(\013;W)1236 3128 y Fn(and)g(b)l(ounde)l(d)h(by)f(a) g(c)l(onstant)f Fp(C)41 b Fq(=)33 b Fp(C)6 b Fq(\(\001)p Fp(;)14 b(\025;)g Fq(\000\))36 b Fp(>)d Fq(1)p Fn(.)57 b(F)-6 b(urther,)722 3228 y Fp(\026)772 3240 y Fl(W)848 3228 y Fq(\()p Fp(P)12 b Fq(\))23 b Fm(\020)g Fn(V)-6 b(ol)1221 3240 y Fl(W)1296 3228 y Fq(\()p Fp(P)12 b Fq(\))1425 3198 y Fl(\013)1503 3228 y Fn(for)30 b(every)h Fp(P)k Fm(2)23 b Fp(})p Fn(;)613 3391 y(3.)42 b(the)35 b(stable)g(lamination)h Fm(F)1592 3361 y Fl(s)1662 3391 y Fn(of)g(a)f(gener)l(alize)l(d)h (horsesho)l(e)h(is)e(absolutely)h(c)l(ontin-)722 3491 y(uous)d(wrt)g(the)h(me)l(asur)l(e)f(classes)h Fm(fD)1920 3503 y Fl(\013;\015)2022 3487 y Fi(u)2065 3491 y Fm(g)2107 3503 y Fl(\015)2146 3487 y Fi(u)2184 3503 y Fk(2)p Fo(\000)2270 3487 y Fi(u)2346 3491 y Fn(and)g(it)g(has)g(a)g(c)l(ontinuous)e(and)722 3591 y(b)l(ounde)l(d)e(Jac)l(obian:)1212 3772 y Fp(d)14 b(H)1345 3742 y Fk(\003)1383 3772 y Fm(D)1447 3784 y Fl(\013;)p 1510 3751 39 3 v 1 x(\015)1548 3766 y Fi(u)p 1212 3809 380 4 v 1269 3885 a Fp(d)g Fm(D)1390 3897 y Fl(\013;\015)1492 3881 y Fi(u)1601 3828 y Fq(\()p Fp(x)p Fq(\))25 b(=)1824 3686 y Fj(")1873 3725 y Fo(+)p Fk(1)1878 3750 y Fj(Y)1877 3926 y Fl(i)p Fo(=1)2084 3772 y Fp(J)8 b Fq(\()p Fp(T)34 b Fm(j)23 b Fp(\015)2347 3742 y Fl(u)2390 3772 y Fq(\)\()p Fp(T)2515 3742 y Fl(i)2542 3772 y Fq(\()p Fp(x)p Fq(\)\))p 2014 3809 744 4 v 2014 3885 a Fp(J)8 b Fq(\()p Fp(T)34 b Fm(j)p 2229 3840 48 4 v 23 w Fp(\015)2277 3854 y Fl(u)2320 3885 y Fq(\)\()p Fp(T)2445 3861 y Fl(i)2472 3885 y Fq(\()p Fp(H)7 b Fq(\()p Fp(x)p Fq(\)\)\))2767 3686 y Fj(#)2815 3701 y Fl(\013)2876 3828 y Fp(:)374 b Fq(\(2\))639 4110 y(The)28 b(follo)n(wing)e(is)i(an)f(useful)h (regularit)n(y)e(prop)r(ert)n(y)g(of)i(the)g(stable)f(lamination.)515 4309 y(Here)h Fp(H)781 4321 y Fl(\015)820 4305 y Fi(u)858 4321 y Fl(;)p 878 4288 39 3 v(\015)917 4303 y Fi(u)988 4309 y Fq(is)h(the)f(holonom)n(y)f(map)h(of)h(the)f(lo)r(cal)g(stable)g (lamination)g Fm(F)2941 4279 y Fl(s)3005 4309 y Fq(de\014ned)g(b)n(y) 515 4408 y(unstable)g(admissible)g(unstable)g(sections)g Fp(\015)1943 4378 y Fl(u)1986 4408 y Fp(;)p 2023 4363 48 4 v 14 w(\015)2071 4377 y Fl(u)2138 4408 y Fm(2)d Fq(\000)2270 4378 y Fl(u)2313 4408 y Fq(\()p Fg(R)p Fq(\):)39 b Fp(H)2562 4420 y Fl(\015)2601 4404 y Fi(u)2640 4420 y Fl(;)p 2660 4387 39 3 v 1 x(\015)2698 4402 y Fi(u)2741 4408 y Fq(\()p Fp(x)p Fq(\))26 b(=)e Fp(W)3057 4378 y Fl(s)3092 4408 y Fq(\()p Fp(x;)14 b(R)q Fq(\))20 b Fm(\\)p 515 4462 48 4 v 515 4508 a Fp(\015)562 4476 y Fl(u)606 4508 y Fq(,)31 b(for)f(ev)n(ery)f Fp(x)g Fm(2)f Fp(\015)1219 4478 y Fl(u)1296 4508 y Fm(\\)1386 4446 y Fj(S)1455 4533 y Fl(x)p Fk(2)p Fo(\012)1602 4508 y Fp(W)1692 4478 y Fl(s)1728 4508 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\).)46 b Fp(J)8 b Fq(\()p Fp(T)39 b Fm(j)28 b Fp(\015)2272 4478 y Fl(u)2315 4508 y Fq(\))j(will)g(denote)f(throughout)g(this)515 4608 y(w)n(ork)22 b(the)j(Jacobian)d(of)i Fp(T)35 b Fq(restricted)24 b(to)g Fp(\015)1885 4578 y Fl(u)1952 4608 y Fq(with)g(resp)r(ect)g(to)g (the)g(in)n(trinsic)g(riemannian)515 4707 y(v)n(olume.)58 b Fp(J)909 4677 y Fl(u)952 4707 y Fp(T)12 b Fq(\()p Fp(x)p Fq(\))35 b(=)g Fp(J)8 b Fq(\()p Fp(T)46 b Fm(j)36 b Fp(W)1589 4677 y Fl(u)1632 4707 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\)\)\()p Fp(x)p Fq(\))37 b(is)d(the)i(unstable)e(Jacobian)g(of)h Fp(T)46 b Fq(wrt)34 b(to)515 4807 y(the)f(in)n(trinsic)g(riemannian)g (v)n(olume)g(of)g(the)h(lo)r(cal)e(unstable)i(manifold)f Fp(W)2950 4777 y Fl(u)2993 4807 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\).)55 b(\(1\))515 4907 y(follo)n(ws)25 b(from)h(\(2\))g (using)g(Billingsley's)f(metho)r(d.)37 b(See)26 b([11)o(])g(and)g (compare)f([8,)h(App)r(endix)515 5006 y(I)r(I)r(I].)i(Being)f(indep)r (enden)n(t)i(of)e Fp(W)35 b Fm(2)24 b(F)1742 4976 y Fl(u)1813 5006 y Fq(w)n(e)j(can)g(in)n(tro)r(duce)g(the)h(follo)n(wing)1926 5255 y(6)p eop %%Page: 7 7 7 6 bop 515 523 a Fh(De\014nition)31 b(2.2.)40 b Fq(dim)1289 535 y Fl(u)1332 523 y Fq(\(\012\))24 b(=)f(dim)1706 535 y Fk(D)r Fl(;W)1855 523 y Fq(\(\012\),)g(for)d(ev)n(ery)g(lo)r(cal)h (unstable)g(manifold)g Fp(W)35 b Fm(2)515 623 y(F)583 593 y Fl(u)654 623 y Fq(and)27 b(is)h(called)f(the)h(unstable)f (dimension)h(of)f(\012.)515 868 y Fh(Theorem)d(B)44 b Fn(L)l(et)25 b Fp(T)34 b Fq(:)23 b(\012)g Fe(\011)h Fn(b)l(e)h(a)g Fp(C)1695 838 y Fo(2)1758 868 y Fn(gener)l(alize)l(d)h(horsesho)l(e)g (with)g(b)l(ounde)l(d)f(distortion)515 968 y(and)34 b(exp)l(ansion)h(c) l(o)l(e\016cient)f(b)l(ounde)l(d)h(fr)l(om)f(b)l(elow.)53 b(Then,)36 b(the)e(fol)t(lowing)j(statements)515 1067 y(ar)l(e)30 b(e)l(quivalent:)613 1213 y(1.)42 b Fp(h)770 1225 y Fl(\026)815 1213 y Fq(\()p Fp(T)12 b Fq(\))22 b(=)1050 1146 y Fj(R)1133 1213 y Fq(log)28 b Fp(J)1322 1183 y Fl(u)1365 1213 y Fp(T)12 b Fq(\()p Fp(x)p Fq(\))p Fp(d\026)p Fq(\()p Fp(x)p Fq(\))p Fn(;)613 1369 y(2.)42 b(V)-6 b(ol)855 1381 y Fl(W)931 1369 y Fq(\(\012)18 b Fm(\\)h Fp(W)12 b Fq(\))23 b Fp(>)g Fq(0)29 b Fn(for)i(some)f(lo)l(c)l (al)h(unstable)e(manifold)j Fp(W)j Fm(2)23 b(F)2876 1338 y Fl(u)2920 1369 y Fn(;)613 1524 y(3.)42 b(the)22 b(volume)g(V)-6 b(ol)1259 1536 y Fl(W)1335 1524 y Fq(\(\012\))22 b Fn(of)h(the)f (unstable)f(Cantor)i(sets)e Fq(\012)15 b Fm(\\)g Fp(W)2681 1494 y Fl(u)2725 1524 y Fq(\()p Fp(x;)f Fg(R)p Fq(\))22 b Fn(is)g(uniformly)722 1624 y(b)l(ounde)l(d)j(away)g(fr)l(om)g(zer)l (o)f(and)g Fm(F)1820 1594 y Fl(s)1880 1624 y Fn(is)g(absolutely)h(c)l (ontinuous)f(wrt)f(L)l(eb)l(esgue)h(with)722 1724 y(uniformly)31 b(b)l(ounde)l(d)f(Jac)l(obians;)613 1879 y(4.)42 b Fp(\026)772 1891 y Fo(\012)864 1879 y Fn(is)f(absolutely)h(c)l(ontinuous)e(wrt)g (the)h(riemannian)h(volume)f(along)g(the)g(lo)l(c)l(al)722 1979 y(unstable)30 b(manifolds.)515 2125 y(In)g(addition,)k(if)d(any)h (of)f(the)g(ab)l(ove)h(c)l(onditions)g(hold)h(the)e(stable)g(invariant) h(lamination)515 2224 y Fm(F)583 2194 y Fl(s)643 2224 y Fn(is)26 b(absolutely)g(c)l(ontinuous)e(wrt)h(L)l(eb)l(esgue)h(and)f (it)g(has)h(a)g(b)l(ounde)l(d)g(Jac)l(obian,)h(namely)1233 2398 y Fp(d)14 b(H)1366 2368 y Fk(\003)1404 2398 y Fm(L)p 1461 2377 39 3 v 13 x Fl(\015)1500 2392 y Fi(u)p 1233 2435 311 4 v 1290 2511 a Fp(d)g Fm(L)1404 2523 y Fl(\015)1443 2507 y Fi(u)1553 2454 y Fq(\()p Fp(x)p Fq(\))24 b(=)1776 2351 y Fo(+)p Fk(1)1782 2375 y Fj(Y)1781 2552 y Fl(i)p Fo(=1)1987 2398 y Fp(J)8 b Fq(\()p Fp(T)35 b Fm(j)23 b Fp(\015)2251 2368 y Fl(u)2294 2398 y Fq(\)\()p Fp(T)2419 2368 y Fl(i)2446 2398 y Fq(\()p Fp(x)p Fq(\)\))p 1917 2435 744 4 v 1917 2511 a Fp(J)8 b Fq(\()p Fp(T)34 b Fm(j)p 2133 2466 48 4 v 24 w Fp(\015)2180 2479 y Fl(u)2224 2511 y Fq(\)\()p Fp(T)2349 2487 y Fl(i)2376 2511 y Fq(\()p Fp(H)7 b Fq(\()p Fp(x)p Fq(\)\)\))3273 2454 y(\(3\))515 2684 y Fn(for)36 b(every)h(p)l(air)g(of)g(admissible)g(unstable)f (manifolds)i Fp(\015)2325 2654 y Fl(u)2368 2684 y Fp(;)p 2405 2638 V 14 w(\015)2452 2652 y Fl(u)2530 2684 y Fm(2)c Fq(\000)2671 2654 y Fl(u)2714 2684 y Fn(.)57 b(Her)l(e)36 b Fm(L)3058 2696 y Fl(\015)3097 2680 y Fi(u)3176 2684 y Fn(is)g(the)515 2784 y(L)l(eb)l(esgue)g(me)l(asur)l(e)g(class)h(of)g Fp(\015)1550 2753 y Fl(u)1593 2784 y Fn(.)58 b(Comp)l(ar)l(e)38 b([10)q(].)59 b(Ther)l(efor)l(e,)40 b(the)d(er)l(go)l(dic)g(b)l(asin)g (of)515 2883 y(the)f(asymptotic)h(me)l(asur)l(e)f(c)l(ontains)g(a)h (set)f(of)g(p)l(ositive)i(volume,)h(so)d Fp(\026)2848 2895 y Fo(\012)2936 2883 y Fn(is)g(physic)l(al)t(ly)515 2983 y(observable.)515 3144 y(Pr)l(o)l(of.)43 b Fq(W)-7 b(e)27 b(notice)g(that)g Fm(D)1404 3156 y Fo(1)p Fl(;\015)1496 3140 y Fi(u)1565 3144 y Fq(represen)n(ts)f(the)h(Leb)r(esgue)f(measure) g(class)g(of)g Fp(\015)3114 3114 y Fl(u)3157 3144 y Fq(.)37 b(This)515 3243 y(is)23 b(a)h(straigh)n(tforw)n(ard)c(consequence)j(of) h(the)g(de\014nitions)g(and)f(the)h(fact)g(that)g Fp(})g Fq(generates)515 3343 y(the)29 b(stable)g(lamination.)40 b(Compare)27 b([11,)i(Theorem)f(C].)h(In)g(particular,)f(V)-7 b(ol)3001 3355 y Fl(\015)3040 3339 y Fi(u)3083 3343 y Fq(\()p Fp(X)7 b Fq(\))25 b Fp(>)f Fq(0)515 3443 y(implies)f(dim)931 3455 y Fk(D)r Fl(;W)1079 3443 y Fq(\()p Fp(X)7 b Fq(\))23 b(=)g(1.)35 b(Th)n(us,)24 b(dim)1798 3455 y Fk(D)r Fl(;W)1947 3443 y Fq(\(\012\))g(=)e(1)h(implies)h(that)f Fp(\026)2750 3455 y Fl(W)2826 3443 y Fq(,)h(the)g(pro)5 b(jection)515 3542 y(of)32 b Fp(\026)g Fq(along)f Fm(F)990 3512 y Fl(s)1058 3542 y Fq(on)n(to)g(the)i(lo)r(cal)f(unstable)g(manifolds)g Fp(W)42 b Fq(=)31 b Fp(W)2622 3512 y Fl(u)2665 3542 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\),)34 b(is)e(equiv)-5 b(alen)n(t)515 3642 y(to)34 b(the)i(restriction)d(of)i(the)g(Leb)r(esgue)g(measure)f (class)f Fm(L)2386 3654 y Fl(W)2497 3642 y Fq(to)i Fp(W)49 b Fm(\\)37 b Fq(\012,)g(b)r(ounded)e(b)n(y)515 3742 y(constan)n(ts)19 b(not)i(dep)r(ending)f(on)g Fp(x)k Fm(2)f Fq(\012)e(neither)f(on)g Fp(W)35 b Fm(2)24 b(F)2387 3711 y Fl(u)2430 3742 y Fq(.)35 b(In)20 b(particular,)h(the)g(v)n(olume)515 3841 y(V)-7 b(ol)648 3853 y Fl(W)724 3841 y Fq(\(\012\))26 b(of)f(the)h(unstable)f (Can)n(tor)f(sets)h(\012)i Fm(\\)h Fp(W)2134 3811 y Fl(u)2177 3841 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\))26 b(is)f(uniformly)g(b)r (ounded)h(a)n(w)n(a)n(y)515 3941 y(from)h(zero,)h(b)n(y)f(the)i (absolute)e(con)n(tin)n(uit)n(y)h(of)g(the)g(lamination)g(and)g(since)g (its)g(Jacobians)515 4040 y(are)37 b(uniformly)i(b)r(ounded.)71 b(As)39 b(V)-7 b(ol)1726 4052 y Fl(W)1802 4040 y Fq(\(\012\))42 b Fp(>)f Fq(0)e(for)f(some)g(lo)r(cal)g(unstable)h(manifold)515 4140 y Fp(W)48 b Fq(=)37 b Fp(W)833 4110 y Fl(u)876 4140 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\))37 b(implies)f(dim)1543 4152 y Fk(D)r Fl(;W)1692 4140 y Fq(\(\012\))h(=)g(1,)g(w)n(e)f(see)f (that)h(all)g(these)g(conditions)f(are)515 4240 y(equiv)-5 b(alen)n(t.)36 b(T)-7 b(o)28 b(\014nish,)g(w)n(e)f(notice)g(that)h(the) g(ergo)r(dic)f(basin)g(of)g Fp(\026)c Fq(=)g Fp(\026)2809 4252 y Fo(\012)2861 4240 y Fq(,)k(de\014ned)h(as)631 4471 y Fp(W)721 4437 y Fl(s)756 4471 y Fq(\()p Fp(\026)p Fq(\))c(=)f Fm(f)p Fp(x)g Fm(2)g Fg(R)g Fq(:)78 b(lim)1295 4521 y Fl(n)p Fk(!)p Fo(+)p Fk(1)1561 4415 y Fq(1)p 1557 4452 50 4 v 1557 4528 a Fp(n)1631 4367 y Fl(n)p Fk(\000)p Fo(1)1634 4392 y Fj(X)1633 4571 y Fl(k)q Fo(=0)1770 4471 y Fp(\036)p Fq(\()p Fp(T)1912 4437 y Fl(k)1953 4471 y Fq(\()p Fp(x)p Fq(\)\))25 b(=)2208 4358 y Fj(Z)2319 4471 y Fp(\036)p Fq(\()p Fp(z)t Fq(\))p Fp(d\026)p Fq(\()p Fp(z)t Fq(\))56 b Fm(8)14 b Fp(\036)22 b Fm(2)h Fp(C)3006 4437 y Fo(0)3044 4471 y Fq(\()p Fp(M)9 b Fq(\))p Fm(g)p Fp(;)515 4707 y Fq(con)n(tains)18 b(the)h(stable)g(set)g(of)g(\012,)i Fp(W)1599 4677 y Fl(s)1634 4707 y Fq(\(\012\))j(=)1870 4645 y Fj(S)1939 4732 y Fl(x)p Fk(2)p Fo(\012)2100 4707 y Fp(W)2190 4677 y Fl(s)2226 4707 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\).)35 b(No)n(w,)20 b Fp(W)2781 4677 y Fl(s)2816 4707 y Fq(\(\012\))g(has)e(p)r(ositiv)n(e)515 4807 y(v)n(olume,)42 b(in)e(view)f(of)h(the)g(absolute)f(con)n(tin)n(uit)n(y)h(of)f(the)h (stable)g(lamination)f(wrt)h(the)515 4907 y(Leb)r(esgue)28 b(measure,)h(using)g(a)f(F)-7 b(ubini's)30 b(theorem)f(argumen)n(t,)f (hence)h(V)-7 b(ol)14 b(\()p Fp(W)3039 4876 y Fl(s)3075 4907 y Fq(\()p Fp(\026)p Fq(\)\))26 b Fp(>)f Fq(0)515 5006 y(and)i(w)n(e)g(are)g(done.)p 3318 5006 4 57 v 3322 4954 50 4 v 3322 5006 V 3372 5006 4 57 v 1926 5255 a(7)p eop %%Page: 8 8 8 7 bop 639 523 a Fq(Next)26 b(Theorem)f(can)g(b)r(e)h(deduced)g(from)f (the)h(ab)r(o)n(v)n(e)e(results,)i(using)f(Moran's)f(co)n(v)n(ers)515 623 y(argumen)n(ts)g(as)h(w)n(e)g(did)h(b)r(efore)f(for)g(h)n(yp)r(erb) r(olic)g(rep)r(ellors.)35 b(Cf.)i([8)o(,)26 b(Chapter)f(7])g(and)h([11) o(,)515 722 y(Theorem)h(D].)515 922 y Fh(Theorem)c(C)43 b Fn(L)l(et)23 b Fp(T)35 b Fq(:)23 b(\012)g Fe(\011)g Fn(b)l(e)h(a)h Fp(C)1691 891 y Fo(2)1752 922 y Fn(gener)l(alize)l(d)h (c)l(onformal)f(horsesho)l(e)h(with)f(b)l(ounde)l(d)515 1021 y(distortion)38 b(and)g(exp)l(ansion)g(c)l(o)l(e\016cient)g(b)l (ounde)l(d)g(fr)l(om)g(b)l(elow.)63 b(Then)38 b(ther)l(e)f(exists)h(a) 515 1121 y(unique)29 b(er)l(go)l(dic)i(me)l(asur)l(e)f Fp(\026)23 b Fq(=)f Fp(\026)1595 1133 y Fo(\012)1677 1121 y Fn(such)29 b(that)613 1287 y(1.)42 b Fq(dim)861 1299 y Fk(H)922 1287 y Fq(\(\012)35 b Fm(\\)h Fp(W)1230 1257 y Fl(u)1273 1287 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\)\))34 b Fn(is)g(c)l(onstant)f(for)i(every)f(lo)l(c)l(al)h(unstable)f (manifold)h Fp(W)42 b Fq(=)722 1386 y Fp(W)812 1356 y Fl(u)856 1386 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\))p Fn(;)613 1553 y(2.)42 b(the)30 b(stable)g(lamination)h Fm(F)1577 1522 y Fl(s)1643 1553 y Fn(is)f(Lipschitz;)613 1719 y(3.)42 b Fp(\026)772 1731 y Fl(W)848 1719 y Fn(,)34 b(the)f(pr)l(oje)l(ction)i (of)e Fp(\026)1587 1731 y Fo(\012)1672 1719 y Fn(along)h(the)f(stable)h (lamination)g(onto)f Fp(W)12 b Fn(,)34 b(is)g(e)l(quiva-)722 1818 y(lent)e(to)h(the)f(Hausdor\013)h(me)l(asur)l(e)f(b)l(ounde)l(d)h (by)g(a)g(c)l(onstant)f(which)i(only)f(dep)l(ends)722 1918 y(on)h(the)h(b)l(ounde)l(d)f(ge)l(ometry)h(of)g(admissible)h (manifolds,)i(the)c(distortion)h(and)g(the)722 2017 y(non)e(line)l(ar)g (exp)l(ansion)h(c)l(o)l(e\016cient;)h(further,)f(the)f(fol)t(lowing)i (str)l(ong)e(F)-6 b(r)l(ostman)e('s)722 2117 y(distribution)31 b(pr)l(op)l(erty)f(holds:)891 2343 y Fp(C)956 2309 y Fk(\000)p Fo(1)1068 2343 y Fm(\024)1166 2287 y Fp(\026)1216 2299 y Fl(W)1291 2287 y Fq(\()p Fp(B)t Fq(\()p Fp(x;)14 b(r)r Fq(\))35 b Fm(\\)d Fp(W)12 b Fq(\))p 1166 2324 656 4 v 1278 2401 a Fp(r)1317 2377 y Fo(dim)1427 2386 y Fb(H)1481 2377 y Fo(\(\012)g Fk(\\)f Fl(W)e Fo(\))1854 2343 y Fm(\024)23 b Fp(C)274 b Fm(8)29 b Fp(x)23 b Fm(2)g Fq(\012)60 b Fp(W)35 b Fq(=)22 b Fp(W)2909 2309 y Fl(u)2953 2343 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\)\))p Fp(:)515 2594 y Fn(Pr)l(o)l(of.)43 b Fq(By)d(b)r(ounded)g(distortion)g(diam)28 b(\()p Fp(})1954 2606 y Fl(n)1999 2594 y Fq(\()p Fp(x)p Fq(\))g Fm(\\)f Fp(W)2310 2563 y Fl(u)2353 2594 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\)\))45 b Fm(\020)f(k)p Fp(J)2837 2563 y Fl(u)2880 2594 y Fp(T)2941 2563 y Fl(n)2985 2594 y Fq(\()p Fp(x)p Fq(\))p Fm(k)3138 2563 y Fk(\000)p Fo(1)3268 2594 y Fq(are)515 2693 y(uniformly)c(b)r(ounded)h(for)e(ev)n(ery)g Fp(x)45 b Fm(2)f Fq(\012)d(and)f Fp(n)k Fm(\025)g Fq(0.)74 b(W)-7 b(e)41 b(use)f(this)g(to)g(in)n(tro)r(duce)515 2793 y(Moran's)21 b(co)n(v)n(ers)g Fp(})1127 2805 y Fl(r)1187 2793 y Fq(for)h(ev)n(ery)g(0)h Fp(<)f(r)k(<)d Fq(1)f(asso)r(ciated)g (to)h Fp(})p Fq(.)35 b(F)-7 b(or)23 b(these)g(Moran's)e(co)n(v)n(ers) 515 2892 y(there)34 b(exists)f(a)h(univ)n(ersal)f(constan)n(t)g Fp(M)42 b(>)34 b Fq(1)f(suc)n(h)h(that)g Fp(B)t Fq(\()p Fp(x;)14 b(r)r Fq(\))39 b Fm(\\)e Fp(\015)2854 2862 y Fl(u)2931 2892 y Fq(in)n(tersects)c(at)515 2992 y(most)28 b(M)g(atoms)f(in)i(the)f(family)g Fp(})1616 3004 y Fl(r)1653 2992 y Fq(\()p Fp(\015)1733 2962 y Fl(u)1776 2992 y Fq(\),)h(for)e(ev)n (ery)g Fp(x)e Fm(2)f Fq(\003,)k(0)23 b Fp(<)h(r)i(<)e Fq(1)k(and)f(admissible)515 3092 y(unstable)g(manifold)h Fp(\015)1239 3062 y Fl(u)1282 3092 y Fq(.)37 b(Moreo)n(v)n(er,)25 b(of)i(Moran's)g(co)n(v)n(ers)e(follo)n(ws)i(b)r(ecause)1081 3274 y Fp(B)t Fq(\()p Fp(x;)14 b(C)1329 3240 y Fk(\000)p Fo(1)1433 3274 y Fp(r)r Fq(\))20 b Fm(\\)e Fp(\015)1645 3240 y Fl(u)1711 3274 y Fm(\022)23 b Fp(})1852 3286 y Fl(r)1889 3274 y Fq(\()p Fp(x)p Fq(\))c Fm(\\)g Fp(\015)2141 3240 y Fl(u)2207 3274 y Fm(\022)k Fp(B)t Fq(\()p Fp(x;)14 b(C)20 b(r)r Fq(\))g Fm(\\)f Fp(\015)2770 3240 y Fl(u)3273 3274 y Fq(\(4\))515 3457 y(for)39 b(ev)n(ery)g(0)44 b Fp(<)f(r)k(<)d Fq(1)39 b(and)h Fp(x)45 b Fm(2)f Fq(\012)c(and)g(some)g (univ)n(ersal)f(constan)n(t)g Fp(C)50 b(>)44 b Fq(1)39 b(only)515 3557 y(dep)r(ending)23 b(on)g(the)g(distortion)f(and)h(the)h (b)r(ounded)f(geometry)f(of)h(admissible)f(manifolds.)515 3656 y(Cf.)37 b([8,)27 b(Chapter)g(7].)639 3756 y(By)i(Theorem)e(A)i (there)g(exists)f(an)g(ergo)r(dic)g(T-in)n(v)-5 b(arian)n(t)27 b(probabilit)n(y)g Fp(\026)e Fq(=)f Fp(\026)3139 3768 y Fo(\012)3219 3756 y Fq(suc)n(h)515 3855 y(that)d Fp(\026)738 3867 y Fl(W)813 3855 y Fq(\()p Fp(})898 3867 y Fl(r)935 3855 y Fq(\()p Fp(x)p Fq(\)\))k Fm(\020)d Fq(V)-7 b(ol)14 b(\()p Fp(})1409 3867 y Fl(r)1446 3855 y Fq(\()p Fp(x)p Fq(\)\))1589 3825 y Fo(dim)1700 3834 y Fi(u)1739 3825 y Fo(\(\012\))1843 3855 y Fq(,)22 b(for)e(ev)n(ery)g Fp(x)k Fm(2)f Fq(\012)e(and)f(0)j Fp(<)g(r)i(<)e Fq(1.)34 b(Therefore,)515 3955 y Fp(\026)565 3967 y Fl(W)640 3955 y Fq(\()p Fp(B)t Fq(\()p Fp(x;)14 b(r)r Fq(\))29 b Fm(\\)e Fp(W)12 b Fq(\))43 b Fm(\020)g Fp(r)1349 3925 y Fl(u)12 b Fo(dim)1510 3933 y Fi(u)1549 3925 y Fo(\(\012\))1693 3955 y Fq(for)39 b(ev)n(ery)f Fp(x)44 b Fm(2)f Fq(\012,)g(0)g Fp(<)g(r)i(<)e Fq(1)c(and)h(a)f(suitable)515 4055 y(constan)n(t)32 b Fp(C)38 b(>)31 b Fq(1)h(indep)r(enden)n(t)i(of)e Fp(x)i Fq(and)e Fp(r)r Fq(.)53 b(So,)34 b(dim)2342 4067 y Fk(H)2403 4055 y Fq(\(\012)23 b Fm(\\)f Fp(W)12 b Fq(\))32 b(=)f Fp(u)14 b Fq(dim)3045 4067 y Fl(u)3088 4055 y Fq(\(\012\))33 b(and)515 4154 y(the)25 b(asymptotic)f(measure)g Fp(\026)1450 4166 y Fo(\012)1502 4154 y Fq(,)h(the)h(dynamical)e(measure)g(and)h (the)g(Hausdor\013)f(measure)515 4254 y(shall)32 b(b)r(e)g(equiv)-5 b(alen)n(t.)51 b(Finally)-7 b(,)33 b(b)n(y)f(F)-7 b(rostman's)32 b(uniform)g(distribution)g(prop)r(ert)n(y)f(and)515 4354 y(the)d(existence)f(of)g(a)g(b)r(ounded)h(Jacobian)e(with)i(resp)r(ect) f(to)h(the)f(Hausdor\013)g(measure)g(for)515 4453 y(the)h(holonomies)e (of)i(the)g(stable)f(lamination)g Fm(F)2048 4423 y Fl(s)2111 4453 y Fq(w)n(e)g(see)h(that)712 4636 y Fm(k)p Fp(x)19 b Fm(\000)f Fp(y)s Fm(k)989 4602 y Fl(\013)1058 4636 y Fm(\020)23 b(H)1216 4648 y Fl(\013;\015)1318 4632 y Fi(u)1361 4636 y Fq(\([)p Fp(x;)14 b(y)s Fq(]\))23 b Fm(\020)g(H)1780 4648 y Fl(\013;)p 1843 4615 39 3 v(\015)1881 4630 y Fi(u)1925 4636 y Fq(\([)p Fp(H)7 b Fq(\()p Fp(x)p Fq(\))p Fp(;)14 b(H)7 b Fq(\()p Fp(y)s Fq(\)]\))24 b Fm(\020)f(k)p Fp(H)7 b Fq(\()p Fp(x)p Fq(\))19 b Fm(\000)f Fp(H)7 b Fq(\()p Fp(y)s Fq(\))p Fm(k)3112 4602 y Fl(\013)3159 4636 y Fp(;)515 4819 y Fq(where)28 b Fp(\013)e Fq(=)g(dim)1064 4831 y Fk(H)1125 4819 y Fq(\(\012)20 b Fm(\\)f Fp(W)12 b Fq(\),)30 b Fm(H)1556 4831 y Fl(\013;\015)1658 4814 y Fi(u)1730 4819 y Fq(is)f(the)h(restriction)e(of)h(the)h(Hausdor\013)e (measure)g(to)515 4918 y(the)21 b(admissible)g(unstable)g(manifold)h Fp(\015)1759 4888 y Fl(u)1823 4918 y Fq(and)f Fp(H)30 b Fq(=)22 b Fp(H)2233 4930 y Fl(\015)2272 4914 y Fi(u)2311 4930 y Fl(;)p 2331 4897 V 1 x(\015)2369 4912 y Fi(u)2434 4918 y Fq(the)g(holonom)n(y)e(map)h(de\014ned)1926 5255 y(8)p eop %%Page: 9 9 9 8 bop 515 523 a Fq(b)n(y)32 b Fp(\015)683 493 y Fl(u)759 523 y Fq(and)p 926 477 48 4 v 33 w Fp(\015)974 491 y Fl(u)1017 523 y Fq(.)53 b(Constan)n(ts)32 b(are)g(uniform,)i(b)n(y)f (previous)e(remarks)h(and)g(the)i(b)r(ounded)515 623 y(geometry)26 b(of)i(admissible)f(manifolds.)36 b(Hence,)996 777 y Fm(k)p Fp(H)7 b Fq(\()p Fp(x)p Fq(\))18 b Fm(\000)g Fp(H)7 b Fq(\()p Fp(y)s Fq(\))p Fm(k)p 996 814 557 4 v 1136 890 a(k)p Fp(x)18 b Fm(\000)g Fp(y)s Fm(k)1585 833 y(\020)23 b Fq(1)193 b Fm(8)27 b Fp(x;)14 b(y)26 b Fm(2)d Fp(\015)2259 798 y Fl(u)2334 833 y Fm(\\)2439 754 y Fj([)2422 932 y Fl(z)r Fk(2)p Fo(\012)2562 833 y Fp(W)2652 798 y Fl(s)2688 833 y Fq(\()p Fp(z)t(;)14 b Fg(R)p Fq(\))p Fp(;)515 1088 y Fq(meaning)28 b(that)i Fp(C)1095 1058 y Fk(\000)p Fo(1)1184 1088 y Fm(k)p Fp(x)20 b Fm(\000)f Fp(y)s Fm(k)25 b(\024)g(k)p Fp(H)7 b Fq(\()p Fp(x)p Fq(\))20 b Fm(\000)f Fp(H)7 b Fq(\()p Fp(y)s Fq(\))p Fm(k)25 b(\024)g Fp(C)6 b Fm(k)p Fp(x)20 b Fm(\000)f Fp(y)s Fm(k)p Fq(,)29 b Fm(F)2716 1058 y Fl(s)2781 1088 y Fq(is)g(Lipsc)n(hitz,)g(and)515 1187 y(w)n(e)e(are)g(done.)p 3318 1187 4 57 v 3322 1135 50 4 v 3322 1187 V 3372 1187 4 57 v 515 1459 a Fr(3)134 b(Bounded)29 b(distortion)j(and)d(b)t (ounded)g(geometry)j(es-)716 1609 y(timates)515 1790 y Fq(Along)25 b(this)i(Section)f Fp(T)34 b Fq(:)23 b(\012)g Fe(\011)j Fq(will)h(b)r(e)f(a)g Fp(C)1908 1760 y Fo(2)1972 1790 y Fq(h)n(yp)r(erb)r(olic)g(horsesho)r(e)e(de\014ned)j(b)n(y)f (coun)n(t-)515 1890 y(ably)d(\(p)r(ossibly)h(in\014nite\))h(man)n(y)e (h)n(yp)r(erb)r(olic)g(branc)n(hes)f Fp(T)2364 1902 y Fl(i)2415 1890 y Fq(:)h Fp(S)2512 1902 y Fl(i)2562 1890 y Fm(\000)-14 b(!)23 b Fp(U)2776 1902 y Fl(i)2804 1890 y Fq(.)35 b(W)-7 b(e)24 b(supp)r(ose)g(in)515 1990 y(addition)30 b(that)h(the)g(collection)f Fp(T)1595 2002 y Fl(i)1652 1990 y Fq(ha)n(v)n(e)g(non-linear)f(expansion)g(b)r(ounded)i(from)f(b)r (elo)n(w)515 2089 y(and)d(b)r(ounded)h(distortion)f Fh(\(BD\))p Fq(.)515 2243 y Fh(Lemma)i(3.1.)40 b Fn(Ther)l(e)c(is)g(c)l(onstant)e Fp(C)39 b(>)32 b Fq(1)p Fn(,)k(only)g(dep)l(ending)g(on)f(the)g (distortion,)j(the)515 2342 y(exp)l(ansion)25 b(c)l(o)l(e\016cient)h (and)f(the)h(b)l(ounde)l(d)f(ge)l(ometry)h(of)g(the)f(admissible)i (manifolds)g(such)515 2442 y(that,)j(for)h(every)f Fp(\015)1110 2412 y Fl(u)1176 2442 y Fm(2)24 b Fq(\000)1307 2412 y Fl(u)1380 2442 y Fn(and)30 b Fp(n)23 b Fm(\025)f Fq(0)30 b Fn(it)f(holds)1203 2596 y Fp(J)8 b Fq(\()p Fp(T)1350 2566 y Fl(n)1418 2596 y Fm(j)23 b Fp(\015)1512 2566 y Fl(u)1555 2596 y Fq(\)\()p Fp(x)p Fq(\))p 1203 2633 497 4 v 1205 2709 a Fp(J)8 b Fq(\()p Fp(T)1352 2685 y Fl(n)1420 2709 y Fm(j)23 b Fp(\015)1514 2685 y Fl(u)1557 2709 y Fq(\)\()p Fp(y)s Fq(\))1732 2652 y Fm(\024)g Fq(exp\()p Fp(C)6 b(d)2087 2664 y Fl(u)2131 2652 y Fq(\()p Fp(T)2224 2618 y Fl(n)2269 2652 y Fq(\()p Fp(x)p Fq(\))p Fp(;)14 b(T)2478 2618 y Fl(n)2523 2652 y Fq(\()p Fp(y)s Fq(\)\))p 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Fi(i)p Fd(+1)e Fm(k)p Fp(D)r(T)1489 4974 y Fl(n)p Fk(\000)p Fl(i)p Fk(\000)p Fo(1)1693 4998 y Fq(\()p Fp(w)r Fq(\))p Fm(k)14 b Fq(inf)1976 5014 y Fl(w)r Fk(2)p Fl(\015)2110 4997 y Fi(i)2153 4998 y Fm(k)p Fp(D)r(T)e Fq(\()p Fp(w)r Fq(\))p Fm(k)2522 4941 y(\001)18 b(k)p Fp(T)2666 4907 y Fl(n)2710 4941 y Fq(\()p Fp(x)p Fq(\))i Fm(\000)e Fp(T)2985 4907 y Fl(n)3029 4941 y Fq(\()p Fp(y)s Fq(\))p Fm(k)1926 5255 y Fq(9)p eop %%Page: 10 10 10 9 bop 1177 590 a Fm(\024)23 b Fp(C)6 b Fq(\(\000\)\001)1544 486 y Fl(n)p Fk(\000)p Fo(1)1547 511 y Fj(X)1553 688 y Fl(i)p Fo(=0)1683 590 y Fp(\025)1731 555 y Fk(\000)p Fo(\()p Fl(n)p Fk(\000)p Fl(i)p Fk(\000)p Fo(1\))2060 590 y Fm(\001)18 b(k)p Fp(T)2204 555 y Fl(n)2248 590 y Fq(\()p Fp(x)p Fq(\))h Fm(\000)f Fp(T)2522 555 y Fl(n)2567 590 y Fq(\()p Fp(y)s Fq(\))p Fm(k)591 896 y(\024)23 b Fp(C)6 b Fq(\(\000\)\001)961 792 y Fo(+)p Fk(1)960 817 y Fj(X)957 993 y Fl(n)p Fo(=0)1096 896 y Fp(\025)1144 862 y Fk(\000)p Fl(n)1260 896 y Fm(\001)19 b(k)p Fp(T)1405 862 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Fq(\()p Fp(T)941 3944 y Fl(i)991 3932 y Fm(j)24 b Fp(\015)1086 3901 y Fl(u)1129 3932 y Fq(\()p Fp(z)t Fq(\)\)\()p Fp(z)t Fq(\).)37 b(Arguing)26 b(similarly)h(as)g(w)n (e)g(did)h(b)r(efore)f(w)n(e)h(get)762 4159 y(log)907 4102 y Fp(J)953 4114 y Fl(i)981 4102 y Fq(\()p Fp(T)1074 4072 y Fl(j)1108 4102 y Fq(\()p Fp(x)p Fq(\)\))p 907 4139 346 4 v 909 4216 a Fp(J)955 4228 y Fl(i)983 4216 y Fq(\()p Fp(T)1076 4192 y Fl(j)1110 4216 y Fq(\()p Fp(y)s Fq(\)\))1313 4159 y Fm(\024)227 b Fq(sup)1429 4233 y Fl(w)r Fk(2)p Fl(W)1595 4216 y Fi(s)1626 4233 y Fo(\()p Fl(T)1700 4216 y Fi(j)1731 4233 y Fo(\()p Fl(x)p Fo(\))p Fl(;)p Fa(R)p Fo(\))1934 4159 y Fm(kr)28 b Fq(log)f Fp(J)2253 4171 y Fl(i)2281 4159 y Fq(\()p Fp(w)r Fq(\))p Fm(k)33 b(\001)g(k)p Fp(T)2640 4124 y Fl(j)2673 4159 y Fq(\()p Fp(x)p Fq(\))20 b Fm(\000)e Fp(T)2948 4124 y Fl(j)2982 4159 y Fq(\()p Fp(y)s Fq(\))p Fm(k)898 4467 y(\024)1051 4399 y Fq(sup)1176 4420 y Fl(w)r Fk(2)p Fl(W)1342 4403 y Fi(s)1373 4420 y Fo(\()p Fl(T)1447 4403 y Fi(j)1478 4420 y Fo(\()p Fl(x)p Fo(\))p Fl(;)p Fa(R)p Fo(\))1685 4399 y Fm(kr)28 b Fq(log)g Fp(J)2005 4411 y Fl(i)2032 4399 y Fq(\()p Fp(w)r Fq(\))p Fm(k)p 1051 4448 1150 4 v 1142 4524 a Fq(inf)1243 4539 y Fl(w)r Fk(2)p Fl(W)1409 4523 y Fi(s)1440 4539 y Fo(\()p Fl(T)1514 4523 y Fi(j)1546 4539 y Fo(\()p Fl(x)p Fo(\))p Fl(;)p Fa(R)p Fo(\))1753 4524 y Fm(k)p Fp(D)r(T)1915 4536 y Fl(i)1941 4524 y Fq(\()p Fp(w)r Fq(\))p Fm(k)2210 4467 y(k)p Fp(T)2313 4432 y Fl(j)s Fo(+1)2431 4467 y Fq(\()p Fp(x)p Fq(\))19 b Fm(\000)f Fp(T)2705 4432 y Fl(j)s Fo(+1)2823 4467 y Fq(\()p Fp(y)s Fq(\))p Fm(k)p Fp(:)515 4680 y Fq(Th)n(us)27 b(log\()p Fp(J)910 4692 y Fl(i)938 4680 y Fq(\()p Fp(T)1031 4650 y Fl(j)1066 4680 y Fq(\()p Fp(x)p Fq(\)\))14 b Fp(=)g(J)1325 4692 y Fl(i)1353 4680 y Fq(\()p Fp(T)1446 4650 y Fl(j)1480 4680 y Fq(\()p Fp(y)s Fq(\)\)\))24 b Fm(\024)f Fq(\001)14 b Fp(C)6 b Fq(\(\000\))p Fp(\025)2076 4650 y Fk(\000)p Fo(\()p Fl(j)s Fo(+1\))2300 4680 y Fp(d)2343 4692 y Fl(s)2378 4680 y Fq(\()p Fp(x;)14 b(y)s Fq(\))29 b(and)e(then)708 4930 y(log)h Fp(h)p Fq(\()p Fp(x;)14 b(y)s Fq(\))24 b Fm(\024)1196 4826 y Fo(+)p Fk(1)1194 4851 y Fj(X)1197 5028 y Fl(j)s Fo(=1)1342 4930 y Fq(\001)14 b Fp(C)6 b Fq(\(\000\))p Fp(\025)1654 4896 y Fk(\000)p Fo(\()p Fl(j)s Fo(+1\))1878 4930 y Fp(d)1921 4942 y Fl(s)1957 4930 y Fq(\()p Fp(x;)14 b(y)s Fq(\))23 b(=)g(\001)14 b Fp(C)6 b Fq(\(1)19 b Fm(\000)f Fp(\025)2632 4896 y Fk(\000)p Fo(1)2721 4930 y Fq(\))p Fp(\025)2801 4896 y Fk(\000)p Fo(2)2891 4930 y Fp(d)2934 4942 y Fl(s)2970 4930 y Fq(\()p Fp(x;)c(y)s Fq(\))p Fp(:)1905 5255 y Fq(10)p eop %%Page: 11 11 11 10 bop 515 523 a Fq(F)-7 b(urther,)27 b Fp(h)p Fq(\()p Fp(x;)14 b(y)s Fq(\))g Fp(=)g(h)p Fq(\()p Fp(x)1276 493 y Fk(0)1300 523 y Fp(;)g(y)1381 493 y Fk(0)1404 523 y Fq(\))23 b Fm(\024)g Fq(exp)13 b(\()p Fp(C)6 b Fq(\()p Fp(d)1859 535 y Fl(s)1896 523 y Fq(\()p Fp(x;)14 b(y)s Fq(\))19 b Fm(\000)f Fp(d)2233 535 y Fl(s)2269 523 y Fq(\()p Fp(x)2348 493 y Fk(0)2372 523 y Fp(;)c(y)2453 493 y Fk(0)2475 523 y Fq(\)\).)38 b(Therefore,)1070 674 y Fp(h)p Fq(\()p Fp(x;)14 b(y)s Fq(\))p 1047 711 288 4 v 1047 787 a Fp(h)p Fq(\()p Fp(x)1174 763 y Fk(0)1198 787 y Fp(;)g(y)1279 763 y Fk(0)1302 787 y Fq(\))1367 730 y Fm(\024)23 b Fq(max)13 b Fm(f)p Fq(exp)o(\()p Fp(C)6 b(d)1931 742 y Fl(u)1976 730 y Fq(\()p Fp(x;)14 b(x)2139 695 y Fk(0)2163 730 y Fq(\)\))p Fp(;)g Fq(exp\()p Fp(C)6 b(d)2531 742 y Fl(u)2575 730 y Fq(\()p Fp(y)s(;)14 b(y)2732 695 y Fk(0)2755 730 y Fq(\)\))g Fp(;)515 937 y Fq(for)27 b(some)g Fp(C)i Fq(=)23 b Fp(C)6 b Fq(\(\001)p Fp(;)14 b(\025;)g Fq(\000\))24 b Fp(>)f Fq(1)k(as)g(claimed.)p 3318 937 4 57 v 3322 885 50 4 v 3322 937 V 3372 937 4 57 v 515 1099 a Fh(Corollary)k(3.2.)40 b Fn(Ther)l(e)30 b(is)e(a)h(c)l(onstant)e Fp(C)j Fq(=)22 b Fp(C)6 b Fq(\(\001)p Fp(;)14 b(\025;)g Fq(\000\))25 b Fp(>)d Fq(1)28 b Fn(such)g(that,)h (for)g(any)g(two)515 1198 y(admissible)j(manifolds)h Fp(\015)1341 1168 y Fl(u)1414 1198 y Fn(and)p 1577 1152 48 4 v 32 w Fp(\015)1624 1166 y Fl(u)1698 1198 y Fn(in)e Fq(\000)1853 1168 y Fl(u)1927 1198 y Fn(it)f(holds)i(V)-6 b(ol)2356 1210 y Fl(\015)2395 1194 y Fi(u)2438 1198 y Fq(\()p Fp(P)12 b Fq(\))25 b Fm(\020)f Fn(V)-6 b(ol)p 2814 1177 39 3 v 2814 1211 a Fl(\015)2852 1192 y Fi(u)2896 1198 y Fq(\()p Fp(P)12 b Fq(\))31 b Fn(for)g(every)515 1298 y Fp(P)k Fm(2)23 b Fp(})p Fn(.)639 1445 y Fq(In)39 b(particular,)h(the)g(v)n(olume)e(of)g(the)h(cylinder)g(V)-7 b(ol)13 b(\()p Fp(P)f Fq(\))40 b(is)e(comparable)f(with)j(the)515 1544 y(v)n(olume)30 b(of)i(its)f(unstable)g(sections)g Fp(\015)1720 1514 y Fl(u)1798 1544 y Fm(\\)k Fp(P)12 b Fq(,)32 b(i.e.)48 b(V)-7 b(ol)2296 1556 y Fl(\015)2335 1540 y Fi(u)2378 1544 y Fq(\()p Fp(P)12 b Fq(\))29 b Fm(\020)g Fq(V)-7 b(ol)14 b(\()p Fp(P)e Fq(\),)33 b(b)r(ounded)e(b)n(y) 515 1644 y(uniform)c(constan)n(ts)g(whic)n(h)g(do)h(not)f(dep)r(end)i (on)e Fp(p)g Fq(neither)h Fp(\015)2497 1614 y Fl(u)2563 1644 y Fm(2)23 b Fq(\000)2693 1614 y Fl(u)2737 1644 y Fq(.)515 1805 y Fn(Pr)l(o)l(of.)43 b Fq(It)28 b(follo)n(ws)e(from)i (Corollary)d(3.1)i(and)g(Lemma)g(3.2)g(that)h(for)f(ev)n(ery)f Fp(n)d(>)g Fq(0)1261 1959 y(V)-7 b(ol)1395 1971 y Fl(\015)1434 1955 y Fi(u)1477 1959 y Fq(\()p Fp(})1562 1971 y Fl(n)1607 1959 y Fq(\()p Fp(x)p Fq(\)\))p 1261 1996 491 4 v 1261 2072 a(V)g(ol)p 1395 2051 39 3 v 1395 2085 a Fl(\015)1433 2066 y Fi(u)1477 2072 y Fq(\()p Fp(})1562 2084 y Fl(n)1607 2072 y Fq(\()p Fp(x)p Fq(\)\))1784 2015 y Fm(\024)23 b Fp(C)1961 1951 y Fq(sup)2086 1972 y Fl(\015)2125 1955 y Fi(u)2164 1972 y Fk(2)p Fo(\000)2250 1955 y Fi(u)2320 1951 y Fq(V)-7 b(ol)14 b(\()p Fp(\015)2534 1921 y Fl(u)2578 1951 y Fq(\))p 1961 1996 649 4 v 1973 2072 a(inf)2074 2084 y Fl(\015)2113 2068 y Fi(u)2152 2084 y Fk(2)p Fo(\000)2238 2068 y Fi(u)2308 2072 y Fq(V)-7 b(ol)14 b(\()p Fp(\015)2522 2048 y Fl(u)2566 2072 y Fq(\))2620 2015 y Fp(;)515 2221 y Fq(and)27 b(similarly)g(so)g(the)h(lo)n(w)n(er)e(b)r(ound.)p 3318 2221 4 57 v 3322 2168 50 4 v 3322 2221 V 3372 2221 4 57 v 639 2382 a(Previous)g(Corollary)f(has)i(t)n(w)n(o)g(imp)r(ortan) n(t)g(consequences:)639 2529 y Fm(\017)41 b Fq(the)36 b(holonom)n(y)f(of)g(the)h(stable)g(lamination)f(is)h(absolutely)e(con) n(tin)n(uous)h(wrt)h(the)722 2629 y(dynamical)27 b(class)g Fm(fD)1422 2641 y Fl(\013;\015)1524 2625 y Fi(u)1566 2629 y Fm(g)1608 2641 y Fl(\015)1647 2625 y Fi(u)1685 2641 y Fk(2)p Fo(\000)1771 2625 y Fi(u)1842 2629 y Fq(and)h(it)g(has)f (a)g(b)r(ounded)h(Jacobian;)639 2786 y Fm(\017)41 b Fq(the)33 b(measure)f(of)h(the)g(unstable)g(Can)n(tor)e(sets)i(cut)g(b)n(y)g Fp(W)43 b Fq(=)31 b Fp(W)2813 2756 y Fl(u)2857 2786 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\))33 b(are)f(uni-)722 2885 y(formly)27 b(b)r(ounded,)h(i.e.)37 b Fp(C)1555 2855 y Fk(\000)p Fo(1)1668 2885 y Fm(\024)23 b(D)1820 2897 y Fl(\013;W)1958 2885 y Fq(\(\012\))h Fm(\024)e Fp(C)34 b Fq(for)27 b(ev)n(ery)g Fp(W)35 b Fm(2)23 b(F)2892 2855 y Fl(u)2935 2885 y Fq(.)515 3032 y(Both)i(conclusions)f(follo)n(ws)g (from)g(the)i(\(ob)n(vious\))e(estimate)h Fm(D)2511 3044 y Fl(\013;\015)2613 3028 y Fi(u)2656 3032 y Fq(\()p Fp(B)t Fq(\))f Fm(\020)e(D)2962 3044 y Fl(\013;)p 3025 3011 39 3 v 1 x(\015)3064 3026 y Fi(u)3107 3032 y Fq(\()p Fp(H)7 b Fq(\()p Fp(B)t Fq(\)\))515 3132 y(b)r(ounded)34 b(b)n(y)g(constan)n(ts)f Fp(C)39 b Fq(=)34 b Fp(C)6 b Fq(\(\001)p Fp(;)14 b(\025;)g Fq(\000\))34 b Fp(>)f Fq(1,)i(for)f(ev)n (ery)e Fp(\015)2564 3102 y Fl(u)2641 3132 y Fq(and)p 2809 3086 48 4 v 34 w Fp(\015)2857 3100 y Fl(u)2900 3132 y Fq(,)k(where)d Fp(H)40 b Fq(=)515 3232 y Fp(H)584 3244 y Fl(\015)623 3227 y Fi(u)661 3244 y Fl(;)p 681 3211 39 3 v(\015)720 3225 y Fi(u)801 3232 y Fq(is)f(the)f(holonom)n(y)f (transformation)g(de\014ned)i(b)n(y)f(these)g(unstable)g(admissible)515 3331 y(manifolds.)639 3431 y(W)-7 b(e)35 b(shall)g(pro)n(v)n(e)e(later) h(that)h Fp(h)g Fq(=)f Fp(h)1840 3443 y Fl(\015)1879 3427 y Fi(u)1918 3443 y Fl(;)p 1938 3410 V(\015)1976 3425 y Fi(u)2054 3431 y Fq(b)r(elo)n(w)h(is)f(the)i(Jacobian)d(of)h Fp(H)42 b Fq(wrt)35 b(the)515 3530 y(dynamical)27 b(measure)f(class:) 1274 3767 y Fp(h)p Fq(\()p Fp(x)p Fq(\))e(=)1544 3625 y Fj(")1593 3663 y Fo(+)p Fk(1)1598 3688 y Fj(Y)1598 3865 y Fl(i)p Fo(=1)1804 3710 y Fp(J)8 b Fq(\()p Fp(T)34 b Fm(j)24 b Fp(\015)2068 3680 y Fl(u)2111 3710 y Fq(\)\()p Fp(T)2236 3680 y Fl(i)2263 3710 y Fq(\()p Fp(x)p Fq(\)\))p 1734 3748 744 4 v 1734 3824 a Fp(J)8 b Fq(\()p Fp(T)34 b Fm(j)p 1949 3778 48 4 v 23 w Fp(\015)1997 3792 y Fl(u)2040 3824 y Fq(\)\()p Fp(T)2165 3800 y Fl(i)2193 3824 y Fq(\()p Fp(H)7 b Fq(\()p Fp(x)p Fq(\)\)\))2487 3625 y Fj(#)2536 3639 y Fl(\013)2597 3767 y Fp(:)515 3998 y Fq(The)27 b(follo)n(wing)g(result)g(is)h(an)f(straigh)n(tforw)n(ard)e (consequence)h(of)i(Lemma)f(3.2.)515 4145 y Fh(Lemma)i(3.3.)40 b Fn(F)-6 b(or)30 b(every)h Fp(\015)1466 4115 y Fl(u)1539 4145 y Fn(and)p 1700 4099 V 30 w Fp(\015)1747 4113 y Fl(u)1821 4145 y Fn(in)e Fq(\000)1974 4115 y Fl(u)2047 4145 y Fn(it)h(holds)1401 4304 y Fp(h)p Fq(\()p Fp(x)p Fq(\))24 b Fm(\024)f Fq(exp\()p Fp(C)6 b(d)1939 4316 y Fl(s)1975 4304 y Fq(\()p Fp(x;)14 b(H)7 b Fq(\()p Fp(x)p Fq(\)\)\))931 b(\(8\))515 4463 y Fn(for)40 b(a)f(c)l(onstant)g Fp(C)47 b Fq(=)39 b Fp(C)6 b Fq(\(\001)p Fp(;)14 b(\025;)g Fq(\000\))42 b Fp(>)e Fq(1)p Fn(.)66 b(Her)l(e)39 b Fp(d)2192 4475 y Fl(s)2228 4463 y Fq(\()p Fp(x;)14 b(y)s Fq(\))41 b(=)f Fp(d)2609 4478 y Fl(W)2680 4461 y Fi(s)2712 4478 y Fo(\()p Fl(x;)p Fa(R)p Fo(\))2866 4463 y Fq(\()p Fp(x;)14 b(y)s Fq(\))40 b Fn(for)g Fp(y)j Fm(2)515 4562 y Fp(W)605 4532 y Fl(s)640 4562 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\))p Fn(.)39 b(A)n(lso,)31 b(for)f(every)h Fp(x)f Fn(and)g Fp(y)j Fn(in)d Fp(\015)1933 4532 y Fl(u)1185 4713 y Fp(h)p Fq(\()p Fp(x)p Fq(\))p 1185 4750 160 4 v 1187 4826 a Fp(h)p Fq(\()p Fp(y)s Fq(\))1378 4769 y Fm(\024)23 b Fq(max)13 b Fm(f)p Fp(e)1715 4735 y Fl(C)t(d)1802 4743 y Fi(u)1840 4735 y Fo(\()p Fl(x;y)r Fo(\)\))2029 4769 y Fp(;)27 b(e)2118 4735 y Fl(C)t(d)2205 4743 y Fi(u)2244 4735 y Fo(\()p Fl(H)t Fo(\()p Fl(x)p Fo(\))p Fl(;H)t Fo(\()p Fl(y)r Fo(\)\)\))2640 4769 y Fm(g)p Fp(;)568 b Fq(\(9\))515 4977 y Fn(wher)l(e)30 b Fp(d)792 4989 y Fl(u)836 4977 y Fq(\()p Fp(x;)14 b(y)s Fq(\))23 b(=)g Fp(d)1182 4992 y Fl(W)1253 4975 y Fi(u)1293 4992 y Fo(\()p Fl(x;)p Fa(R)p Fo(\))1446 4977 y Fq(\()p Fp(x;)14 b(y)s Fq(\))p Fn(,)31 b(for)g Fp(y)25 b Fm(2)f Fp(W)2062 4947 y Fl(u)2105 4977 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\))p Fn(.)1905 5255 y Fq(11)p eop %%Page: 12 12 12 11 bop 515 523 a Fr(4)134 b(Asymptotic)47 b(unstable)f(measures)h (in)f(generalized)716 672 y(horsesho)t(es)515 854 y Fh(Lemma)29 b(4.1.)40 b Fn(L)l(et)26 b Fq(\012)g Fn(b)l(e)h(a)f Fp(C)1501 824 y Fo(2)1565 854 y Fn(hyp)l(erb)l(olic)i(horsesho)l(e)g(de\014ne)l (d)f(by)g(in\014nitely)f(many)h(hy-)515 954 y(p)l(erb)l(olic)34 b(br)l(anches)f Fp(T)1212 966 y Fl(i)1266 954 y Fq(:)28 b Fp(S)1368 966 y Fl(i)1423 954 y Fm(\000)-14 b(!)28 b Fp(U)1642 966 y Fl(i)1701 954 y Fn(satisfying)34 b(b)l(ounde)l(d)f (distortion)g(c)l(ondition)g Fh(\(BD\))p Fn(.)515 1054 y(Supp)l(ose)23 b(in)g(addition)h(that)f(the)g(dynamic)l(al)h(me)l (asur)l(e)f(class)g(in)g(dimension)h Fp(\013)f Fq(=)g(dim)3211 1066 y Fl(u)3254 1054 y Fq(\(\012\))515 1153 y Fn(is)29 b(nontrivial,)h(i.e.)39 b Fq(0)23 b Fp(<)g Fm(D)1377 1165 y Fl(\013;\015)1479 1149 y Fi(u)1521 1153 y Fq(\(\012\))h Fp(<)f Fq(+)p Fm(1)p Fn(.)37 b(Then,)30 b(for)g(every)f(lo)l(c)l(al)h (unstable)e(manifold)515 1253 y Fp(W)35 b Fq(=)22 b Fp(W)805 1223 y Fl(u)849 1253 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\))29 b Fn(ther)l(e)h(is)f(a)h(unique)e(Bor)l(el)j(pr)l(ob)l(ability)g Fp(\026)2383 1265 y Fl(W)2488 1253 y Fn(e)l(quivalent)e(to)g(the)h (dynam-)515 1352 y(ic)l(al)36 b(me)l(asur)l(e)e Fm(D)1063 1364 y Fl(\013;W)1236 1352 y Fn(and)h(a)h(c)l(onstant)e Fp(C)k Fq(=)32 b Fp(C)6 b Fq(\(\001)p Fp(;)14 b(\015)5 b(;)14 b Fq(\000\))32 b Fp(>)g Fq(1)i Fn(only)i(dep)l(ending)g(in)f (the)515 1452 y(distortion)30 b(and)h(the)f(b)l(ounde)l(d)g(ge)l (ometry)g(of)h(the)e(admissible)j(manifolds)g(such)e(that)1436 1672 y Fp(C)1501 1637 y Fk(\000)p Fo(1)1613 1672 y Fm(\024)1848 1615 y Fp(\026)1898 1627 y Fl(W)1973 1615 y Fq(\()p Fp(P)12 b Fq(\))p 1711 1652 529 4 v 1711 1730 a(\()p Fn(V)-6 b(ol)28 b Fp(P)12 b Fq(\))1987 1706 y Fo(dim)2097 1714 y Fi(u)2136 1706 y Fo(\(\012\))2273 1672 y Fm(\024)22 b Fp(C)q(;)788 b Fq(\(10\))515 1893 y Fn(for)30 b(every)h(ge)l(ometric) l(al)g(cylinder)g Fp(P)k Fm(2)23 b(P)7 b Fn(.)639 2054 y Fq(W)-7 b(e)31 b(will)g(pro)n(v)n(e)d(this)j(result)f(through)g(sev)n (eral)e(lemmas)j(b)r(elo)n(w.)44 b(F)-7 b(or)30 b(this)h(w)n(e)f(b)r (egin)515 2154 y(recalling)22 b(the)i(de\014nitions)g(of)g(a)g (measurable)e(transformation)g(of)i(a)f(measure)g(space.)35 b(Let)515 2253 y(\()p Fp(X)r(;)14 b(d)p Fq(\))38 b(b)r(e)h(a)e (complete)h(metric)g(space.)67 b Fp(T)51 b Fq(:)41 b(\()p Fp(X)r(;)14 b Fm(A)p Fp(;)g(\026)p Fq(\))41 b Fe(\011)c Fq(is)h(a)g Fn(me)l(asur)l(e-the)l(or)l(etic)l(al)515 2353 y(exp)l(anding)30 b(tr)l(ansformation)f Fq(if)f(there)f(exists)g (a)h(sequence)f(of)g(partitions)g Fp(})2923 2365 y Fl(n)2996 2353 y Fq(suc)n(h)g(that:)616 2514 y(1.)p 722 2442 332 4 v 722 2452 a Fj(S)791 2539 y Fl(P)9 b Fk(2)p Fl(})929 2547 y Fi(n)989 2514 y Fp(P)34 b Fq(=)23 b Fp(X)62 b(;)69 b Fq(inf)1488 2526 y Fl(P)9 b Fk(2)p Fl(})1626 2534 y Fi(n)1685 2514 y Fp(\026)p Fq(\()p Fp(T)j Fq(\()p Fp(P)g Fq(\)\))23 b Fp(>)g Fq(0;)616 2677 y(2.)41 b(for)31 b(eac)n(h)f Fp(n)e Fm(\025)g Fq(0)j(and)f Fp(P)41 b Fm(2)29 b Fp(})1682 2689 y Fl(n)p Fo(+1)1811 2677 y Fq(,)j Fp(T)12 b Fq(\()p Fp(P)g Fq(\))30 b(is)h(a)f(union)h(of)g(atoms)f(of)h Fp(})2976 2689 y Fl(n)3083 2677 y Fq(mo)r(d)g(\(0\))722 2777 y(and)d Fp(T)34 b Fm(j)23 b Fp(P)39 b Fq(is)28 b(one-to-one;)616 2940 y(3.)41 b(there)28 b(are)e(0)d Fp(<)f(\025)i(<)f Fq(1)k(,)h Fp(K)g(>)23 b Fq(0)k(suc)n(h)g(that)1496 3117 y Fp(d)p Fq(\()p Fp(x;)14 b(y)s Fq(\))24 b Fm(\024)e Fp(K)6 b(\025)1967 3082 y Fl(n)2026 3117 y Fp(d)p Fq(\()p Fp(T)2162 3082 y Fl(n)2207 3117 y Fq(\()p Fp(x)p Fq(\))p Fp(;)14 b(T)2416 3082 y Fl(n)2461 3117 y Fq(\()p Fp(x)p 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b(,)25 b(this)f(is)g(a)g(v)n(ersion)f(of)h(a)g(remark)-5 b(able)515 5006 y(and)27 b(w)n(ell)h(kno)n(wn)e(result)i(due)g(to)f(D.)h(Ruelle.) 1905 5255 y(12)p eop %%Page: 13 13 13 12 bop 639 523 a Fq(No)n(w,)44 b(w)n(e)c(pro)5 b(jects)40 b(the)i(dynamics)e(in)h(\012)g(along)e(the)j(stable)e(lamination)h(to)f (get)515 623 y(an)k(expanding)g(transformation.)87 b(Namely)-7 b(,)49 b(w)n(e)c(let)g Fp(\034)9 b Fq(\()p Fp(x)p Fq(\))53 b(=)e Fp(H)2726 638 y Fl(W)9 b Fo(\()p Fl(T)g Fo(\()p Fl(x)p Fo(\)\))p Fl(;W)3112 623 y Fm(\016)30 b Fp(T)12 b Fq(\()p Fp(x)p Fq(\),)515 722 y(where)41 b Fp(W)59 b Fq(=)46 b Fp(W)1107 692 y Fl(u)1151 722 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\))42 b(and)g Fp(W)12 b Fq(\()p Fp(T)g Fq(\()p Fp(x)p Fq(\)\))47 b(=)f Fp(W)2145 692 y Fl(u)2189 722 y Fq(\()p Fp(T)12 b Fq(\()p Fp(x)p Fq(\))p Fp(;)i Fg(R)p Fq(\).)80 b(W)-7 b(e)42 b(shall)f(denote)h(b)r(e-)515 822 y(lo)n(w)c(\003)j(=)g(\012)26 b Fm(\\)g Fp(W)1138 792 y Fl(u)1182 822 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\))39 b(the)g(unstable)g(Can)n(tor)e(set)i(cut)g(b)n(y)g Fp(W)53 b Fq(=)42 b Fp(W)2961 792 y Fl(u)3004 822 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\))39 b(and)515 922 y Fp(\034)32 b Fq(:)24 b(\003)e Fm(\000)-14 b(!)23 b Fq(\003)i(the)g(pro)5 b(jection)23 b(of)i Fp(T)34 b Fm(j)23 b Fp(W)1792 891 y Fl(u)1836 922 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\))25 b(along)e(the)i(stable)g(lea)n (v)n(es.)34 b(No)n(w)24 b(put)i(the)515 1021 y(metric)j Fp(d)818 991 y Fk(\003)856 1021 y Fq(\()p Fp(x;)14 b(y)s Fq(\))26 b(=)f(dist)1312 1033 y Fk(H)1387 1021 y Fp(d)p Fq(\()p Fp(W)1552 991 y Fl(s)1588 1021 y Fq(\()p Fp(x;)14 b(R)q Fq(\))p Fp(;)g(W)1927 991 y Fl(s)1963 1021 y Fq(\()p Fp(y)s(;)g(R)q Fq(\)\))29 b(on)g(\003,)g(where)f(dist)2849 1033 y Fk(H)2939 1021 y Fq(is)h(the)h(Haus-)515 1121 y(dor\013)24 b(distance)g(of)g(sets.)36 b(\(\003)p Fp(;)14 b(d)1483 1091 y Fk(\003)1521 1121 y Fq(\))25 b(is)f(a)g(compact)g (metric)g(space.)35 b(The)25 b(Mark)n(o)n(v)d(partitions)515 1220 y Fp(})p Fq(\()p Fp(n;)14 b(W)e Fq(\))25 b(and)g(non-linear)e (expansion)h(can)h(b)r(e)h(used)f(to)f(v)n(erify)h(the)g(ab)r(o)n(v)n (e)f(axioms)g(pro)n(v-)515 1320 y(ing)31 b(that)h Fp(\034)40 b Fq(:)30 b(\(\003)p Fp(;)14 b Fm(D)1161 1332 y Fl(\013;W)1299 1320 y Fq(\))31 b Fe(\011)g Fq(satis\014es)g(the)h(ab)r(o)n(v)n(e)f (conditions)g(except)h(p)r(ossibly)f(b)n(y)h(the)515 1420 y(H\177)-42 b(older)33 b(prop)r(ert)n(y)g(of)h(the)g(Jacobian.)55 b(Actually)-7 b(,)36 b(b)n(y)e(conformalit)n(y)e(of)i(the)h(dynamical) 515 1519 y(measure)27 b(class)h(and)h(the)g(absolute)f(con)n(tin)n(uit) n(y)g(of)h(the)g(lamination)f(implies)h(that)g Fp(\034)39 b Fq(has)515 1619 y(a)27 b(Jacobian)f(wrt)h Fm(D)1147 1631 y Fl(\013;W)1286 1619 y Fq(.)37 b(In)28 b(fact,)1220 1810 y Fp(J)1266 1822 y Fl(\013)1314 1810 y Fp(\034)9 b Fq(\()p Fp(x)p Fq(\))24 b(=)1630 1754 y Fp(d)14 b(H)1763 1724 y Fk(\003)1801 1754 y Fm(D)1865 1766 y Fl(\013;W)p 1592 1791 450 4 v 1592 1867 a Fp(d)g Fm(D)1713 1882 y Fl(\013;W)9 b Fo(\()p Fl(T)g Fo(\()p Fl(x)p Fo(\)\))2051 1810 y Fq(\()p Fp(T)j Fq(\()p Fp(x)p Fq(\)\)[)p Fp(J)2364 1776 y Fl(u)2408 1810 y Fp(T)g Fq(\()p Fp(x)p Fq(\)])2603 1776 y Fl(\013)2650 1810 y Fp(;)515 2012 y Fq(for)20 b(ev)n(ery)g(lo)r(cal)g(unstable)h(manifold)g Fp(W)35 b Fq(=)22 b Fp(W)1990 1982 y Fl(u)2034 2012 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\),)23 b(where)d(b)n(y)h(conformalit)n(y)e(w)n(e)i (refer)515 2112 y(to)27 b(the)h(follo)n(wing)515 2247 y Fh(Lemma)h(4.2.)40 b Fn(L)l(et)d Fp(h)g Fq(:)g Fp(W)49 b Fm(\000)-14 b(!)37 b Ff(R)1691 2217 y Fl(m)1797 2247 y Fn(b)l(e)g(a)h Fp(C)2051 2217 y Fo(1+)p Fl(\017)2205 2247 y Fn(emb)l(e)l(dding,)j(then)c Fm(D)2895 2259 y Fl(\013;W)3033 2247 y Fq(\()p Fp(h)p Fq(\()p Fp(B)t Fq(\)\))h(=)515 2280 y Fj(R)554 2377 y Fl(B)625 2347 y Fq(\()p Fp(J)8 b(h)p Fq(\()p Fp(x)p Fq(\)\))902 2317 y Fl(\013)965 2347 y Fp(d)p Fm(D)1072 2359 y Fl(\013;W)1210 2347 y Fq(\()p Fp(x)p Fq(\))p Fn(,)31 b(for)g(every)f(Bor)l(el)h(subset)d Fp(B)g Fm(\022)22 b Fq(\012)32 b Fm(\\)g Fp(W)12 b Fn(.)39 b Fp(J)8 b(h)29 b Fn(is)h(the)g(Jac)l(obian)515 2447 y(of)g(h)g(wrt)g(intrinsic)g(riemannian)h(volume)f(in)g(W.)639 2582 y Fq(This)c(w)n(as)e(pro)n(v)n(ed)g(in)h([11)o(])h(for)f(h)n(yp)r (erb)r(olic)g(rep)r(ellors.)35 b(The)25 b(pro)r(of)g(for)g(horsesho)r (es)e(of)515 2682 y(saddle)g(t)n(yp)r(e)i(is)f(similar)f(and)h(w)n(e)g (omit)g(it.)36 b(Notice)24 b(to)r(o)g(for)g(further)g(use)g(that)g(the) h(sev)n(eral)515 2782 y(metrics)j(that)i(w)n(e)e(are)g(using)h(are)f (indeed)h(comparable,)f(b)r(ounded)h(b)n(y)g(constan)n(ts)f(whic)n(h) 515 2881 y(only)f(dep)r(end)h(on)g(the)f(b)r(ounded)h(geometry)f(of)g (the)h(admissible)g(manifolds,)f(i.e.)927 3026 y Fp(d)970 3038 y Fl(u)1014 3026 y Fq(\()p Fp(x;)14 b(y)s Fq(\))24 b Fm(\020)e(k)p Fp(x)d Fm(\000)f Fp(y)s Fm(k)k(\020)h Fp(d)1747 2992 y Fk(\003)1785 3026 y Fq(\()p Fp(x;)14 b(y)s Fq(\))111 b(for)27 b(ev)n(ery)54 b Fp(y)26 b Fm(2)d Fp(W)2697 2992 y Fl(u)2741 3026 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\))p Fp(:)265 b Fq(\(11\))639 3171 y(One)29 b(can)f(k)n(eep)h(trac)n(k)f(the)h(pro)r(of)f(of)h(Ruelle's)g(theorem)g (using)f(b)r(ounded)i(distortion)515 3270 y(estimates)36 b(in)h(previous)f(Section,)j(in)e(spite)f(that)h(Jacobians)e(migh)n(t)i (not)g(b)r(e)g(H\177)-42 b(older)36 b(.)515 3370 y(Compare)26 b([6,)h(Chapter)h(I)r(I)r(I])g(and)f([12)o(,)h(Theorem)f(8].)639 3470 y(Actually)-7 b(,)43 b(let)c Fp( )45 b Fq(=)d Fm(\000)14 b Fq(log)28 b Fp(J)1604 3482 y Fl(\013)1651 3470 y Fp(\034)49 b Fq(and)39 b Fm(L)1966 3482 y Fl( )2016 3470 y Fp(h)p Fq(\()p Fp(x)p Fq(\))44 b(=)2326 3407 y Fj(P)2413 3495 y Fk(f)p Fl(y)r Fo(:)p Fl(\034)7 b Fo(\()p Fl(y)r Fo(\)=)p Fl(x)p Fk(g)2767 3470 y Fp(e)2806 3439 y Fl( )r Fo(\()p Fl(y)r Fo(\))2944 3470 y Fp(h)p Fq(\()p Fp(y)s Fq(\))39 b(b)r(e)h(the)515 3569 y(Ruelle)26 b(op)r(erator.)34 b(Then,)26 b Fm(L)1429 3539 y Fk(\003)1429 3593 y Fl( )1480 3569 y Fm(D)1544 3581 y Fl(\013;W)1705 3569 y Fq(=)d Fm(D)1857 3581 y Fl(\013;W)1995 3569 y Fq(,)k(where)e Fm(L)2340 3539 y Fk(\003)2340 3593 y Fl( )2416 3569 y Fq(is)h(the)g(adjoin)n(t)f(of)h(the)g(Ruelle)515 3669 y(op)r(erator)34 b(acting)i(o)n(v)n(er)f(the)i(Borel)e(measures.)63 b(W)-7 b(e)36 b(seek)g(for)g(a)g Fp(\034)9 b Fq(-in)n(v)-5 b(arian)n(t)36 b(measure)515 3768 y Fp(\026)565 3780 y Fl(W)664 3768 y Fq(absolutely)23 b(con)n(tin)n(uous)g(wrt)h Fm(D)1680 3780 y Fl(\013;W)1842 3768 y Fq(whic)n(h)g(is)g(obtained)g(b) n(y)f(redistributing)g(it)i(mass)515 3868 y(through)i(the)h(dynamics,)f (that)h(is)1267 4087 y Fp(\026)1317 4099 y Fl(W)1393 4087 y Fq(\()p Fp(B)t Fq(\))c(=)77 b(lim)1635 4137 y Fl(n)p Fk(!)p Fo(+)p Fk(1)1901 4031 y Fq(1)p 1897 4068 50 4 v 1897 4144 a Fp(n)1971 3983 y Fl(n)p Fk(\000)p Fo(1)1974 4008 y Fj(X)1974 4187 y Fl(k)q Fo(=0)2121 4031 y Fm(D)2185 4043 y Fl(\013;W)2323 4031 y Fq(\()p Fp(\034)2400 4001 y Fk(\000)p Fl(k)2494 4031 y Fp(B)t Fq(\))p 2121 4068 473 4 v 2195 4144 a Fm(D)2259 4156 y Fl(\013;W)2397 4144 y Fq(\(\003\))2603 4087 y Fp(;)515 4328 y Fq(for)18 b(ev)n(ery)f(Borel)g(subset)h Fp(B)27 b Fm(\022)c Fq(\003.)34 b(Indeed,)20 b(let)f Fp(\026)2038 4340 y Fl(n)2106 4328 y Fq(=)k(1)14 b Fp(=)g(n)2382 4266 y Fj(P)2470 4286 y Fl(n)p Fk(\000)p Fo(1)2470 4353 y Fl(k)q Fo(=0)2600 4328 y Fq(\()p Fp(\034)2677 4298 y Fl(k)2719 4328 y Fq(\))2751 4298 y Fk(\003)2789 4328 y Fm(D)2853 4340 y Fl(\013;W)3005 4328 y Fp(=)g Fm(D)3125 4340 y Fl(\013;W)3263 4328 y Fq(\(\003\).)515 4428 y Fp(\026)565 4440 y Fl(n)641 4428 y Fq(is)32 b(absolutely)e(con)n(tin)n(uous)g(wrt)i Fm(D)1767 4440 y Fl(\013;W)1937 4428 y Fq(and)f(it)g(has)g(a)g(Jacobian,)g (namely)g Fp(h)3128 4440 y Fl(n)3173 4428 y Fq(\()p Fp(x)p Fq(\))f(=)515 4539 y(1)14 b Fp(=)g(n)703 4477 y Fj(P)791 4497 y Fl(n)p Fk(\000)p Fo(1)791 4564 y Fl(k)q Fo(=0)935 4477 y Fj(P)1022 4564 y Fl(y)r Fk(2)p Fl(\034)1141 4547 y Fb(\000)p Fi(n)1226 4564 y Fl(x)1295 4539 y Fp(e)1334 4509 y Fl(S)1375 4517 y Fi(n)1415 4509 y Fl( )r Fo(\()p Fl(y)r Fo(\))1553 4539 y Fq(,)28 b(where)f Fp(S)1895 4551 y Fl(n)1940 4539 y Fp( )s Fq(\()p Fp(x)p Fq(\))d(=)2220 4477 y Fj(P)2307 4497 y Fl(n)p Fk(\000)p Fo(1)2307 4564 y Fl(i)p Fo(=0)2465 4539 y Fp( )s Fq(\()p Fp(\034)2599 4509 y Fl(i)2628 4539 y Fq(\()p Fp(x)p Fq(\)\).)639 4638 y(The)19 b(follo)n(wing)f(conditions)g(are)g(su\016cien)n(t)h(to)g(pro) n(v)n(e)e(the)i(existence)f(of)h(an)g(asymptotic)515 4738 y(measure.)1423 4804 y Fj(X)1362 4987 y Fl(y)r Fk(2)p Fl(\034)1481 4970 y Fb(\000)p Fi(n)1566 4987 y Fl(x)1631 4883 y Fp(e)1670 4849 y Fl(S)1711 4857 y Fi(n)1751 4849 y Fl( )r Fo(\()p Fl(y)r Fo(\))1912 4883 y Fm(\024)k Fp(K)116 b Fm(8)27 b Fp(x)51 b Fm(2)23 b Fq(\003)p Fp(;)713 b Fq(\(12\))1905 5255 y(13)p eop %%Page: 14 14 14 13 bop 515 523 a Fq(for)27 b(some)g(constan)n(t)g Fp(K)h(>)23 b Fq(0)k(and)g(that)1055 759 y Fp(C)1114 771 y Fl( )1165 759 y Fq(\()p Fp(x;)14 b(y)s Fq(\))23 b(=)g(sup)1468 827 y Fl(n>)p Fo(0)1693 759 y Fq(sup)1635 834 y Fl(y)r Fk(2)p Fl(\034)1754 817 y Fb(\000)p Fi(n)1839 834 y Fl(x)1904 656 y(n)p Fk(\000)p Fo(1)1907 680 y Fj(X)1913 857 y Fl(i)p Fo(=0)2044 692 y Fj(\002)2079 759 y Fp( )s Fq(\()p Fp(\034)2213 725 y Fl(i)2241 759 y Fq(\()p Fp(x)p Fq(\)\))d Fm(\000)e Fp( )s Fq(\()p Fp(\034)2621 725 y Fl(i)2650 759 y Fq(\()p Fp(y)s Fq(\)\))2790 692 y Fj(\003)3231 759 y Fq(\(13\))515 1012 y(exists,)41 b(is)d(b)r(ounded)h(b)n(y)f(a)h (constan)n(t)e Fp(C)1836 1024 y Fl( )1887 1012 y Fq(,)k(and)e Fp(C)2183 1024 y Fl( )2233 1012 y Fq(\()p Fp(x;)14 b(y)s Fq(\))42 b Fm(\000)-14 b(!)41 b Fq(0)d(as)g Fp(d)p Fq(\()p Fp(x;)14 b(x)3041 981 y Fk(0)3066 1012 y Fq(\))41 b Fm(\000)-14 b(!)41 b Fq(0.)515 1111 y(Cf.)h([12)o(,)30 b(Theorem)e(8].)42 b(Condition)29 b(\(12\))g(and)g(\(13\))f(imply)i(that)g(the)f(sequence) g Fm(f)p Fp(h)3208 1123 y Fl(n)3252 1111 y Fm(g)g Fq(is)515 1211 y(uniformly)d(b)r(ounded)h(and)f(equicon)n(tin)n(uous.)35 b Fp(h)26 b Fq(will)h(b)r(e)f(a)g(limit)h(p)r(oin)n(t)g(of)f(this)h (sequence)515 1310 y(whic)n(h)g(exists)f(b)n(y)h(Arzel\023)-42 b(a-Ascoli)26 b(theorem.)36 b(Standard)26 b(argumen)n(ts)g(sho)n(w)g (that)i Fp(h)f Fq(is)g(the)515 1410 y(unique)37 b(p)r(ositiv)n(e)f (eigenfunction)h(with)1827 1343 y Fj(R)1910 1410 y Fp(h)p Fq(\()p Fp(x)p Fq(\))p Fp(d)14 b Fm(D)2190 1422 y Fl(\013;W)2330 1410 y Fq(\()p Fp(x)p Fq(\))39 b(=)f Fm(D)2647 1422 y Fl(\013;W)2785 1410 y Fq(\(\003\).)65 b Fp(d\026)3088 1422 y Fl(W)3164 1410 y Fq(\()p Fp(x)p Fq(\))39 b(=)515 1510 y Fp(h)p Fq(\()p Fp(x)p Fq(\))p Fp(d)p Fm(D)781 1522 y Fl(\013;W)920 1510 y Fq(\()p Fp(x)p Fq(\))14 b Fp(=)g Fm(D)1165 1522 y Fl(\013;W)1304 1510 y Fq(\(\003\))28 b(is)f(the)h(asymptotic)f(measure)g(w)n(e)g(are)g(lo)r(oking)f(for.)639 1609 y(Inequalities)j(\(12\))g(and)g(\(13\))g(ab)r(o)n(v)n(e)f(will)i (b)r(e)g(consequence)e(of)h(the)h(follo)n(wing)e(result)515 1709 y(and)f(argumen)n(ts)f(in)i([6,)g(Chapter)f(I)r(I)r(I,)h(Lemma)f (1.4].)515 1869 y Fh(Lemma)i(4.3.)40 b Fn(Ther)l(e)31 b(exists)e(a)h(c)l(onstant)f Fp(C)h Fq(=)22 b Fp(C)6 b Fq(\(\001)p Fp(;)14 b(\025;)g Fq(\000\))25 b Fp(>)d Fq(1)29 b Fn(such)h(that)1135 2044 y Fp(J)1181 2056 y Fl(\013)1228 2044 y Fp(\034)1273 2009 y Fl(n)1319 2044 y Fq(\()p Fp(x)p Fq(\))14 b Fp(=)g(J)1546 2056 y Fl(\013)1594 2044 y Fp(\034)1639 2009 y Fl(n)1685 2044 y Fq(\()p Fp(y)s Fq(\))23 b Fm(\024)f Fq(exp\()p Fp(C)f(d)2185 2009 y Fk(\003)2223 2044 y Fq(\()p Fp(\034)2300 2009 y Fl(n)2346 2044 y Fq(\()p Fp(x)p Fq(\))p Fp(;)14 b(\034)2539 2009 y Fl(n)2586 2044 y Fq(\()p Fp(y)s Fq(\)\)\))515 2219 y Fn(and)1120 2318 y Fm(j)p Fp(J)1189 2330 y Fl(\013)1236 2318 y Fp(\034)1281 2284 y Fl(n)1327 2318 y Fq(\()p Fp(x)p Fq(\))g Fp(=)g(J)1554 2330 y Fl(\013)1602 2318 y Fp(\034)1647 2284 y Fl(n)1693 2318 y Fq(\()p Fp(y)s Fq(\))k Fm(\000)g Fq(1)p Fm(j)23 b(\024)g Fp(C)d(d)2200 2284 y Fk(\003)2238 2318 y Fq(\()p Fp(\034)2315 2284 y Fl(n)2361 2318 y Fq(\()p Fp(x)p Fq(\))p Fp(;)14 b(\034)2554 2284 y Fl(n)2601 2318 y Fq(\()p Fp(y)s Fq(\)\)\))515 2463 y Fn(whenever)30 b Fp(})p Fq(\()p Fp(n;)14 b(W)e Fq(\)\()p Fp(x)p Fq(\))25 b(=)d Fp(})p Fq(\()p Fp(n;)14 b(W)e Fq(\)\()p Fp(y)s Fq(\))p Fn(.)515 2623 y(Pr)l(o)l(of.)43 b Fq(W)-7 b(e)34 b(shall)f(denote)g(b)r(elo)n(w)g Fp(C)39 b Fq(=)32 b Fp(C)6 b Fq(\(\000\))34 b(all)f(the)h(v)-5 b(arious)32 b(constan)n(ts)g(whic)n(h)h(only)515 2722 y(dep)r(ends)23 b(on)g(the)g(b)r(ounded)h(geometry)d(of)i(the)h(admissible)f (submanifolds.)35 b(No)n(w,)23 b(denote)515 2822 y Fp(J)569 2792 y Fl(u)612 2822 y Fq(\()p Fp(x)p Fq(\))h(=)f Fp(J)889 2792 y Fl(u)932 2822 y Fp(T)12 b Fq(\()p Fp(x)p Fq(\))28 b(and)f Fp(h)p Fq(\()p Fp(z)t Fq(\))c(=)1559 2760 y Fj(Q)1637 2780 y Fo(+)p Fk(1)1637 2847 y Fl(i)p Fo(=1)1772 2822 y Fp(J)1826 2792 y Fl(u)1870 2822 y Fq(\()p Fp(T)1963 2792 y Fl(i)1990 2822 y Fp(z)t Fq(\))14 b Fp(=)g(J)2189 2792 y Fl(u)2231 2822 y Fq(\()p Fp(T)2324 2792 y Fl(i)2351 2822 y Fp(H)7 b Fq(\()p Fp(z)t Fq(\)\).)37 b(Then,)1121 3028 y Fp(J)1167 3040 y Fl(\013)1214 3028 y Fp(\034)1259 2998 y Fl(n)1305 3028 y Fq(\()p Fp(x)p Fq(\))p 1121 3065 297 4 v 1123 3141 a Fp(J)1169 3153 y Fl(\013)1216 3141 y Fp(\034)1261 3117 y Fl(n)1307 3141 y Fq(\()p Fp(y)s Fq(\))1450 3084 y(=)1538 2942 y Fj(")1586 2981 y Fl(n)p Fk(\000)p Fo(1)1596 3006 y Fj(Y)1589 3184 y Fl(k)q Fo(=0)1736 3028 y Fp(h)p Fq(\()p Fp(T)12 b Fq(\()p Fp(\034)1954 2998 y Fl(k)1995 3028 y Fq(\()p Fp(x)p Fq(\)\)\))p 1736 3065 436 4 v 1738 3141 a Fp(h)p Fq(\()p Fp(T)g Fq(\()p Fp(\034)1956 3117 y Fl(k)1997 3141 y Fq(\()p Fp(y)s Fq(\)\)\))2214 3084 y Fm(\001)2279 3028 y Fp(J)2333 2998 y Fl(u)2377 3028 y Fq(\()p Fp(\034)2454 2998 y Fl(k)2496 3028 y Fq(\()p Fp(x)p Fq(\)\))p 2279 3065 361 4 v 2281 3141 a Fp(J)2335 3117 y Fl(u)2379 3141 y Fq(\()p Fp(\034)2456 3117 y Fl(k)2498 3141 y Fq(\()p Fp(y)s Fq(\)\))2650 2942 y Fj(#)2698 2955 y Fl(\013)2760 3084 y Fp(:)515 3347 y Fq(No)n(w)21 b Fm(j)14 b Fq(log)28 b Fp(J)924 3317 y Fl(u)967 3347 y Fq(\()p Fp(\034)1044 3317 y Fl(k)1086 3347 y Fq(\()p Fp(x)p Fq(\)\))7 b Fm(\000)g Fq(log)29 b Fp(J)1498 3317 y Fl(u)1541 3347 y Fq(\()p Fp(\034)1618 3317 y Fl(k)1660 3347 y Fq(\()p Fp(y)s Fq(\)\))p Fm(j)24 b(\024)f Fq(sup)2060 3367 y Fl(z)2112 3347 y Fm(kr)14 b Fp(J)2291 3317 y Fl(u)2334 3347 y Fq(\()p Fp(z)t Fq(\))p Fm(k)20 b(\001)h(k)p Fp(\034)2634 3317 y Fl(k)2675 3347 y Fq(\()p Fp(x)p Fq(\))7 b Fm(\000)g Fp(\034)2910 3317 y Fl(k)2952 3347 y Fq(\()p Fp(y)s Fq(\))p Fm(k)p Fq(.)35 b(Hence)658 3522 y Fm(j)14 b Fq(log)27 b Fp(J)883 3487 y Fl(u)927 3522 y Fq(\()p Fp(\034)1004 3487 y Fl(k)1046 3522 y Fq(\()p Fp(x)p Fq(\)\))19 b Fm(\000)f Fq(log)28 b Fp(J)1480 3487 y Fl(u)1524 3522 y Fq(\()p Fp(\034)1601 3487 y Fl(k)1643 3522 y Fq(\()p Fp(y)s Fq(\)\))p Fm(j)23 b(\024)g Fp(C)34 b Fq(sup)2055 3587 y Fl(z)2149 3522 y Fm(kr)14 b Fp(J)2328 3487 y Fl(u)2371 3522 y Fq(\()p Fp(z)t Fq(\))p Fm(k)31 b(\001)i(k)p Fp(T)2710 3487 y Fl(k)2749 3522 y Fq(\()p Fp(x)p Fq(\))19 b Fm(\000)f Fp(T)3023 3487 y Fl(k)3063 3522 y Fq(\()p Fp(y)s Fq(\))p Fm(k)p Fp(;)515 3750 y Fq(since)i Fm(k)p Fp(\034)798 3720 y Fl(k)840 3750 y Fq(\()p Fp(x)p Fq(\))5 b Fm(\000)g Fp(\034)1071 3720 y Fl(k)1113 3750 y Fq(\()p Fp(y)s Fq(\))p Fm(k)23 b(\020)f Fp(d)1416 3720 y Fk(\003)1455 3750 y Fq(\()p Fp(\034)1532 3720 y Fl(k)1574 3750 y Fq(\()p Fp(x)p Fq(\))p Fp(;)14 b(\034)1767 3720 y Fl(k)1809 3750 y Fq(\()p Fp(y)s Fq(\)\))24 b(=)e Fp(d)2103 3720 y Fk(\003)2142 3750 y Fq(\()p Fp(T)2235 3720 y Fl(k)2275 3750 y Fq(\()p Fp(x)p Fq(\))p Fp(;)14 b(T)2484 3720 y Fl(k)2525 3750 y Fq(\()p Fp(y)s Fq(\)\))23 b Fm(\020)g(k)p Fp(T)2879 3720 y Fl(k)2918 3750 y Fq(\()p Fp(x)p Fq(\))5 b Fm(\000)g Fp(T)3165 3720 y Fl(k)3206 3750 y Fq(\()p Fp(y)s Fq(\))p Fm(k)p Fq(,)515 3849 y(b)n(y)37 b(the)h(b)r(ounded)g(geometry)f(of)h (the)g(admissible)f(manifolds.)67 b(Therefore,)39 b(using)f(the)515 3949 y(b)r(ounded)31 b(distortion)e(condition)h(as)g(w)n(e)g(did)h(in)g (the)f(ab)r(o)n(v)n(e)f(Section)i(w)n(e)f(conclude)g(that)515 4049 y(for)d(ev)n(ery)f Fp(n)d(>)g Fq(0,)685 4194 y Fl(n)p Fk(\000)p Fo(1)688 4219 y Fj(X)687 4398 y Fl(k)q Fo(=0)839 4298 y Fm(j)14 b Fq(log)27 b Fp(J)1064 4264 y Fl(u)1108 4298 y Fq(\()p Fp(\034)1185 4264 y Fl(k)1227 4298 y Fq(\()p Fp(x)p Fq(\)\))19 b Fm(\000)f Fq(log)28 b Fp(J)1661 4264 y Fl(u)1705 4298 y Fq(\()p Fp(\034)1782 4264 y Fl(k)1824 4298 y Fq(\()p Fp(y)s Fq(\)\))p Fm(j)23 b(\024)g Fp(C)2177 4194 y Fl(n)p Fk(\000)p Fo(1)2180 4219 y Fj(X)2180 4398 y Fl(k)q Fo(=0)2317 4298 y Fp(\025)2365 4264 y Fk(\000)p Fl(k)2458 4298 y Fp(d)2501 4264 y Fk(\003)2539 4298 y Fq(\()p Fp(\034)2616 4264 y Fl(n)2663 4298 y Fq(\()p Fp(x)p Fq(\))p Fp(;)14 b(\034)2856 4264 y Fl(n)2902 4298 y Fq(\()p Fp(y)s Fq(\)\))1937 4511 y Fm(\024)23 b Fp(C)6 b Fq(\(1)18 b Fm(\000)g Fp(\025)2313 4477 y Fk(\000)p Fo(1)2403 4511 y Fq(\))p Fp(d)2478 4477 y Fk(\003)2516 4511 y Fq(\()p Fp(\034)2593 4477 y Fl(n)2639 4511 y Fq(\()p Fp(x)p Fq(\))p Fp(;)c(\034)2832 4477 y Fl(n)2879 4511 y Fq(\()p Fp(y)s Fq(\)\))p Fp(:)515 4686 y Fq(Therefore,)969 4940 y(log)1090 4798 y Fj(")1138 4836 y Fl(n)p Fk(\000)p Fo(1)1148 4861 y Fj(Y)1141 5039 y Fl(k)q Fo(=0)1288 4883 y Fp(J)1342 4853 y Fl(u)1386 4883 y Fq(\()p Fp(\034)1463 4853 y Fl(k)1504 4883 y Fq(\()p Fp(x)p Fq(\)\))p 1288 4920 V 1290 4997 a Fp(J)1344 4973 y Fl(u)1387 4997 y Fq(\()p Fp(\034)1464 4973 y Fl(k)1506 4997 y Fq(\()p Fp(y)s Fq(\)\))1659 4798 y Fj(#)1707 4811 y Fl(\013)1777 4940 y Fm(\024)23 b Fq(2)p Fp(C)6 b Fq(\(1)18 b Fm(\000)g Fp(\025)2195 4905 y Fk(\000)p Fo(1)2285 4940 y Fq(\))p Fp(d)2360 4905 y Fk(\003)2398 4940 y Fq(\()p Fp(\034)2475 4905 y Fl(n)2522 4940 y Fq(\()p Fp(x)p Fq(\))p Fp(;)c(\034)2715 4905 y Fl(n)2761 4940 y Fq(\()p Fp(y)s Fq(\)\))p Fp(:)1905 5255 y Fq(14)p eop %%Page: 15 15 15 14 bop 515 523 a Fq(Also,)27 b(log)h Fp(h)p Fq(\()p Fp(T)12 b Fq(\()p Fp(\034)1078 493 y Fl(k)1119 523 y Fq(\()p Fp(x)p Fq(\)\)\))i Fp(=)g(h)p Fq(\()p Fp(T)e Fq(\()p Fp(\034)1582 493 y Fl(k)1624 523 y Fq(\()p Fp(y)s Fq(\)\)\))28 b(is)g(b)r(ounded)g(from)f(ab)r(o)n(v)n(e)f(b)n(y)948 698 y Fp(C)20 b Fq(max)13 b Fm(f)p Fp(d)1280 710 y Fl(u)1323 698 y Fq(\()p Fp(T)f Fq(\()p Fp(\034)1493 664 y Fl(k)1534 698 y Fq(\()p Fp(x)p Fq(\)\))p Fp(;)i(T)e Fq(\()p Fp(\034)1852 664 y Fl(k)1895 698 y Fq(\()p Fp(y)s Fq(\)\)\))i Fp(;)28 b(d)2175 710 y Fl(u)2219 698 y Fq(\()p Fp(\034)2296 664 y Fl(k)q Fo(+1)2421 698 y Fq(\()p Fp(x)p Fq(\))p Fp(;)14 b(\034)2614 664 y Fl(k)q Fo(+1)2741 698 y Fq(\()p Fp(y)s Fq(\)\))p Fm(g)p Fp(;)515 873 y Fq(and)25 b(this,)i(in)f(turn,)g(is)g (b)r(ounded)g(b)n(y)g(2)p Fp(C)6 b(d)1842 843 y Fk(\003)1880 873 y Fq(\()p Fp(\034)1957 843 y Fl(k)q Fo(+1)2083 873 y Fq(\()p Fp(x)p Fq(\))p Fp(;)14 b(\034)2276 843 y Fl(k)q Fo(+1)2402 873 y Fq(\()p Fp(y)s Fq(\)\).)37 b(Th)n(us,)26 b(for)f(ev)n(ery)g Fp(n)e(>)f Fq(0)515 973 y(w)n(e)27 b(ha)n(v)n(e)813 1206 y(log)948 1064 y Fj(")996 1102 y Fl(n)p Fk(\000)p Fo(1)1006 1127 y Fj(Y)999 1305 y Fl(k)q Fo(=0)1146 1149 y Fp(h)p Fq(\()p Fp(T)12 b Fq(\()p Fp(\034)1364 1119 y Fl(k)1405 1149 y Fq(\()p Fp(x)p Fq(\)\)\))p 1146 1187 436 4 v 1148 1263 a Fp(h)p Fq(\()p Fp(T)g Fq(\()p Fp(\034)1366 1239 y Fl(k)1407 1263 y Fq(\()p Fp(y)s Fq(\)\)\))1591 1064 y Fj(#)1663 1206 y Fm(\024)23 b Fq(2)p Fp(C)1871 1102 y Fl(n)p Fk(\000)p Fo(1)1874 1127 y Fj(X)1874 1305 y Fl(k)q Fo(=0)2011 1206 y Fp(\025)2059 1171 y Fk(\000)p Fl(n)p Fk(\000)p Fl(k)q Fk(\000)p Fo(1)2330 1206 y Fp(d)2373 1171 y Fk(\003)2411 1206 y Fq(\()p Fp(\034)2488 1171 y Fl(n)2535 1206 y Fq(\()p Fp(x)p Fq(\))p Fp(;)14 b(\034)2728 1171 y Fl(n)2774 1206 y Fq(\()p Fp(y)s Fq(\)\))1809 1419 y Fm(\024)23 b Fp(C)6 b Fq(\(1)18 b Fm(\000)g Fp(\025)2185 1384 y Fk(\000)p Fo(1)2275 1419 y Fq(\))p Fp(d)2350 1384 y Fk(\003)2388 1419 y Fq(\()p Fp(\034)2465 1384 y Fl(n)2512 1419 y Fq(\()p Fp(x)p Fq(\))p Fp(;)c(\034)2705 1384 y Fl(n)2751 1419 y Fq(\()p Fp(y)s Fq(\)\))p Fp(;)515 1594 y Fq(absorbing)22 b(the)i(v)-5 b(arious)22 b(constan)n(ts)h(in)n(to)h (some)f Fp(C)29 b Fq(=)23 b Fp(C)6 b Fq(\(\000\).)36 b(That)24 b(\014nish)g(the)g(pro)r(of.)p 3318 1594 4 57 v 3322 1541 50 4 v 3322 1594 V 3372 1594 4 57 v 639 1758 a(The)k(ab)r(o)n(v)n(e)f(Lemma)h(and)f(argumen)n(ts)g(in)h([12)o (])h(or)e([6)o(])h(pro)n(v)n(es)e(the)j(existence)e(of)h Fp(\026)3280 1770 y Fl(W)3356 1758 y Fq(.)515 1858 y(W)-7 b(e)28 b(use)f(somewhere)g(that)g Fm(D)1465 1870 y Fl(\013;W)1604 1858 y Fq(\(\012\))d Fm(\020)e Fq(1.)37 b(In)27 b(fact,)1380 1965 y Fj(Z)1426 2154 y Fo(\003)1490 2078 y Fp(h)1538 2090 y Fl(n)1583 2078 y Fq(\()p Fp(x)p Fq(\))p Fp(d)p Fm(D)1801 2090 y Fl(\013;W)1941 2078 y Fq(\()p Fp(x)p Fq(\))d(=)e Fm(D)2227 2090 y Fl(\013;W)2366 2078 y Fq(\(\012\))p Fp(;)515 2308 y Fq(hence,)j(b)n(y)f(Lemma)g(4.3)g Fp(e)1341 2278 y Fk(\000)p Fl(\016)1429 2308 y Fm(D)1493 2320 y Fl(\013;W)1631 2308 y Fq(\(\012\))g Fm(\024)f(j)p Fp(h)1938 2320 y Fl(n)1983 2308 y Fq(\()p Fp(x)p Fq(\))p Fm(j)h(\024)f Fp(e)2268 2278 y Fl(\016)2304 2308 y Fm(D)2368 2320 y Fl(\013;W)2506 2308 y Fq(\(\012\),)j(for)e(some)g Fp(\016)i(>)c Fq(0)i(only)515 2407 y(dep)r(ending)40 b(on)g(the)g(distortion,)i (expanding)e(co)r(e\016cien)n(t)f(b)r(ounds)i(and)e(the)i(b)r(ounded) 515 2507 y(geometry)25 b(of)h(admissible)g(manifolds,)h(pro)n(ving)d (\(12\).)36 b(Lemma)27 b(4.3)e(and)h(\(12\))g(p)r(ermit)h(to)515 2607 y(pro)n(v)n(e)f(prop)r(ert)n(y)g(\(13\).)639 2706 y Fp(\026)689 2718 y Fl(W)805 2706 y Fq(is)39 b(the)h(unique)g(ergo)r (dic)f Fp(\034)49 b Fq(in)n(v)-5 b(arian)n(t)39 b(measure)g(absolutely) g(con)n(tin)n(uous)f(wrt)515 2806 y Fm(D)579 2818 y Fl(\013;W)717 2806 y Fq(.)f(F)-7 b(urthermore,)27 b(as)g Fp(\034)37 b Fq(is)28 b(top)r(ologically)d(mixing,)j(it)g(holds)1377 3036 y Fp(\026)1427 3048 y Fl(W)1502 3036 y Fq(\()p Fp(B)t Fq(\))c(=)77 b(lim)1745 3085 y Fl(n)p Fk(!)p Fo(+)p Fk(1)2007 2979 y Fm(D)2071 2991 y Fl(\013;W)2209 2979 y Fq(\()p Fp(\034)2286 2949 y Fk(\000)p Fl(n)2384 2979 y Fp(B)t Fq(\))p 2007 3017 477 4 v 2083 3093 a Fm(D)2147 3105 y Fl(\013;W)2285 3093 y Fq(\(\003\))2494 3036 y Fp(:)515 3258 y Fq(The)27 b(asymptotic)g(measure)g Fp(\026)1486 3270 y Fl(W)1589 3258 y Fq(satis\014es)1355 3433 y Fp(e)1394 3398 y Fk(\000)p Fo(2)p Fl(\016)1538 3433 y Fm(\024)c Fp(\026)1676 3445 y Fl(W)1752 3433 y Fq(\()p Fp(})1837 3445 y Fl(n)1882 3433 y Fq(\()p Fp(x)p Fq(\)\))p Fp(J)2071 3445 y Fl(\026)2117 3433 y Fp(\034)2162 3398 y Fl(n)2208 3433 y Fq(\()p Fp(x)p Fq(\))h Fm(\024)e Fp(e)2469 3398 y Fo(2)p Fl(\016)3231 3433 y Fq(\(14\))515 3607 y(for)i(ev)n(ery)f Fp(n)g Fm(\025)g Fq(0,)i Fp(\026)p Fq(-a.e.)35 b Fp(x)23 b Fm(2)h Fq(\003.)35 b(Here)25 b Fp(J)1849 3619 y Fl(\026)1893 3607 y Fp(\034)34 b Fq(is)25 b(the)g(Jacobian)e(of)i Fp(\034)34 b Fq(wrt)25 b(the)g(asymptotic)515 3707 y(measure)20 b Fp(\026)882 3719 y Fl(W)957 3707 y Fq(.)35 b(W)-7 b(e)21 b(get)g(these)g(b)r(ounds)g(using)f(an)h(standard)f(argumen)n(t.)33 b(In)21 b(fact,)i Fp(J)3156 3719 y Fl(\026)3200 3707 y Fp(\034)3245 3677 y Fl(n)3314 3707 y Fq(=)515 3807 y Fp(J)561 3819 y Fl(\013)608 3807 y Fp(\034)653 3777 y Fl(n)717 3807 y Fm(\001)c Fp(h)f Fm(\016)g Fp(\034)930 3777 y Fl(n)990 3807 y Fp(=)c(h)p Fq(,)27 b(hence:)1008 3974 y Fp(J)1054 3986 y Fl(\026)1098 3974 y Fp(\034)1143 3944 y Fl(n)1189 3974 y Fq(\()p Fp(x)p Fq(\))p 1008 4011 294 4 v 1010 4087 a Fp(J)1056 4099 y Fl(\026)1100 4087 y Fp(\034)1145 4063 y Fl(n)1191 4087 y Fq(\()p Fp(y)s Fq(\))1334 4030 y Fm(2)d Fq([)1446 3974 y(inf)1547 3986 y Fl(x)p Fk(2)p Fo(\003)1693 3974 y Fp(h)p Fq(\()p Fp(x)p Fq(\)])p 1446 4011 431 4 v 1446 4087 a(sup)1571 4108 y Fl(x)p Fk(2)p Fo(\003)1717 4087 y Fp(h)p Fq(\()p Fp(x)p Fq(\))1886 4030 y Fp(;)1933 3973 y Fq(sup)2058 3994 y Fl(x)p Fk(2)p Fo(\003)2204 3973 y Fp(h)p Fq(\()p Fp(x)p Fq(\)])p 1933 4011 454 4 v 1957 4087 a(inf)2058 4099 y Fl(x)p Fk(2)p Fo(\003)2204 4087 y Fp(h)p Fq(\()p Fp(x)p Fq(\))2397 4030 y(])f Fm(2)g Fq([)p Fp(e)2583 3996 y Fk(\000)p Fo(2)p Fl(\016)2705 4030 y Fp(;)14 b(e)2781 3996 y Fo(2)p Fl(\016)2850 4030 y Fq(])p Fp(;)515 4258 y Fq(for)29 b(ev)n(ery)f Fp(x)14 b(;)28 b(y)g Fm(2)f Fp(W)45 b Fm(\\)34 b Fp(P)41 b Fq(and)29 b Fp(P)38 b Fm(2)27 b Fp(})1823 4270 y Fl(n)1868 4258 y Fq(.)42 b(As)29 b Fp(\034)2102 4228 y Fl(n)2148 4258 y Fq(\()p Fp(P)12 b Fq(\))27 b(=)e(\012)34 b Fm(\\)g Fp(W)12 b Fq(,)29 b(then)h Fp(\026)2960 4270 y Fl(W)3036 4258 y Fq(\()p Fp(\034)3113 4228 y Fl(n)3159 4258 y Fq(\()p Fp(P)12 b Fq(\))26 b(=)515 4358 y(1,)44 b(for)d(ev)n(ery)f Fp(P)58 b Fm(2)46 b Fp(})p Fq(\()p Fp(n;)14 b(W)e Fq(\))41 b(the)h(Mark)n(o)n (v)d(partition)i(of)g(n-cylinders)g(of)g(\003.)78 b(Since)515 4457 y Fm(D)579 4469 y Fl(\013;W)717 4457 y Fq(\(\012\))24 b Fm(\020)e Fq(1)28 b(w)n(e)f(get)g(a)g(constan)n(t)g Fp(C)j Fq(=)22 b Fp(C)6 b Fq(\(\001)p Fp(;)14 b(\025;)g Fq(\000\))24 b Fp(>)f Fq(1)k(suc)n(h)g(that,)h(for)g(ev)n(ery)e Fp(n)d Fm(\025)f Fq(0)987 4632 y Fp(C)1052 4598 y Fk(\000)p Fo(1)1142 4632 y Fm(D)1206 4644 y Fl(\013;W)1344 4632 y Fq(\(\012\))i Fm(\024)e(D)1643 4644 y Fl(\013;W)1782 4632 y Fq(\()p Fp(})1867 4644 y Fl(n)1912 4632 y Fq(\()p Fp(x)p Fq(\)\))p Fp(J)2101 4644 y Fl(\013)2150 4632 y Fp(\034)2195 4598 y Fl(n)2241 4632 y Fq(\()p Fp(x)p Fq(\))i Fm(\024)e Fp(C)f Fm(D)2607 4644 y Fl(\013;W)2745 4632 y Fq(\(\012\))p Fp(;)339 b Fq(\(15\))515 4807 y(for)28 b(ev)n(ery)f(admissible)i(unstable)f(manifold)h Fp(W)37 b Fm(2)25 b Fq(\000)2192 4777 y Fl(u)2264 4807 y Fq(and)k(for)f(ev)n (ery)f(stable)i(n-cylinder)515 4907 y(in)21 b Fp(})658 4919 y Fl(n)703 4907 y Fq(.)34 b(The)21 b(follo)n(wing)e(sho)n(ws)h (that)g Fp(\026)1721 4919 y Fl(W)1818 4907 y Fq(is)g(equiv)-5 b(alen)n(t)20 b(to)h Fm(D)2440 4919 y Fl(\013;W)2599 4907 y Fq(b)r(ounded)g(b)n(y)f(constan)n(ts)515 5006 y(whic)n(h)27 b(do)h(not)f(dep)r(end)h(on)g Fp(W)35 b Fm(2)23 b(F)1678 4976 y Fl(u)1721 5006 y Fq(.)1905 5255 y(15)p eop %%Page: 16 16 16 15 bop 515 523 a Fh(Lemma)29 b(4.4.)40 b Fn(The)30 b(asymptotic)g(me)l(asur)l(e)f Fp(\026)2000 535 y Fl(W)2104 523 y Fn(is)g(e)l(quivalent)h(to)e(the)h(dynamic)l(al)i(me)l(a-)515 623 y(sur)l(e)39 b Fm(D)766 635 y Fl(\013;W)905 623 y Fn(.)69 b(F)-6 b(urther,)42 b(ther)l(e)e(is)g(a)h(c)l(onstant)e Fp(C)48 b Fq(=)41 b Fp(C)6 b Fq(\(\001)p Fp(;)14 b(\025;)g Fq(\000\))42 b Fp(>)f Fq(1)f Fn(such)g(that)g(the)515 722 y(r)l(atios)30 b Fp(\026)797 734 y Fl(W)873 722 y Fq(\()p Fp(B)t Fq(\))14 b Fp(=)g Fm(D)1138 734 y Fl(\013;W)1276 722 y Fq(\()p Fp(B)t Fq(\))31 b Fn(ar)l(e)g(b)l(ounde)l(d)f(in)h Fm(D)2063 734 y Fl(\013;W)2201 722 y Fq(\(\012\)[)p Fp(C)2413 692 y Fk(\000)p Fo(1)2503 722 y Fp(;)14 b(C)6 b Fq(])30 b Fn(for)h(every)g(Bor)l(el)h(sub-)515 822 y(set)d Fp(B)e Fm(\022)c Fq(\003)29 b Fn(of)i(p)l(ositive)g(me)l(asur)l(e.)515 969 y(Pr)l(o)l(of.)43 b Fq(Observ)n(e)23 b(that)i Fp(J)1314 981 y Fl(\026)1358 969 y Fp(\034)1403 939 y Fl(n)1449 969 y Fq(\()p Fp(x)p Fq(\))14 b Fp(=)g(J)1676 981 y Fl(\013)1724 969 y Fp(\034)1769 939 y Fl(n)1815 969 y Fq(\()p Fp(x)p Fq(\))24 b Fm(2)f Fq([)p Fp(e)2090 939 y Fk(\000)p Fo(2)p Fl(\016)2211 969 y Fp(;)14 b(e)2287 939 y Fo(2)p Fl(\016)2356 969 y Fq(].)36 b(Therefore,)24 b(com)n(bining)g(\(15\))515 1068 y(and)j(\(14\))g(yields)892 1183 y Fj(\022)1040 1244 y Fp(e)1079 1214 y Fo(2)p Fl(\016)1148 1244 y Fp(C)p 963 1281 327 4 v 963 1357 a Fm(D)1027 1369 y Fl(\013;W)1165 1357 y Fq(\(\012\))1300 1183 y Fj(\023)1361 1194 y Fk(\000)p Fo(1)1473 1300 y Fm(\024)c Fp(\026)1611 1312 y Fl(W)1686 1300 y Fq(\()p Fp(})1771 1312 y Fl(n)1817 1300 y Fq(\()p Fp(x)p Fq(\)\))14 b Fp(=)g Fm(D)2094 1312 y Fl(\013;W)2233 1300 y Fq(\()p Fp(})2318 1312 y Fl(n)2363 1300 y Fq(\()p Fp(x)p Fq(\)\))24 b Fm(\024)2705 1244 y Fp(e)2744 1214 y Fo(2)p Fl(\016)2813 1244 y Fp(C)p 2628 1281 V 2628 1357 a Fm(D)2692 1369 y Fl(\013;W)2830 1357 y Fq(\(\012\))2965 1300 y Fp(;)243 b Fq(\(16\))515 1510 y(So,)27 b(for)g(ev)n(ery)f(Borel) h(subset)h Fp(\026)1525 1522 y Fl(W)1600 1510 y Fq(\()p Fp(B)t Fq(\))c Fm(\020)f(D)1907 1522 y Fl(\013;W)2045 1510 y Fq(\()p Fp(B)t Fq(\),)28 b(since)g Fp(})p Fq(\()p Fp(W)12 b Fq(\))28 b(is)f(a)h(generator.)p 3318 1510 4 57 v 3322 1457 50 4 v 3322 1510 V 3372 1510 4 57 v 515 1671 a Fh(Lemma)h(4.5.)40 b Fn(L)l(et)32 b Fp(\013)27 b Fq(=)f(dim)1499 1683 y Fl(u)1542 1671 y Fq(\(\012\))p Fn(.)46 b(Then)32 b(ther)l(e)g(is)g(a)g(c)l(onstant)g Fp(C)h Fq(=)26 b Fp(C)6 b Fq(\(\001)p Fp(;)14 b(\025;)g Fq(\000\))28 b Fp(>)e Fq(1)515 1771 y Fn(such)k(that)f(for)i(every)f Fp(n)23 b(>)g Fq(0)29 b Fn(and)h Fp(P)35 b Fm(2)24 b Fp(})1838 1783 y Fl(n)1913 1771 y Fn(the)29 b(fol)t(lowing)k(holds)1116 1973 y Fp(C)1181 1938 y Fk(\000)p Fo(1)1271 1973 y Fm(D)1335 1985 y Fl(\013;W)1473 1973 y Fq(\(\012\))24 b Fm(\024)1718 1917 y(D)1782 1929 y Fl(\013;W)1921 1917 y Fq(\()p Fp(P)44 b Fm(\\)33 b Fq(\003\))p 1718 1954 510 4 v 1780 2030 a Fn(V)-6 b(ol)1913 2042 y Fl(W)1989 2030 y Fq(\()p Fp(P)12 b Fq(\))2118 2006 y Fl(\013)2261 1973 y Fm(\024)22 b Fp(C)e Fm(D)2491 1985 y Fl(\013;W)2630 1973 y Fq(\(\012\))p Fp(:)515 2175 y Fn(Pr)l(o)l(of.)1007 2321 y Fm(D)1071 2333 y Fl(\013;W)1210 2321 y Fq(\(\012\))p 823 2358 697 4 v 823 2434 a(sup)948 2454 y Fl(\015)987 2438 y Fi(u)1025 2454 y Fk(2)p Fo(\000)1111 2438 y Fi(u)1182 2434 y Fq(V)-7 b(ol)14 b(\()p Fp(\015)1396 2410 y Fl(u)1439 2434 y Fq(\))1471 2410 y Fl(\013)1552 2377 y Fm(\024)1650 2321 y(D)1714 2333 y Fl(\013;W)1852 2321 y Fq(\()p Fp(T)1945 2291 y Fl(n)1990 2321 y Fq(\()p Fp(P)44 b Fm(\\)33 b Fq(\003\)\))p 1650 2358 680 4 v 1711 2434 a(V)-7 b(ol)1845 2446 y Fl(W)1921 2434 y Fq(\()p Fp(T)2014 2410 y Fl(n)2058 2434 y Fq(\()p Fp(P)12 b Fq(\)\))2219 2410 y Fl(\013)2362 2377 y Fm(\024)22 b Fp(C)2539 2321 y Fm(D)2603 2333 y Fl(\013;W)2741 2321 y Fq(\()p Fp(P)44 b Fm(\\)33 b Fq(\003\))p 2539 2358 510 4 v 2600 2434 a(V)-7 b(ol)2734 2446 y Fl(W)2810 2434 y Fq(\()p Fp(P)12 b Fq(\))2939 2410 y Fl(\013)3058 2377 y Fp(:)515 2591 y Fq(Lik)n(ewise)26 b(w)n(e)i(pro)n(v)n(e)1240 2736 y Fm(D)1304 2748 y Fl(\013;W)1442 2736 y Fq(\()p Fp(P)44 b Fm(\\)33 b Fq(\003\))p 1240 2773 V 1301 2849 a(V)-7 b(ol)1435 2861 y Fl(W)1511 2849 y Fq(\()p Fp(P)12 b Fq(\))1640 2825 y Fl(\013)1782 2792 y Fm(\024)23 b Fp(C)2131 2736 y Fm(D)2195 2748 y Fl(\013;W)2334 2736 y Fq(\(\012\))p 1959 2773 673 4 v 1959 2849 a(inf)2060 2861 y Fl(\015)2099 2845 y Fi(u)2138 2861 y Fk(2)p Fo(\000)2224 2845 y Fi(u)2294 2849 y Fq(V)-7 b(ol)14 b(\()p Fp(\015)2508 2825 y Fl(u)2551 2849 y Fq(\))2583 2825 y Fl(\013)2641 2792 y Fp(;)515 2998 y Fq(where)39 b Fp(C)45 b Fq(only)39 b(dep)r(ends)h(on)f(the)h(distortion)f(\001,)k(the)d(expansion)e(co)r (e\016cien)n(t)h Fp(\025)h Fq(and)515 3098 y(the)30 b(b)r(ounded)h (geometry)e(of)h(the)g(admissible)g(manifolds.)45 b(W)-7 b(e)30 b(\014nish)h(the)f(pro)r(of,)g(using)515 3197 y(previous)c(lemma)i(and)f(the)h(b)r(ounded)g(geometry)e(of)i(the)g (admissible)f(manifolds.)p 3318 3197 4 57 v 3322 3145 50 4 v 3322 3197 V 3372 3197 4 57 v 515 3458 a(Lemma)41 b(4.4)f(and)h(Lemma)f(4.5)h(sho)n(w)f(that)h(the)g(ratios)f Fm(D)2481 3470 y Fl(\013;W)2620 3458 y Fq(\()p Fp(P)12 b Fq(\))i Fp(=)g Fq(V)-7 b(ol)2952 3470 y Fl(W)3027 3458 y Fq(\()p Fp(P)12 b Fq(\))3156 3428 y Fl(\013)3245 3458 y Fq(and)515 3558 y Fp(\026)565 3570 y Fl(W)640 3558 y Fq(\()p Fp(P)g Fq(\))i Fp(=)g Fm(D)903 3570 y Fl(\013;W)1041 3558 y Fq(\()p Fp(P)e Fq(\))33 b(are)f(b)r(oth)h(comparable)e(with)j Fm(D)2255 3570 y Fl(\013;W)2393 3558 y Fq(\(\012\),)h(b)r(ounded)e(b)n (y)f(constan)n(ts)515 3657 y Fp(C)40 b Fq(=)33 b Fp(C)6 b Fq(\(\001)p Fp(;)14 b(\025;)g(\015)5 b(;)14 b Fg(R)p Fq(\))34 b Fp(>)f Fq(1,)i(for)f(ev)n(ery)e(lo)r(cal)i(unstable)g (manifold)g Fp(W)45 b Fq(=)33 b Fp(W)2965 3627 y Fl(u)3009 3657 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\))34 b(and)515 3757 y(ev)n(ery)24 b(cylinder)i Fp(P)34 b Fm(2)24 b Fp(})h Fq(of)h(the)g(generating)f(net.)36 b(Therefore,)25 b Fp(\026)2552 3769 y Fl(W)2628 3757 y Fq(\()p Fp(P)12 b Fq(\))23 b Fm(\020)g Fq(V)-7 b(ol)3002 3769 y Fl(W)3077 3757 y Fq(\()p Fp(P)12 b Fq(\))3206 3727 y Fl(\013)3280 3757 y Fq(for)515 3857 y(ev)n(ery)28 b(cylinder)h Fp(P)37 b Fm(2)27 b Fp(})p Fq(,)i(b)r(ounded)h(b)n(y)f(constan)n(ts)f(only)h (dep)r(ending)h(on)f(the)h(distortion,)515 3956 y(the)g(expansion)f(co) r(e\016cien)n(t)h(and)g(the)g(b)r(ounded)h(geometry)d(of)i(\012.)44 b(That)30 b(concludes)g(the)515 4056 y(pro)r(of)d(of)g(Lemma)h(4.1.)515 4326 y Fr(5)134 b(Pro)t(of)45 b(of)h(Theorem)f(A)515 4508 y Fq(In)30 b(view)f(of)h(argumen)n(ts)e(in)i(preceding)f(Section,) i(to)e(pro)n(v)n(e)f(Theorem)h(A,)i(w)n(e)e(only)g(ha)n(v)n(e)515 4608 y(to)i(sho)n(w)g(that)h(the)g(unstable)g(Can)n(tor)e(sets)i(of)f (a)h Fp(C)2185 4578 y Fo(2)2254 4608 y Fq(horsesho)r(e)e(with)i (in\014nitely)h(man)n(y)515 4707 y(branc)n(hes)c(ha)n(v)n(e)h (non-trivial)g(dynamical)g(measure.)46 b(F)-7 b(or)30 b(this)h(w)n(e)g(will)g(sho)n(w)f(that)h(\012)g(is)515 4807 y(the)d(limit)g(of)g(a)f(sequence)g(of)g(horsesho)r(es)f(with)i (\014nitely)g(man)n(y)f(branc)n(hes)g(\012)2967 4819 y Fl(n)3040 4807 y Fq(and)g(sho)n(w)515 4907 y(that)f(the)g(asymptotic) f(measure)g Fp(\026)1631 4919 y Fo(\012)1709 4907 y Fq(is)g(a)h(limit)h (of)e(certain)h(Gibbs)g(measures)e(supp)r(orted)515 5006 y(on)j(\012)690 5018 y Fl(n)735 5006 y Fq(.)1905 5255 y(16)p eop %%Page: 17 17 17 16 bop 515 523 a Fh(Lemma)29 b(5.1.)40 b Fn(L)l(et)c Fq(\012)e Fm(\022)g Ff(R)1440 493 y Fl(m)1545 523 y Fn(b)l(e)i(a)g Fp(C)1796 493 y Fo(2)1869 523 y Fn(hyp)l(erb)l(olic)i(horsesho)l(e)g (with)f(in\014nitely)f(many)515 623 y(br)l(anches)e(and)f(b)l(ounde)l (d)h(distortion.)50 b(Then)34 b(ther)l(e)g(exists)e(a)i(se)l(quenc)l(e) f Fq(\012)2904 635 y Fl(n)2978 623 y Fm(\022)c Fq(\012)k Fn(of)i(hy-)515 722 y(p)l(erb)l(olic)h(horsesho)l(es)g(with)f (\014nitely)g(many)g(br)l(anches)g(and)g(er)l(go)l(dic)h(me)l(asur)l (es)e Fp(\026)3140 734 y Fl(n)3220 722 y Fn(such)515 822 y(that)613 983 y(1.)42 b Fq(\012)23 b(=)p 893 911 319 4 v 893 921 a Fj(S)962 1008 y Fl(n>)p Fo(0)1106 983 y Fq(\012)1166 995 y Fl(n)1211 983 y Fn(;)613 1152 y(2.)42 b Fp(\026)772 1164 y Fl(n;W)909 1152 y Fq(\()p Fp(P)12 b Fq(\))27 b Fm(\020)g Fn(V)-6 b(ol)1290 1164 y Fl(W)1365 1152 y Fq(\()p Fp(P)12 b Fq(\))1494 1122 y Fo(dim)1605 1131 y Fi(u)1644 1122 y Fo(\(\012)1717 1130 y Fi(n)1758 1122 y Fo(\))1820 1152 y Fn(b)l(ounde)l(d)32 b(by)h(some)f(c)l(onstant) g Fp(C)h(>)27 b Fq(1)k Fn(for)i(every)722 1252 y(lo)l(c)l(al)40 b(unstable)f(manifold)i Fp(W)52 b Fq(=)39 b Fp(W)1927 1222 y Fl(u)1970 1252 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\))40 b Fn(and)f(for)h(every)g Fp(P)51 b Fn(in)39 b(a)g(family)i(of)722 1352 y(stable)31 b(cylinders)h(gener)l(ating)e(the)g(stable)h (lamination)h(of)f Fq(\012)2655 1364 y Fl(n)2730 1352 y Fn(wher)l(e)g Fp(\026)3015 1364 y Fl(n;W)3181 1352 y Fn(is)g(the)722 1451 y(pr)l(oje)l(ction)g(of)g Fp(\026)1255 1463 y Fl(n)1330 1451 y Fn(onto)f Fp(W)41 b Fn(along)31 b(the)f(stable)g(lamination)h Fm(F)2714 1421 y Fl(s)2750 1451 y Fn(.)639 1613 y Fq(Supp)r(ose)g(pro)n(v)n(ed)d(Lemma)j(5.1)e (and)h(let)h Fp(\026)2013 1583 y Fk(\003)2081 1613 y Fq(some)f(limit)h(measure)f(of)g(the)h(sequence)515 1712 y Fp(\026)565 1724 y Fl(n)610 1712 y Fq(,)42 b(then:)60 b Fp(\026)969 1682 y Fk(\003)969 1735 y Fl(W)1045 1712 y Fq(\()p Fp(P)12 b Fq(\))43 b Fm(\025)e Fp(C)1388 1682 y Fk(\000)p Fo(1)1478 1712 y Fq(V)-7 b(ol)1611 1724 y Fl(W)1687 1712 y Fq(\()p Fp(P)12 b Fq(\))1816 1682 y Fl(\013)1903 1712 y Fq(and)39 b Fp(\026)2126 1682 y Fk(\003)2126 1735 y Fl(W)2201 1712 y Fq(\()p Fp(int)14 b(P)e Fq(\))42 b Fm(\024)g Fp(C)20 b Fq(V)-7 b(ol)2815 1724 y Fl(W)2890 1712 y Fq(\()p Fp(P)12 b Fq(\))3019 1682 y Fl(\013)3067 1712 y Fq(,)42 b(for)d(ev-)515 1812 y(ery)d Fp(P)48 b Fq(in)37 b(the)g(generating)e(net)i Fp(})f Fq(.)64 b(This)37 b(implies)g(that)f(0)i Fp(<)g Fm(D)2692 1824 y Fl(\013;W)2830 1812 y Fq(\(\012\))h Fp(<)e Fq(+)p Fm(1)p Fq(,)i(so)515 1912 y(dim)653 1924 y Fl(u)697 1912 y Fq(\(\012\))23 b(=)g(lim)1047 1924 y Fl(n)p Fk(!)p Fo(+)p Fk(1)1289 1912 y Fq(dim)1428 1924 y Fl(u)1471 1912 y Fq(\(\012)1563 1924 y Fl(n)1609 1912 y Fq(\))k(and)h(there)f(is)g(an)g(asymptotic)g (measure)g Fp(\026)3041 1924 y Fl(W)3144 1912 y Fq(for)g(ev-)515 2011 y(ery)37 b(lo)r(cal)g(unstable)g(manifold)h Fp(W)51 b Fq(=)40 b Fp(W)1892 1981 y Fl(u)1935 2011 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\).)67 b(Actually)38 b Fp(\026)2624 1981 y Fk(\003)2624 2034 y Fl(W)2739 2011 y Fq(=)h Fp(\026)2893 2023 y Fl(W)3007 2011 y Fq(since)e(they)515 2111 y(are)i Fp(\034)50 b Fq(in)n(v)-5 b(arian)n(t,)43 b(ergo)r(dic)c(and)i Fp(\026)1666 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Fm(g)3167 2422 y Fl(W)9 b Fk(2F)3337 2406 y Fi(u)515 2509 y Fq(is)31 b(a)g(family)g(of)g(conditional)g(measures)e(of)j Fp(\026)1971 2521 y Fo(\012)2053 2509 y Fq(resp)r(ect)f(to)g(the)h(measurable)e (decomp)r(o-)515 2609 y(sition)19 b(de\014ned)i(b)n(y)e Fp(\030)t Fq(\()p Fp(x)p Fq(\))24 b(=)f(\012)17 b Fm(\\)g Fp(W)1621 2579 y Fl(u)1664 2609 y Fq(\()p Fp(x;)d Fg(R)p Fq(\).)35 b(That)20 b(completes)f(the)i(pro)r(of)e(of)h(Theorem)f(A.) 515 2808 y Fh(Pro)s(of)42 b(of)g(Lemma)e(5.1)p Fq(:)54 b(let)37 b(us)g(tak)n(e)f(\012)1951 2820 y Fl(n)2033 2808 y Fq(the)h(horsesho)r(e)e(generated)h(b)n(y)g(the)h(\014rst)515 2908 y Fp(n)31 b Fq(branc)n(hes)e Fp(T)992 2920 y Fl(i)1048 2908 y Fq(:)g Fp(S)1151 2920 y Fl(i)1208 2908 y Fm(\000)-15 b(!)29 b Fp(U)1427 2920 y Fl(i)1486 2908 y Fq(,)j Fp(i)c Fq(=)g(1)p Fp(;)14 b Fm(\001)g(\001)g(\001)27 b Fp(;)14 b(n)p Fq(.)47 b(Clearly)-7 b(,)31 b(\012)g(is)g(the)h(top)r(ological)d (limit)j(of)515 3007 y(this)c(sequence.)639 3107 y(Let)19 b(\012)f(b)r(e)h(a)f 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Fp(W)35 b Fq(=)22 b Fp(W)2124 3806 y Fl(u)2168 3836 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\);)616 3993 y(3.)41 b(let)31 b Fp(\026)895 4005 y Fl(\013)972 3993 y Fq(b)r(e)g(equilibrium)f(state) g(for)f(the)i(p)r(oten)n(tial)f Fm(\000)p Fp(\013)14 b Fq(log)g Fp(J)2672 3963 y Fl(u)2715 3993 y Fp(T)e Fq(.)44 b(Then,)31 b(the)f(pro-)722 4093 y(jection)23 b(of)g Fp(\026)1127 4105 y Fl(\013)1197 4093 y Fq(on)n(to)f Fp(W)35 b Fq(=)23 b Fp(W)1670 4063 y Fl(u)1713 4093 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\))24 b(along)e(the)h(stable)f(lea)n(v)n (es)g(is)g(the)i(asymptotic)722 4193 y(measure)j Fp(\026)1096 4205 y Fl(W)1171 4193 y Fq(.)515 4342 y(T)-7 b(o)21 b(pro)n(v)n(e)e (this)j(w)n(e)f(recall)f(that)i(the)f(unstable)h(\014b)r(er)f(bundle)h (of)f(a)g(horsesho)r(e)f(with)i(\014nitely)515 4441 y(man)n(y)i(branc)n (hes)f(satis\014es)g(a)h(H\177)-42 b(older)24 b(condition.)36 b(In)24 b(particular)f(the)i(unstable)g(Jacobian)515 4541 y Fp(J)569 4511 y Fl(u)612 4541 y Fp(T)42 b Fq(is)30 b(H\177)-42 b(older)30 b(con)n(tin)n(uous.)45 b(So)30 b(w)n(e)g(apply)h(Ruelle's)f(theorem)g(to)h(the)g(p)r(oten)n(tial)f Fp(\036)e Fq(=)515 4640 y Fm(\000)p Fp(\013)14 b Fq(log)g Fp(J)822 4610 y Fl(u)865 4640 y Fp(T)37 b Fq(to)26 b(get)f(a)h(Gibbs)g (measure)f Fp(\026)1866 4652 y Fl(\013)1913 4640 y Fq(.)36 b(That)26 b(is,)g(there)g(is)g(a)f(constan)n(t)g Fp(C)k(>)23 b Fq(1)i(suc)n(h)515 4740 y(that)1352 4929 y Fp(C)1417 4894 y Fk(\000)p Fo(1)1529 4929 y Fm(\024)1829 4872 y Fp(\026)p Fq(\()p Fp(})1964 4884 y Fl(n)2009 4872 y Fq(\()p Fp(x)p Fq(\)\))p 1627 4909 730 4 v 1627 4985 a(exp)o(\()p Fm(\000)p Fp(P)12 b(n)18 b Fq(+)g Fp(S)2117 4997 y Fl(n)2163 4985 y Fp(\036)p Fq(\()p Fp(x)p Fq(\)\))2389 4929 y Fm(\024)23 b Fp(C)695 b Fq(\(17\))1905 5255 y(17)p eop %%Page: 18 18 18 17 bop 515 523 a Fq(for)36 b(ev)n(ery)f Fp(x)k Fm(2)f Fq(\012,)h Fp(n)f Fm(\025)g Fq(0,)g(where)e Fp(P)50 b Fq(=)38 b Fp(P)12 b Fq(\()p Fp(T)7 b(;)14 b(\036)p Fq(\))36 b(is)h(the)g(top)r(ological)e(pressure)g(and)515 631 y Fp(S)566 643 y Fl(n)611 631 y Fp(\036)p Fq(\()p Fp(x)p Fq(\))24 b(=)883 569 y Fj(P)971 590 y Fl(n)p Fk(\000)p Fo(1)971 656 y Fl(k)q Fo(=0)1115 631 y Fp(\036)p Fq(\()p Fp(T)1257 601 y Fl(k)1297 631 y Fq(\()p Fp(x)p Fq(\)\).)36 b Fp(\026)1549 643 y Fl(\036)1615 631 y Fq(is)21 b(an)f(equilibrium)i (state)f(for)f Fp(\036)i Fq(in)f(the)h(follo)n(wing)e(sense)515 731 y Fp(P)12 b Fq(\()p Fp(T)7 b(;)14 b(\036)p Fq(\))28 b(=)f Fp(h)954 743 y Fl(\026)998 731 y Fq(\()p Fp(T)12 b Fq(\))20 b(+)1228 664 y Fj(R)1311 731 y Fp(\036)p Fq(\()p Fp(x)p Fq(\))p Fp(d\026)p Fq(\()p Fp(x)p Fq(\).)48 b(Cf.)d([2])30 b(for)g(de\014nitions)g(and)g(pro)r(ofs.)45 b(Prop)r(erties)515 831 y(\(1\))31 b(and)f(\(2\))h(are)f(en)n(tirely)g(similar)g(to)g (those)h(pro)n(v)n(ed)e(for)h(h)n(yp)r(erb)r(olic)g(rep)r(ellors)g(in)h ([11)o(])515 930 y(and)k(w)n(e)g(omit)g(the)h(pro)r(of.)59 b(No)n(w,)37 b(let)e Fp(\026)1837 942 y Fl(\013)1920 930 y Fq(b)r(e)g(the)h(Gibbs)g(measure)e(for)h(the)g(p)r(oten)n(tial) 515 1030 y Fp(\036)k Fq(=)g Fm(\000)p Fp(\013)28 b Fq(log)g Fp(J)1042 1000 y Fl(u)1085 1030 y Fp(T)12 b Fq(.)66 b(As)37 b Fp(\013)j Fq(=)e(dim)1702 1042 y Fl(u)1745 1030 y Fq(\(\012\))g(is)f (the)h(solution)f(the)h(Bo)n(w)n(en)e(equation)g(one)515 1129 y(sho)n(ws)f(that)h Fp(\026)1000 1141 y Fl(\013)1047 1129 y Fq(\()p Fp(})1132 1141 y Fl(n)1177 1129 y Fq(\()p Fp(x)p Fq(\)\))j Fm(\020)d Fq(V)-7 b(ol)14 b(\()p Fp(})1679 1141 y Fl(n)1724 1129 y Fq(\()p Fp(x)p Fq(\)\))1867 1099 y Fl(\013)1916 1129 y Fq(,)38 b(using)e(the)g(Gibbs)g(prop)r(ert)n(y)f (and)h(v)n(olume)515 1229 y(lemma)31 b(\(15\).)46 b(Therefore,)31 b(the)g(pro)5 b(jection)30 b(of)h Fp(\026)2099 1241 y Fl(\013)2178 1229 y Fq(on)n(to)f Fp(W)40 b Fq(=)29 b Fp(W)2670 1199 y Fl(u)2713 1229 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\))32 b(and)e Fp(\026)3161 1241 y Fl(W)3268 1229 y Fq(are)515 1329 y(equiv)-5 b(alen)n(t.)36 b(Being)27 b(ergo)r(dic)g(and)g Fp(\034)38 b Fq(in)n(v)-5 b(arian)n(t)26 b(w)n(e)h(conclude)h(that)g(are)e(equal.)639 1428 y(No)n(w,)36 b(let)f Fp(\026)1037 1440 y Fl(n)1117 1428 y Fq(b)r(e)g(the)g(Gibbs)g (measure)e(of)i Fp(\036)g Fq(=)f Fm(\000)14 b Fq(dim)2467 1440 y Fl(u)2510 1428 y Fq(\(\012)2602 1440 y Fl(n)2648 1428 y Fq(\))g(log)g Fp(J)2869 1398 y Fl(u)2913 1428 y Fp(T)e Fq(.)57 b(The)34 b(pro-)515 1528 y(jection)f(of)f Fp(\026)939 1540 y Fl(n)1017 1528 y Fq(on)n(to)g(an)n(y)g(lo)r(cal)g (unstable)h(manifold)f Fp(W)44 b Fq(=)31 b Fp(W)2566 1498 y Fl(u)2609 1528 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\))33 b(is)g(equal)f(to)h(the)515 1628 y(asymptotic)k(measure)g Fp(\026)1336 1640 y Fl(W)1449 1628 y Fq(asso)r(ciated)f(to)i(\012)2026 1640 y Fl(n)2071 1628 y Fq(.)67 b(So)38 b Fp(\026)2337 1640 y Fl(n;W)2473 1628 y Fq(\()p Fp(P)12 b Fq(\))40 b Fm(\020)g Fq(V)-7 b(ol)2881 1640 y Fl(W)2956 1628 y Fq(\()p Fp(P)12 b Fq(\))3085 1597 y Fo(dim)3196 1606 y Fi(u)3235 1597 y Fo(\(\012)3308 1605 y Fi(n)3349 1597 y Fo(\))515 1727 y Fq(b)r(ounded)36 b(b)n(y)g(constan)n(ts)f Fp(C)43 b Fq(=)37 b Fp(C)6 b Fq(\(\001)p Fp(;)14 b(\025;)g Fq(\000\))38 b Fp(>)e Fq(1)g(only)f(dep)r(ending)i(on)e(the)i (distortion)515 1827 y(prop)r(erties)30 b(and)i(the)g(b)r(ounded)g (geometry)f(of)g(the)h(admissible)g(manifolds)f(of)h(\012,)g(so)f(w)n (e)515 1926 y(are)26 b(done.)515 2201 y Fr(6)134 b(Computing)38 b(the)g(Jacobian)h(of)e(the)h(stable)h(lamina-)716 2350 y(tion)515 2532 y Fq(In)c(this)h(Section)g(\012)f(alw)n(a)n(ys)f (denote)h(a)g Fp(C)1881 2502 y Fo(2)1955 2532 y Fq(generalized)f (horsesho)r(e)g(constructed)h(in)g(a)515 2632 y(cub)r(e)h Fg(R)p Fq(.)62 b(\000)910 2602 y Fl(u)989 2632 y Fq(\(resp.)g(\000)1307 2602 y Fl(s)1342 2632 y Fq(\))37 b(are)e(the)h(admissible)g(unstable)g (\(resp.)61 b(stable\))36 b(manifolds.)515 2732 y Fm(F)583 2701 y Fl(s)641 2732 y Fq(=)23 b Fm(f)p Fp(W)861 2701 y Fl(s)896 2732 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\))23 b(:)g Fp(x)h Fm(2)f Fq(\012)p Fm(g)h Fq(is)f(the)h(stable)g(lamination) f(and)h Fp(W)2556 2701 y Fl(s)2591 2732 y Fq(\(\012\))g(=)2826 2669 y Fj(S)2895 2756 y Fl(x)p Fk(2)p Fo(\012)3043 2732 y Fp(W)3133 2701 y Fl(u)3176 2732 y Fq(\()p Fp(x;)14 b Fg(R)p Fq(\))515 2831 y(is)29 b(the)i Fn(stable)h(set)g(of)g Fq(\012.)44 b(As)30 b(b)r(efore)g Fm(D)1778 2843 y Fl(\013;\015)1880 2827 y Fi(u)1952 2831 y Fq(is)g(the)g(dynamical)g(unstable)f(measure)g (and)515 2931 y Fp(\013)23 b Fq(=)g(dim)817 2943 y Fl(u)861 2931 y Fq(\(\012\).)37 b Fp(J)1091 2943 y Fl(\013)1166 2931 y Fq(is)28 b(the)g(Jacobian)e(wrt)h(the)h(dynamical)f(measure)g (class)f Fm(D)3016 2943 y Fl(\013;\015)3118 2927 y Fi(u)3161 2931 y Fq(.)515 3097 y Fh(Lemma)j(6.1.)40 b Fn(L)l(et)g Fq(\003)i(=)g Fp(W)1494 3067 y Fl(s)1529 3097 y Fq(\(\012\))e Fm(\\)h Fp(\015)1837 3067 y Fl(u)1920 3097 y Fn(and)g(let)f Fp(h)i Fq(:)g(\003)g Fm(\000)-15 b(!)42 b Ff(R)2703 3067 y Fl(m)2813 3097 y Fn(b)l(e)e(a)h(top)l(olo)l(gic)l(al)515 3196 y(emb)l(e)l(dding.)d(Supp)l(ose)27 b Fm(f)p Fp(g)1341 3208 y Fl(n)1408 3196 y Fq(:)c Fp(W)35 b Fm(\000)-14 b(!)23 b Ff(R)1778 3166 y Fl(m)1847 3196 y Fm(g)j Fn(is)g(a)h(se)l (quenc)l(e)e(of)i Fp(C)2561 3166 y Fo(2)2625 3196 y Fn(emb)l(e)l (ddings)g(such)f(that)1285 3379 y Fp(g)1325 3391 y Fl(n)1393 3379 y Fm(j)d Fq(\003)f Fe(\023)h Fp(h)119 b Fn(and)g Fp(J)2088 3391 y Fl(\013)2136 3379 y Fq(\()p Fp(g)2208 3391 y Fl(n)2253 3379 y Fq(\))23 b Fm(j)g Fq(\003)g Fe(\023)g Fp(J)o(;)515 3562 y Fn(uniformly.)39 b(Then,)31 b(h)f(is)h(absolutely)f (c)l(ontinuous)f(r)l(esp)l(e)l(ct)h(to)f Fm(D)2562 3574 y Fl(\013;\015)2664 3558 y Fi(u)2737 3562 y Fn(and)h Fp(J)2944 3574 y Fl(\013)2992 3562 y Fq(\()p Fp(h)p Fq(\))23 b(=)g Fp(J)8 b Fn(.)515 3728 y(Pr)l(o)l(of.)43 b Fq(It)28 b(is)f(an)h(straigh)n(tforw)n(ard)c(adaptation)j(of)g([9,)h(Lemma)f (3.2].)p 3318 3728 4 57 v 3322 3675 50 4 v 3322 3728 V 3372 3728 4 57 v 639 3894 a(Let)e Fp(\015)833 3864 y Fl(u)876 3894 y Fp(;)p 913 3848 48 4 v 14 w(\015)961 3862 y Fl(u)1027 3894 y Fm(2)f Fq(\000)1158 3864 y Fl(u)1201 3894 y Fq(\()p Fg(R)p Fq(\).)36 b(They)25 b(are)f(cross-sections)f(of)i (the)g(stable)g(lamination)g(so)f(they)515 3993 y(de\014ne)30 b(a)g(holonom)n(y)f(transformation)g Fp(h)e Fq(=)g Fp(H)2015 4005 y Fl(\015)2054 3989 y Fi(u)2093 4005 y Fl(;)p 2113 3972 39 3 v 1 x(\015)2151 3987 y Fi(u)2194 3993 y Fq(.)45 b(No)n(w)30 b(w)n(e)g(compute)g(its)h(Jacobian.)515 4093 y(Fix)c Fp(p)c Fm(2)h Fp(W)897 4063 y Fl(s)932 4093 y Fq(\(\012\))33 b Fm(\\)g Fp(\015)1225 4063 y Fl(u)1295 4093 y Fq(and)28 b Fp(q)e Fq(=)c Fp(h)p Fq(\()p Fp(p)p Fq(\))i Fm(2)f Fp(W)1953 4063 y Fl(s)1989 4093 y Fq(\(\012\))32 b Fm(\\)p 2233 4047 48 4 v 33 w Fp(\015)2281 4061 y Fl(u)2324 4093 y Fq(.)639 4193 y(First)j(w)n(e)f(tak)n(e)f(a)h Fp(C)1305 4163 y Fo(2)1377 4193 y Fq(foliation)g Fm(G)40 b Fq(of)34 b Fg(R)g Fq(b)n(y)g(admissible)g(stable)g(manifolds.)58 b(No)n(w,)515 4292 y(de\014ne)23 b(a)g(sequence)g(of)g(foliations)g Fm(G)1653 4262 y Fl(n)1721 4292 y Fq(as)g(the)h(families)f(of)g(discs)g (obtained)g(from)g(iteration)515 4392 y(b)n(y)k Fp(T)691 4362 y Fk(\000)p Fl(n)787 4392 y Fq(,)h(namely:)632 4575 y Fp(p)23 b Fm(7\000)-14 b(!)23 b(G)908 4540 y Fl(n)953 4575 y Fq(\()p Fp(p)p Fq(\))h(:=)f(connected)k(comp)r(onen)n(t)g(of)h Fp(T)2161 4540 y Fk(\000)p Fl(n)2257 4575 y Fm(G)5 b Fq(\()p Fp(T)2404 4540 y Fl(n)2449 4575 y Fq(\()p Fp(p)p Fq(\)\))33 b Fm(\\)f Fg(R)c Fq(con)n(taining)e(p)p Fp(:)639 4757 y Fq(Note)g(that)g Fm(G)1070 4727 y Fl(n)1141 4757 y Fq(is)g(a)f Fp(C)1355 4727 y Fo(2)1419 4757 y Fq(foliation)g(of)h (the)g(\(disjoin)n(t\))g(reunion)f(of)h(stable)g(geometrical)515 4857 y(cylinder)32 b(of)i(generation)d Fp(n)p Fq(,)1456 4795 y Fj(S)1525 4882 y Fl(P)9 b Fk(2)p Fl(})1663 4890 y Fi(n)1722 4857 y Fp(P)j Fq(.)54 b(By)33 b(construction)f(w)n(e)h(ha)n (v)n(e)f(holonom)n(y)f(maps)515 4970 y Fp(g)555 4982 y Fl(n)623 4970 y Fq(=)22 b Fp(G)775 4939 y Fl(n)775 4992 y(\015)814 4975 y Fi(u)853 4992 y Fl(;)p 873 4958 39 3 v(\015)911 4973 y Fi(u)978 4970 y Fq(:)h Fp(D)1093 4982 y Fl(n)1161 4970 y Fm(\000)-14 b(!)1337 4949 y Fq(\026)1318 4970 y Fp(D)1387 4982 y Fl(n)1432 4970 y Fq(,)21 b(where)f Fp(D)1778 4982 y Fl(n)1846 4970 y Fq(=)j Fp(\015)1982 4939 y Fl(u)2042 4970 y Fm(\\)2114 4907 y Fj(S)2184 4994 y Fl(P)9 b Fk(2)p Fl(})2322 5002 y Fi(n)2381 4970 y Fp(P)32 b Fq(and)2639 4949 y(\026)2619 4970 y Fp(D)2688 4982 y Fl(n)2756 4970 y Fq(=)p 2844 4924 48 4 v 23 w Fp(\015)2892 4938 y Fl(u)2952 4970 y Fm(\\)3025 4907 y Fj(S)3094 4994 y Fl(P)9 b Fk(2)p Fl(})3232 5002 y Fi(n)3291 4970 y Fp(P)j Fq(.)1905 5255 y(18)p eop %%Page: 19 19 19 18 bop 515 523 a Fh(Lemma)29 b(6.2.)1522 623 y Fm(G)1576 588 y Fl(n)1644 623 y Fm(\000)-14 b(!)23 b(F)1869 588 y Fl(s)2023 623 y Fn(uniformly)639 789 y Fq(This)28 b(follo)n(ws)e (from)i(Graph)f(T)-7 b(ransform)26 b(theory)-7 b(.)36 b(Therefore:)639 955 y Fm(\017)41 b Fp(g)762 967 y Fl(n)830 955 y Fm(\000)-14 b(!)23 b Fp(h)k Fq(uniformly)h(and)f Fp(Q)1668 967 y Fl(n)1736 955 y Fq(=)c Fp(g)1864 967 y Fl(n)1909 955 y Fq(\()p Fp(p)p Fq(\))g Fm(\000)-14 b(!)23 b Fp(q)s Fq(;)639 1121 y Fm(\017)41 b Fq(eac)n(h)25 b Fp(g)947 1133 y Fl(n)1017 1121 y Fq(admits)g(a)g(Jacobian)f Fp(J)1745 1133 y Fl(\013)1792 1121 y Fq(\()p Fp(g)1864 1133 y Fl(n)1909 1121 y Fq(\))i(wrt)f Fm(D)2181 1133 y Fl(\013;W)2345 1121 y Fq(since)g(the)g(dynamical)g(measure)722 1220 y(class)i(is)g(conformal.)639 1386 y(By)38 b(a)f(expansion)g (along)f(admissible)h(unstable)h(manifolds)f(or,)j(if)e(y)n(ou)f (prefer,)i(b)n(y)515 1486 y Fp(\025)p Fq(-lemma)29 b(argumen)n(ts)f(w)n (e)g(can)h(\014nd)h(t)n(w)n(o)e(compact)g(discs)h Ff(D)2457 1498 y Fl(n)2534 1486 y Fm(\022)c Fp(D)2693 1498 y Fl(n)2767 1486 y Fq(\(resp.)3021 1465 y(\026)3012 1486 y Ff(D)3066 1498 y Fl(n)3143 1486 y Fm(\022)3252 1465 y Fq(\026)3233 1486 y Fp(D)3302 1498 y Fl(n)3347 1486 y Fq(\))515 1586 y(con)n(taining)h Fp(p)d Fm(2)h Fp(W)1152 1556 y Fl(s)1187 1586 y Fq(\(\012\))33 b Fm(\\)f Fp(\015)1479 1556 y Fl(u)1550 1586 y Fq(\(resp.)37 b Fp(q)26 b Fm(2)d Fp(W)2022 1556 y Fl(s)2058 1586 y Fq(\(\012\))32 b Fm(\\)p 2302 1540 48 4 v 33 w Fp(\015)2350 1554 y Fl(u)2421 1586 y Fq(suc)n(h)27 b(that)725 1768 y Fp(T)786 1734 y Fl(n)830 1768 y Fq(\()p Ff(D)916 1780 y Fl(n)968 1768 y Fq(\))c(=)50 b(connected)28 b(comp)r(onen)n(t)f(of)h Fp(T)2106 1734 y Fl(n)2150 1768 y Fq(\()p Fp(\015)2230 1734 y Fl(u)2273 1768 y Fq(\))g(con)n(taining)f Fp(p)2779 1780 y Fl(n)2847 1768 y Fq(=)22 b Fp(T)2995 1734 y Fl(n)3039 1768 y Fq(\()p Fp(p)p Fq(\))p Fp(;)515 1951 y Fq(resp)r(ectiv)n(ely)j(for)h(the)h(other)e(p)r(oin)n(t)i Fp(q)1702 1963 y Fl(n)1770 1951 y Fq(=)c Fp(T)1919 1921 y Fl(n)1963 1951 y Fq(\()p Fp(Q)2061 1963 y Fl(n)2106 1951 y Fq(\))k(and)f Fp(T)2386 1921 y Fl(n)2430 1951 y Fq(\()p 2462 1905 V Fp(\015)2510 1919 y Fl(u)2554 1951 y Fq(\).)37 b(Indeed,)26 b Ff(D)2991 1963 y Fl(n)3043 1951 y Fq(\()p Fp(p)p Fq(\))g(\(resp.)524 2029 y(\026)515 2051 y Ff(D)569 2063 y Fl(n)620 2051 y Fq(\))i(is)f(the)h(connected)g (comp)r(onen)n(t)f(of)h Fp(D)1882 2063 y Fl(n)1954 2051 y Fq(con)n(taining)f Fp(p)g Fq(\(resp.)37 b Fp(Q)2734 2063 y Fl(n)2779 2051 y Fq(\).)g(F)-7 b(urther,)1360 2233 y(dist)1494 2248 y Fl(C)1546 2231 y Fd(1)1583 2233 y Fq(\()p Fp(T)1676 2199 y Fl(n)1720 2233 y Fq(\()p Ff(D)1806 2245 y Fl(n)1858 2233 y Fq(\))p Fp(;)14 b(T)1988 2199 y Fl(n)2032 2233 y Fq(\()2073 2212 y(\026)2064 2233 y Ff(D)2118 2245 y Fl(n)2169 2233 y Fq(\)\))24 b Fm(\000)-14 b(!)23 b Fq(0)2456 2199 y Fo(+)2511 2233 y Fp(:)697 b Fq(\(18\))515 2416 y(No)n(w,)27 b(factorize)g Fp(g)1102 2428 y Fl(n)1174 2416 y Fq(as)g(follo)n(ws)1402 2599 y Fp(g)1442 2611 y Fl(n)1510 2599 y Fq(=)c Fp(T)1659 2564 y Fk(\000)p Fl(n)1774 2599 y Fm(\016)17 b Fp(G)1898 2564 y Fo(0)1898 2619 y Fl(p)1932 2627 y Fi(n)1974 2619 y Fl(;q)2024 2627 y Fi(n)2087 2599 y Fm(\016)h Fq(\()p Fp(T)2240 2564 y Fl(n)2308 2599 y Fm(j)23 b Ff(D)2408 2611 y Fl(n)2459 2599 y Fq(\))515 2781 y(where)c Fp(G)812 2751 y Fo(0)812 2802 y Fl(p)846 2810 y Fi(n)887 2802 y Fl(;q)937 2810 y Fi(n)1005 2781 y Fq(=)k Fp(G)1158 2751 y Fo(0)1158 2813 y Fl(T)1206 2796 y Fi(n)1247 2813 y Fo(\()p Fc(D)1309 2821 y Fi(n)1355 2813 y Fo(\))12 b Fl(;)f(T)1472 2796 y Fi(n)1512 2813 y Fo(\()1542 2798 y(\026)1538 2813 y Fc(D)1574 2821 y Fi(n)1621 2813 y Fo(\))1651 2781 y Fq(.)34 b(Using)20 b(that)f(the)h(dynamical)f (measure)f(is)h(conformal)515 2893 y(w)n(e)27 b(conclude)515 3076 y Fp(J)561 3088 y Fl(\013)608 3076 y Fq(\()p Fp(g)680 3088 y Fl(n)725 3076 y Fq(\)\()p Fp(y)s Fq(\))d(=)e(\()p Fp(J)8 b Fq(\()p Fp(T)1155 3042 y Fk(\000)p Fl(n)1275 3076 y Fm(j)23 b Fp(T)1382 3042 y Fl(n)1427 3076 y Fq(\()1468 3054 y(\026)1459 3076 y Ff(D)1513 3088 y Fl(n)1564 3076 y Fq(\)\))1628 3042 y Fl(\013)1676 3076 y Fq(\()p Fp(T)1769 3042 y Fl(n)1813 3076 y Fq(\()p Fp(g)1885 3088 y Fl(n)1930 3076 y Fq(\()p Fp(y)s Fq(\)\)\))14 b Fp(J)8 b Fq(\()p Fp(G)2267 3042 y Fo(0)2267 3096 y Fl(p)2301 3104 y Fi(n)2344 3096 y Fl(;q)2394 3104 y Fi(n)2439 3076 y Fq(\))2471 3042 y Fl(\013)2518 3076 y Fq(\()p Fp(T)2611 3042 y Fl(n)2656 3076 y Fq(\()p Fp(y)s Fq(\)\))14 b Fp(J)8 b Fq(\()p Fp(T)2957 3042 y Fl(n)3025 3076 y Fm(j)23 b Ff(D)3125 3088 y Fl(n)3176 3076 y Fq(\))3208 3042 y Fl(\013)3256 3076 y Fq(\()p Fp(y)s Fq(\))515 3259 y(for)31 b(ev)n(ery)f Fp(y)j Fm(2)e Ff(D)1082 3271 y Fl(n)1134 3259 y Fq(\()p Fp(p)p Fq(\).)49 b(By)32 b(\(18\))f Fp(G)1691 3228 y Fo(0)1691 3279 y Fl(p)1725 3287 y Fi(n)1767 3279 y Fl(;q)1817 3287 y Fi(n)1892 3259 y Fm(\000)-15 b(!)30 b Fp(id)i Fq(uniformly)-7 b(.)49 b(Th)n(us)32 b Fp(J)8 b Fq(\()p Fp(G)2941 3228 y Fo(0)2941 3279 y Fl(p)2975 3287 y Fi(n)3016 3279 y Fl(;q)3066 3287 y Fi(n)3111 3259 y Fq(\))31 b Fm(\000)-15 b(!)30 b Fq(1)515 3358 y(uniformly)-7 b(,)27 b(to)r(o.)37 b(No)n(w,)977 3541 y Fp(J)8 b Fq(\()p Fp(T)1124 3507 y Fk(\000)p Fl(n)1243 3541 y Fm(j)24 b Fp(T)1351 3507 y Fl(n)1395 3541 y Fq(\()1436 3519 y(\026)1427 3541 y Ff(D)1481 3553 y Fl(n)1532 3541 y Fq(\)\)\()p Fp(T)1689 3507 y Fl(n)1734 3541 y Fq(\()p Fp(g)1806 3553 y Fl(n)1851 3541 y Fq(\()p Fp(y)s Fq(\)\))g(=)f([)p Fp(J)8 b Fq(\()p Fp(T)2273 3507 y Fl(n)2340 3541 y Fm(j)2396 3519 y Fq(\026)2386 3541 y Ff(D)2440 3553 y Fl(n)2492 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