Inverse pressure estimates and the independence of stable dimension for non-invertible maps

Transcription

1 Invere preure etimate and the independence of table dimenion for non-invertible map Eugen Mihailecu and Mariuz Urbańki Abtract We tudy the cae of an Axiom A holomorphic non-degenerate (hence non-invertible) map f : P 2 C P 2 C, where P 2 C tand for the complex projective pace of dimenion 2. Let Λ denote a baic et for f of untable index 1, and x an arbitrary point of Λ; we denote by δ (x) the Haudorff dimenion of W r (x) Λ, where r i ome fixed poitive number and W r (x) i the local table manifold at x of ize r; δ (x) i called the table dimenion at x. In [5], Mihailecu and Urbańki introduced a notion of invere topological preure, denoted by P, which take into conideration preimage of point. In [9], Verjovky and Wu tudied the cae of Henon diffeomorphim for which they proved that the table dimenion i given by a Bowen equality, hence it i independent of the point in the invertible cae. Our non-invertible ituation i different ince the local untable manifold are not uniquely determined by their bae point, intead they depend in general on whole prehitorie of the bae point. Hence our method are different and are baed on uing a equence of invere preure for the iterate of f, in order to give upper and lower etimate of the table dimenion (Theorem 2). A a Corollary, we obtain an etimate of the ocillation of the table dimenion on Λ. When each point x from Λ ha the ame number d of preimage in Λ, then we how in Theorem 3 that δ (x) i independent of x; in fact δ (x) i hown to be equal in thi cae with the unique zero of the map t P (tφ log d ). We alo prove the Lipchitz continuity of the table vector pace over Λ; thi proof i again different than the one for diffeomorphim (however, the untable ditribution i not alway Lipchitz for conformal non-invertible map). Keyword: Haudorff dimenion, table manifold, baic et, invere topological preure. 1. Introduction and notation. Invere topological preure. 2. Etimate from above and below for the table dimenion uing the invere preure of iterate. 3. Independence of δ (x) (of x), when the map f i open on Λ. AMS 2000 Subject Claification: 37D20, 37A35, 37F35 1

2 1 Introduction and notation. Invere topological preure For the cae of hyperbolic automorphim on C 2 (Henon map), Verjovky and Wu ([9]) howed that the Haudorff dimenion of the interection between local table manifold and the Julia et i given a the unique zero of a preure function. For non-invertible conformal map f (for example holomorphic map on the projective complex pace P 2 ) which are hyperbolic on a baic et Λ, the ituation i completely different, and a hown in [2] and [4], thi table dimenion (precie definition will be given later) i not equal to the unique zero of the correponding preure function. At the ame time, we do not have a uniquely determined untable manifold going through a given point of the baic et Λ. In order to deal with the non-invertible cae, Mihailecu and Urbanki have introduced a notion of invere preure ([5]), which take into conideration all the invere iterate of point (intead of the forward iterate from the cae of uual topological preure). In thi paper we will obtain a theorem (Theorem 2) giving lower etimate of the table dimenion by uing zero of invere preure of iterate of f. A a Corollary we obtain an etimate of the maximum poible ocillation of the table dimenion on Λ. Then, when the map i open on the baic et Λ, we will prove (Theorem 3) that the table dimenion i independent of the point; in the proof we ue again idea and concept related to invere preure. Although thee proof and reult may work for a more general etting (finite-toone conformal map with hyperbolic tructure on a baic et, and with the real dimenion of the table vector pace equal to 2), we preffer to tate them in the cae of holomorphic map on P 2, hyperbolic on a baic et Λ. Note alo that in Theorem 1 we actually ue the holomorphicity at the end of the proof; Theorem 1 i ued later in the proof of Theorem 2. A a final remark, we notice that all the proof work when Λ i jut a compact connected invariant et, f i hyperbolic on Λ, f i tranitive on Λ and Λ ha local product tructure. In thi ection we recall ome definition and propertie of invere preure, which will be ued later. We conider the following etting: X i a compact metric pace, f : X X i a continuou urjective map on X, and Y X i a ubet of X. Due to the urjectivity of f, for any point y of X, and any poitive integer m, there exit y m X uch that f m (y m ) = y. By prehitory of length m (or m-prehitory, or branch of length m) of y, we will undertand a collection of conecutive preimage of y, C = (y, y 1,..., y m ), where f(y i ) = y i+1, i = 1,.., m, y 0 = y. Given a prehitory C, we hall denote by n(c) it length. Fix ε > 0. Denote by C m the et of all m-prehitorie of point from X. For uch an m-prehitory C, let X(C, ε) be the et of point ε-hadowed by C (in backward time) i.e: X(C, ε) := {z B(y 0, ε) : z 1 f 1 (z).t. d(z 1, y 1 ) < ε,..., z m f 1 (z m+1 ).t. d(z m, y m ) < ε} Given the m-prehitory of y, C = (y, y 1,..., y m ) and a real continuou function φ on X, (we denote the et of real continuou function on X, by C(X, R)), one can define the conecutive um of φ on C, Smφ(C) = φ(y) + φ(y 1 ) φ(y m ) We may alo ue the notation S mφ(y m ) intead of S mφ(c). We will define now the invere 2

3 preure P by a procedure imilar to that ued in the cae of Haudorff outer meaure. Let φ be an arbitrary continuou function, φ C(X, R); let alo λ a real number and N a poitive integer. Denote by C := C m. We ay that a ubet Γ C, ε-cover X if X = X(C, ε). Then define m 0 C Γ the following expreion M f (λ, φ,y, N, ε) := inf{ exp( λn(c) + S n(c) φ(c)), n(c) N, C Γ, C Γ and Γ C.t Y C Γ X(C, ε)} When N increae, the et of acceptable candidate Γ which ε-cover X get maller, therefore the infimum increae in the previou expreion. Hence lim M N f (λ, φ, Y, N, ε) exit and will be denoted by M f (λ, φ, Y, ε). Now, let P f (φ, Y, ε) := inf{λ : M f (λ, φ, Y, ε) = 0}. Conider two poitive number ε 1 < ε 2 and let u compare P f (φ, Y, ε 1) and P f (φ, Y, ε 2). Given any prehitory C, we have that X(C, ε 1 ) X(C, ε 2 ), o if Γ C ε 1 -cover Y, then Γ alo ε 2 -cover Y. Therefore there are more candidate Γ in the expreion of M f (λ, φ, Y, N, ε 2) than in the expreion of M f (λ, φ, Y, N, ε 1). Thi how that for any N M f (λ, φ, Y, N, ε 2) M f (λ, φ, Y, N, ε 1) Hence 0 M f (λ, φ, Y, ε 2) M f (λ, φ, Y, ε 1), and then from definition, P f (φ, Y, ε 2) P f (φ, Y, ε 1). Thi prove that, when ε decreae to 0, P f (φ, Y, ε) increae, o the limit lim P ε 0 f (φ, Y, ε) doe exit and i denoted by P f (φ, Y ). P f (φ, Y ) i called the invere preure (or invere upper preure) of φ on Y. P f (φ, Y, ε) i called the ε-invere preure of φ on Y. Thi notion ha been introduced in [5], although here we have ued different notation. When the map f will be clear from the context, we may drop the index f from the notation for P f (φ, Y ), P f (φ, Y, ε), M f (λ, φ, Y, N, ε), etc. Alo, we will denote by P f (φ), P f (φ, ε), M f (λ, φ, N, ε), etc., the quantitie P f (φ, X), P f (φ, X, ε), M f (λ, φ, X, N, ε), etc., repectively. The following propoition provide ome propertie of P. Propoition 1. Let f : X X be a continuou urjective map on the compact metric pace X, ε a poitive number and φ a function from C(X, R). i) If Y 1 Y 2 X, then P f (φ, Y 1) P f (φ, Y 2) and P f (φ, Y 1, ε) P f (φ, Y 2, ε). ii) If Y = j J Y j i a finite or countable union of ubet of X, then P f and P f (φ, Y ) = up P f (φ, Y j). j J (φ, Y, ε) = up P f (φ, Y j, ε) j J iii) If f i a homeomorphim on X, then P f (φ) = P f (φ), where P f (φ) denote the uual (forward) topological preure of φ with repect to the map f. iv) P f (φ, Y ) i invariant to topological conjugacy, i.e if f : X X, g : X X are continuou urjective map and Ψ : X X P f (φ, Y ) = P g (φ Ψ 1, Ψ(Y )), for any ubet Y X. i a homeomorphim uch that Ψ f = g Ψ, then 3

4 Proof. We will prove only part ii), the other are traightforward. Aume that Y = Y j i a j J finite or countable union of ubet of X. We will how that, given ome ε > 0, P f (φ, Y, ε) = up P f (φ, Y j, ε), for any function φ C(X, R); the other equality, P f (φ, Y ) = up P f (φ, Y j) j j will follow imilarly. Firt, directly from the definition of P, it follow that P f (φ, Y, ε) up P f (φ, Y j, ε). Take now t > up P f (φ, Y j, ε). Then there exit ome number α > 0 o mall j j that t α > P f (φ, Y j, ε), j J. So M f (t α, φ, Y j, ε) = 0 for all j J. But from the fact that M f (t α, φ, Y j, N, ε) grow with N, we obtain that M f (t α, φ, Y j, N, ε) = 0, j J, N > 0. So, if N i fixed, then for any j J there exit a et Γ j C uch that Y j n(c) N, C Γ j and we have C Γ j exp( (t α)n(c) + S n(c) φ(c)) 1 2 j X(C, ε) and C Γj Now, if we conider the collection Γ := Γ j, then Y = Y j X(C, ε), n(c) N, C Γ, j J j J C Γ and exp( (t α)n(c) + S n(c) φ(c)) 1 C Γ Thi mean that M f (t α, φ, Y, N, ε) 1, hence M f (t, φ, Y, N, ε) e αn. Thu M f (t, φ, Y, ε) = 0 and t P f (φ, Y, ε). In concluion, ince t ha been taken arbitrarily larger than up P f (φ, Y j, ε), j J we obtain the required equality, P f (φ, Y, ε) = up P f (φ, Y j, ε). j J Here are alo ome additional propertie of P, whoe proof can partly be found in [5]; the proof of the propertie for ε-invere preure are imilar. Propoition 2. Let f : X X be a continuou urjective map on the compact metric pace X, Y a ubet of X and φ, ψ C(X, R). Then: i) P f (φ + α, Y ) = P f (φ, Y ) + α. ii) If φ ψ on Y and ε i a poitive number, then P f (φ, Y ) P f (ψ, Y ) and P f (ψ, Y, ε). P f (φ, Y, ε) iii) P f (, Y ) i either finitely valued or contantly. iv) P f (φ, Y ) P f (ψ, Y ) φ ψ if P f the correponding ε-invere preure. (, Y ) i finitely valued; a imilar inequality hold for v) P f (φ + ψ f ψ, Y ) = P f (φ, Y ). vi) If φ i a trictly negative function on X, then the mapping t P f (tφ, Y ) i trictly decreaing if P f (, Y ) i finitely valued. Alo the mapping t P f (tφ, Y, ε) i trictly decreaing. 4

5 The invere entropy h obtained by definition a P (0) i maller or equal than the preimage entropy h i ([5]) and actually, in the cae of homeomorphim, they both coincide with the uual topological entropy (definition and ueful propertie of h i are given, for example, in [6], [5], etc). Another intereting property of P give an alternative way of calculating the invere preure, by uing prehitorie of the ame length at each tep: Propoition 3. [[5]] Let f : X X be a continuou urjective map on a compact metric pace X, and φ C(X, R). Denote by Q m(φ, ε) := inf{ C Γ exp(s mφ(c)), Γ C m, Γ ε covering X}, Then P (φ) = lim 1 lim ε 0 m m log Q m(φ, ε). So, Propoition 3 ay that we can alo obtain P (φ) if in the original definition we conider at tep m only m-prehitorie, then let m converging to, etc. Thi i due to the way in which X(C, ε) wa defined and alo to the poibility of concatenating two prehitorie in order to obtain a longer prehitory. In the equel, we will focu on the cae of a holomorphic non-degenerate map f : P 2 P 2, where P 2 repreent the 2-dimenional complex projective pace P 2 C. Any holomorphic map f on P 2 i given a f([z : w : t]) = [P (z, w, t) : Q(z, w, t) : R(z, w, t)], with P, Q, R homogeneou polynomial in z, w, t, all having the ame degree d. If d 2, then f i called non-degenerate; in thi cae f i non-invertible. We hall aume in the equel that f : P 2 P 2 i non-degenerate and ha Axiom A; let Λ be one of it baic et of untable index 1, meaning that Df ha on Λ both table and untable direction. For definition and dicuion of Axiom A for non-invertible map [7] or [2] are good reference. An important point to remember i that, ince f i not invertible on the invariant et Λ, one ha to define hyperbolicity with repect to the natural extenion of Λ. We recall briefly thi notion and alo how to define hyperbolicity in thi non-invertible cae. Denote firt by ˆΛ := {ˆx = (x, x 1,...)where x i Λ and f(x i 1 ) = x i, i 0, x 0 = x} and call thi et the natural extenion of Λ with repect to f. ˆΛ i a compact metric pace endowed with the metric d(ˆx, ŷ) = i 0 d(x i, y i ) 2 i More general, we can define a metric d K on ˆΛ for any K > 1 by etting d K (ˆx, ŷ) = i 0 d(x i, y i ) K i. A above, we will not pecify the contant K in the notation d K when K = 2. Alo, it can be noticed that for all K > 1, d K give the ame topology on ˆΛ, namely the topology induced on the 5

6 ubet ˆΛ by the product topology on the larger pace Λ N. We denote by π : ˆΛ Λ the canonical projection π(ˆx) = x and by ˆf the homeomorphim ˆf : ˆΛ ˆΛ, ˆf(ˆx) = (fx, x, x 1,...). The hyperbolicity of f on Λ mean that there exit contant C > 0, λ > 1, and for every ˆx ˆΛ, a vector pace E ṷ x T xp 2, and a vector pace E x T x P 2 uch that Df(E ṷ x ) Eu c fx, Df(E x) E fx and we have the inequalitie Df k x (v) C(λ ) k v, Df k x (w) C(λ ) k w for every x Λ, k 0 and all vector v Ex, w Ex ṷ. In the definition of hyperbolicity on ˆΛ we aume alo that E x E ṷ x = T xp 2, ˆx ˆΛ and that E x depend continuouly on x, while E ṷ x depend continuouly on ˆx. E x i called the table tangent vector pace (or the table pace) at x. E ṷ x i called the untable tangent vector pace (or untable pace) correponding to the prehitory ˆx. Like in the diffeomorphim cae, it i poible ([7]) to how that, if r i mall enough (for example 0 < r < r 0 ), there exit table and untable local manifold paing through x: W r (x) := {y P 2, d(f i x, f i y) < r, i 0} W u r (ˆx) := {y P 2, ŷ π 1 (y) with d(y i, x i ) < r, i 0}. If moreover f i holomorphic on P 2, the local (un)table manifold on a baic et of untable index 1, are analytic dik. Now, given a point x Λ and a mall fixed number 0 < r < r 0 < diamλ 2, denote by δ (x) := HD(W r (x) Λ), where HD tand for the Haudorff dimenion of a et. We hall call δ (x), the table dimenion at x. In the equel we hall uppoe alo that C f Λ =, where C f denote the critical et of f. Hence, one can define the negative function φ (y) := log Df E y, y Λ; a a notational remark, E y i a one-dimenional complex pace and Df E y denote the norm of Df retricted to thi table pace. We tudied the table dimenion in [2], [4], [5]. In [2], the firt author howed that δ (x) t, where t i the unique zero of the preure function t P (tφ ) (the topological preure being calculated with repect to the map f Λ ). However in the above inequality we do not have equality in general. Indeed the gap between δ (x) and t i influenced by the number of preimage that a point from Λ ha in Λ, a wa explained in [4], where we obtained a better upper etimate t 0 : Theorem. In the above etting, aume that the map f Λ ha the property that every point x Λ ha at leat d d preimage in Λ. Then δ (x) t 0, where t 0 i the unique zero of the function t P (t log Df E y log d ) and a a conequence, δ (x) h(f Λ) log d log up y Λ Df E y. Another etimate for δ (x) wa given in [5], by uing the unique zero t of the invere preure function t P (tφ ), calculated again with repect to f Λ : Theorem. In the ame etting a before, δ (x) t. Moreover, in cae Λ can be written a the union of countably many compact, pathwie connected and imply connected ubet, t = δ (x), x Λ. 6

7 Let u focu now on the zero t n(ε) of the ε-invere preure function for the iterate f n Λ. If Λ i a baic et for f, then f(λ) = Λ, hence f n (Λ) = Λ, n > 0 integer. Let u denote by Df (y) the linear map Df E y, which can be identified with a complex number ince E y ha complex dimenion 1; imilarly, Df n (y) denote Df n E y, y Λ. Since f i holomorphic, Df n (y) = Df (y) Df (fy)... Df (f n 1 y), y Λ. φ n(y) := log Df n (y), y Λ, o φ n i a trictly negative function on Λ, which ha finite value ince C f Λ =. From Propoition 2 vi) applied to f n Λ : Λ Λ, it follow that the function t P f n (tφ n, ε) i trictly decreaing; ince P f n (0, ε) 0, and P f n (tφ n, ε) < 0 for t > 0 large enough, it follow that thi trictly decreaing function ha a unique zero, denoted by t n(ε). The ame i true for the function t P f n (tφ n) which ha a unique zero t n. When n = 1 we denote t 1 (ε) by t (ε), and t 1 by t. We hall prove in the equel that t n(ε) t np(ε) and t n = t, for any poitive integer n, p and any ε > 0. Firt, we will prove that the table pace E y depend Lipchitz continuouly on y Λ. addition we will how the Lipchitz continuity of y E y when y range in W r (x) ( x Λ), and moreover, that the Lipchitz contant on thee table leave can be choen independently of the point x Λ. Remark alo that the untable pace do not depend Lipchitz on their bae point ince in general they depend on whole prehitorie. In [2], one of the author howed that the untable pace E ṷ x depend Holder continuouly on ˆx, with repect to a fixed metric d K on ˆΛ; the repective Holder exponent depend on the choen contant K > 1. The following theorem wa known in the cae of conformal diffeomorphim, but up to our knowledge it ha never appeared in the cae of non-degenerate holomorphic map on P 2 (which are non-invertible). A it turn out below, the non-invertible cae require it own proof, different than the one given for diffeomorphim. (for example, in thi cae we cannot ue the invere iterate f 1, and on the natural extenion ˆΛ we cannot ue a differentiable tructure). Theorem 1. Conider f : P 2 P 2 a holomorphic Axiom A map, and let Λ be one of it baic et of untable index 1, uch that C f Λ =. Then the map x E x i Lipchitz continuou a a map from Λ to the bundle G 1 (Λ) of pace of complex dimenion 1 in the tangent bundle over Λ, i.e. there exit a poitive contant Υ uch that for all x, y from Λ, d(e x, E y) Υd(x, y). In particular, if φ (y) := log Df E y, y Λ, then φ i Lipchitz continuou. Moreover, there exit a mall r > 0 and Ξ > 0 uch that for any x Λ and any point y, z W r (x), we have φ (y) φ (z) Ξ d(y, z). Proof. For every K > 1 conider the metric d K on ˆΛ, given by the formula d K (ˆx, ŷ) := d(x, y) + d(x 1, y 1 ) K + d(x 2, y 2 ) K Notice that the topology given by d K on ˆΛ i independent of K and i induced by the product topology on a countable product of Λ. In the equel we hall ue a Pointwie Hölder Section Theorem from [10]. Theorem (Pointwie Hölder Section Theorem). Let E = X Y be a vector bundle over a metric pace X, where Y i a cloed, bounded ubet of a Banach pace, and let π : E X be the canonical projection. Let F : E E be a bundle map covering a homeomorphim h : X X, i.e π F = h π. Suppoe that F atifie the following condition: In 7

8 1) F contract the fiber of E in the ene that, for all x X there exit a contant 0 λ x < 1 uch that d(f (x, y), F (x, z)) λ x d(y, z), y, z Y. 2) There exit contant L 1 and α > 0 uch that for all x, x X and y Y, F (x, y) F (x, y) L d(x, x ) α. 3) There exit ome poitive number η uch that up λ x µ α x =: ρ(α) < 1 where µ x denote: x X µ x := inf{ d(hx, hx ) d(x, x ), x, x X, x x, d(x, x ) < η} Alo, let u denote by µ := inf x X µ x and aume that µ > 0. Then we have the following: i) there exit a unique ection σ : X E whoe image i invariant under F, i.e σ h(x) = F σ(x), x X. ii) σ i Hölder continuou with exponent α, i.e σ(x) σ(x ) Hd(x, x ) α, x, x X. iii) Aume that the diameter of Y i bounded by R, then we can bound the Hölder contant H by: H LR µη α (1 up λ x µ α x ) Let u now return to our etting and ee how we can apply thi theorem. By definition of hyperbolicity of f, there exit a continuou plitting of the tangent bundle to P 2 over ˆΛ, given by TˆΛP 2 = E E u, where Ex depend continuouly on x Λ and Ex ṷ depend continuouly on ˆx ˆΛ. The table pace Ex and the table manifold of ize r > 0 at x depend only on the forward iterate of x, wherea the untable pace Ex ṷ and the untable manifold W r u (ˆx) depend on the entire prehitory ˆx of x. Let u take an arbitrary contant K > 1 and conider the metric d K on ˆΛ. Since continuou map can be approximated by Lipchitz continuou map, there exit a plitting F F u (K) of TˆΛP 2 uch that the linear ubpace of complex dimenion 1, F x, depend Lipchitz continuouly on x Λ and the ubpace of dimenion 1, F ṷ x (K) depend Lipchitz on ˆx ˆΛ; alo we aume that Fx approximate Ex, and Fx ṷ(k) approximate Eṷ x uniformly in x, repectively ˆx. A a remark, the pace Fx ṷ (K) depend in general on K ince they have to vary Lipchitz continuouly with repect to the metric d K, wherea the pace Fx are Lipchitz only with repect to the uual euclidian metric induced on Λ, therefore they do not depend on K. Let u aume that d(f x, E x) < ε, d(f ṷ x (K), Eṷ x ) < ε, for all ˆx in ˆΛ, where ε i a mall poitive number. From the above Lipchitz condition, there exit poitive contant τ and τ K uch that d(f x, F y ) τd(x, y), x, y Λ 8

9 and d(f ṷ x (K), F ṷ y (K)) τ Kd K (ˆx, ŷ), ˆx, ŷ ˆΛ In thi cae, Ex can be interpreted a the image of a linear map from Fx to Fx ṷ (K), for any prehitory ˆx of x Λ. Conider therefore Lˆx (K) := L(Fx, Fx ṷ (K)) be the pace of linear map from Fx to Fx ṷ(k). L(K) will denote the vector bundle over ˆΛ given by Lˆx (K), ˆx ˆΛ, where we conider the metric d K on ˆΛ. The pace X of the Hölder Section Theorem will be ˆΛ endowed with d K and the homeomorphim h from the tatement of the ame theorem i the map ˆf 1 : ˆΛ ˆΛ. We will alo conider the bundle map Ψ : L(K) L(K) induced by the graph tranform aociated to the derivative Df 1 (ˆx) : F x F ṷ x (K) F x 1 F ṷ f 1 ˆx (K), where ˆx = (x, x 1,...) ˆΛ. The mapping Df 1 (ˆx) repreent the derivative at x of the local branch of f 1 which take x into x 1, in cae ˆx = (x, x 1,...) i an arbitrary point of ˆΛ; thi derivative doe exit becaue we aumed that the critical et of f doe not interect Λ. In the equel we hall ue alo the notation Df 1 (ˆx) a being the invere of the iomorphim Df (x 1 ) : Ex 1 Ex; imilarly for the notation Dfu 1 ((ˆx). The notion of) graph tranform ued above i explained in [8]. If we aume that Df 1 (ˆx) = Aˆx Cˆx (K) Bˆx (K) Gˆx (K), then Aˆx : F x F x 1, Bˆx (K) : F ṷ x (K) F x 1, Cˆx (K) : Fx F ṷ (K), Gˆx(K) : F ṷ f 1 ˆx x (K) F ṷ (K); let u notice that from the decompoition above, f 1 ˆx Bˆx (K), Cˆx (K) and Gˆx (K) depend on K, but Aˆx doe not, ince the bundle F i independent of K. From the definition of graph tranform, Ψˆx (g) = (Cˆx (K) + Gˆx (K)g) (Aˆx + Bˆx (K)g) 1, (1) for any linear map g Lˆx (K). So it can be noticed that Ψˆx (g) L ˆf 1 ˆx (K), for any ˆx ˆΛ. From contruction, Aˆx and Gˆx (K) approximate Df 1 (ˆx), repectively Dfu 1 (ˆx), while Bˆx (K) < a 1 (ε), Cˆx (K) < a 1 (ε), where a 1 ( ) i a poitive continuou function with a 1 (0) = 0. Hence, if ε i mall enough, then the Lipchitz contant of Ψˆx i maller or equal than λˆx (K), where: λˆx (K) := Df 1 u (ˆx) Df (x 1 ) + a 2 (ε) = Df (x 1 ) Df u (x 1 ) + a 2(ε) < 1, (2) and where a 2 (ε) i a poitive continuou function in ε, with a 2 (0) = 0. Let u recall now that the metric on ˆΛ i d K which depend on the contant K > 1. In the ame pirit a in [8], we can alo aume that the bundle E := L(K) i trivial, otherwie we can replace it with E E, for ome complementary bundle E. Thi replacement doe not depend on the metric d K, ince the metric on E i already induced by the product of the metric d K on ˆΛ and the uual euclidian metric on the pace of linear map. We will etimate the local Lipchitz contant µˆx (K) of h at ˆx ˆΛ, where h = ˆf 1 i our bae homeomorphim. Thu, a in the tatement of the Pointwie Hölder Section Theorem, let µˆx (K) := inf{ d K(hˆx,hŷ) d K (ˆx,ŷ), ˆx ŷ, ˆx, ŷ ˆΛ and d K (ˆx, ŷ) < η} for ome mall η > 0. Denote alo by µ(k) := inf µˆx (K). Then we have: ˆx ˆΛ d K (ˆx, ŷ) = d(x, y) + d(x 1, y 1 ) K = d(x, y) + 1 K d( ˆf 1ˆx, ˆf 1 ŷ) + d(x 2, y 2 ) K = (3) 9

10 Let u denote by ε 0 a poitive contant depending only on f uch that f i injective on ball of radiu ε 0 (inf Df ) 1 centered on Λ and uch that we can apply the Mean Value Inequality on ball Λ of radiu ε 0 (inf Df ) 1. Suppoe that 0 < η < ε 0. If d K (ˆx, ŷ) < η, and d K ( ˆf 1ˆx, ˆf 1 ŷ) > η, then Λ d K (ˆx, ŷ) < ( Df u (x 1 ) + 1 K )d K( ˆf 1ˆx, ˆf 1 ŷ) ince Df u (x 1 ) + 1 K > 1. So, with the aumption that d K (ˆx, ŷ) < η, let u uppoe alo that d K ( ˆf 1ˆx, ˆf 1 ŷ) < η. Hence d(x 1, y 1 ) < η and, from our aumption it follow alo that d(x, y) < η, o, uing the Mean Value Inequality, we obtain that: d K (ˆx, ŷ) ( Df u (x 1) + 1 K )d K( ˆf 1ˆx, ˆf 1 ŷ) = ( Df u (x 1) + 1 K )d K(hˆx, hŷ), (4) where x 1 i ome point with d(x 1, x 1 ) < η. Thi implie that the contant µ x which appear in the Pointwie Hölder Section Theorem i repreented in our ituation by µˆx (K) and, a we aw in ( 4), µˆx (K) ( Df u (x 1 ) + 1 K + ω( Df u, η)) 1, (5) where ω( Df u, η) i the maximum ocillation of Df u on a ball of radiu η centered at an arbitrary point of Λ, and we ued above that Df u (x 1 ) Df u(x 1 ) + ω( Df u, η). Next, we how that Ψˆx i Lipchitz in ˆx; recall that we aumed that L(K) i a trivial bundle, o we can identify all the 1-dimenional complex pace Lˆx (K) with C, and do thi independently of K. We wih to prove that there exit a contant Θ K > 0 uch that Ψˆx (g) Ψŷ(g) Θ K d K (ˆx, ŷ), ˆx, ŷ ˆΛ, g C, g 1 (6) From the fact that f i mooth and F depend Lipchitz in x Λ, while Fx ṷ (K) depend Lipchitz in ˆx ˆΛ, it follow that Aˆx depend Lipchitz in x (with repect to the euclidian metric induced on Λ) and Bˆx (K), Cˆx (K), Gˆx (K) depend Lipchitz in ˆx (with repect to the metric d K ). Recall from ( 1) that Ψˆx (g) = (Cˆx (K) + Gˆx (K)g) (Aˆx + Bˆx (K)g) 1, for any linear map g Lˆx (K). But in our cae, g, Aˆx, Bˆx (K), Cˆx (K), Gˆx (K) are jut complex number. It i enough to how that ˆx (Aˆx + Bˆx (K)g) 1 i Lipchitz. But ince we work with complex number we have (Aˆx + Bˆx (K)g) 1 (Aŷ + Bŷ(K)g) 1 =. Now we ue the fact that (A ŷ Aˆx )+(Bŷ(K) Bˆx (K))g (Aˆx +Bˆx (K)g)(Aŷ+Bŷ(K)g) Aˆx, Bˆx (K) depend Lipchitz in ˆx and Bˆx (K) < a 1 (ε) << 1, ˆx ˆΛ. Thu, for g 1 we get that Aˆx + Bˆx (K)g i uniformly (in ˆx) bounded away from 0, ince Aˆx approximate Df 1 (ˆx) (and we know that Df 1 (ˆx) (up Df ) 1 > 0), and Bˆx (K) i very mall in comparion to Aˆx. In Λ concluion we obtained the Lipchitz continuity of Ψ, hence inequality ( 6). Let u check now the condition 3) of the Pointwie Hölder Section Theorem with α = 1. Uing the relation in ( 2) and ( 5), we have that: ρ(1, K) := up λˆx µˆx (K) 1 ( Df (x 1 ) Df ˆx ˆΛ u (x 1 ) + a 2(ε)) ( Df u (x 1 ) + 1 K + ω( Df u, η)) = = ( Df (x 1 ) Df u (x 1 ) + a 2(ε)) ( 1 K + ω( Df u, η)) + Df (x 1 ) Df u (x 1 ) Df u(x 1 ) + a 2 (ε) Df u (x 1 ) Df (x 1 ) + M(ε, η, K) < 1, 10 (7)

11 where M(ε, η, K) i a poitive continuou function in ε, η, and K with M(0, 0, ) = 0. Thi i why in the lat inequality of ( 7) we were able to take M(ε, η, K) < 1 up Df, for ε and η mall enough and K large enough. The value of uch ε, η, K depend only on f. Therefore, we found that in thi cae condition 3) of the Pointwie Section Theorem i atified for α = 1. Now, according to ( 6), it follow that condition 2) from the tatement of the Pointwie Section Theorem i atified a well, o all the condition of the Pointwie Hölder Section Theorem hold and we get that the unique invariant ection σ i Lipchitz. But in our cae thi unique invariant ection σ i jut the table bundle, σ(ˆx) = E x, ˆx ˆΛ, hence there exit a contant C K depending on K uch that: d(e x, E y) C K d K (ˆx, ŷ), ˆx, ŷ ˆΛ (8) Let u denote now by λ := inf z Λ Df (z), and take ε 0 := λ ε 0, where the number ε 0 ha been introduced earlier; clearly ε 0 0 ince the critical et of f avoid Λ. We want to prove that ( 8) implie that, in fact, x E x i Lipchitz. Cae 1: Let u then aume firt that x, y Λ with d(x, y) ε 0. If 0 denote the diameter of Λ, then Λ d K (ˆx, ŷ) d(x, y) K d(x, y) d(x, y) K ε 0 d(x, y)( ) < d(x, y)( ) C d(x, y), K ε 0 ε 0 (9) with C > 0 a contant independent of K. Cae 2: Now uppoe that 0 < d(x, y) < ε 0 for ome x, y Λ. We conider here the map f retricted to Λ. We will ay that (x, x 1..., x n ) are conecutive preimage of x in Λ if f(x 1 ) = x, f(x 2 ) = x 1,..., f(x n ) = x n+1 and x j Λ, j = 1..n. Conider n = n(x, y) to be the larget poitive integer uch that there exit conecutive preimage of x and of y, (x, x 1,..., x n) and (y, y 1,..., y n) with d(x i, y i ) < ε 0, i = 1,.., n. Since n i the larget uch integer, it follow that, for ome x n 1 f 1 (x n) and y n 1 f 1 (y n), with d(x n 1, y n 1 ) < ε 0λ 1, we have: We alo obtain ε 0 < d(x n 1, y n 1) λ 1 d(x n, y n) (10) d(x i, y i) λ i d(x, y), i = 1,.., n (11) From ( 10) and ( 11), we obtain that d(x n 1, y n 1 ) λ n 1 d(x, y). Thi implie that, for any complete prehitorie ˆx, ŷ of x, y, which tart with the conecutive preimage (x, x 1,..., x n), (y, y 1,..., y n) conidered above, we have d K ( ˆx, ŷ ) = d(x, y) + d(x 1, y 1 ) +... K d(x, y) d(x, y) λ K λ n K n d(x, y) + 2 (12) 0 K n+1 11

12 Aume that K i fixed uch that K > λ 2 ε < 1 and ome η < ε 0. Then, from ( 10) and ( 11), ε 0 < λ n 1 and uch that M(ε, η, K) < 1 up Df for ome Λ d(x, y) < K n+1 d(x, y), which 1 implie that K n+1 contant C uch that for our choen prehitorie ˆx, ŷ, of x, repectively y, < d(x,y) ε 0. Introducing thi inequality in ( 12), one ee that there exit a poitive d K ( ˆx, ŷ ) C d(x, y) (13) By conidering now both Cae 1, ( 9), and Cae 2, ( 13), together with ( 8), we obtain the Lipchitz continuity of the table pace with repect to their bae point i.e there exit a poitive contant Υ uch that for all x, y from Λ, d(e x, E y) Υd(x, y). Thi implie immediately that alo φ i Lipchitz on Λ. Now, we will prove the uniform Lipchitz continuity of the table ditribution and of φ along the table leave. We notice that, ince Λ i compact, one can contruct local table manifold of ize r at all point of Λ, if r > 0 i mall enough. If y i a point in a manifold W r (x), but y i not necearily in Λ, we hall call table pace at y, denoted by E y, the tangent pace at W r (x) at y. We ee that the pace E y vary moothly when y move inide W r (x) for x fixed. So the exitence of a contant Ξ like in the tatement i conditioned only on the boundedne of the curvature of thee local table manifold. Aume then that there exit a equence z n Λ uch that the Lipchitz contant L n of the map g n converge to infinity, where g n (y) := E y, y W r (z n ). Since Λ i compact, the equence (z n ) n ha at leat one convergent ubequence and without lo of generality we can aume that thi ubequence i again (z n ) n and z n z. If x i an arbitrary point in Λ, then W r (x) i an analytic dik which i given a the image of an analytic map h x from the unit dik to C 2. We denote by h n the map h zn, for n poitive integer. But from the hyperbolicity condition, the analytic map h x vary continuouly in x Λ, hence alo h n vary continuouly in n. The norm on of the econd derivative of h n bound the Lipchitz contant L n of the map g n, for all n. Notice however that, ince h n are holomorphic and vary continuouly in n, alo the econd derivative of the map h n vary continuouly in n. Therefore, ince we aumed z n z Λ, we obtain that L n are bounded by ome finite poitive contant L. So the map y E y i L-Lipchitz on W r (x), x Λ. Then, due to the moothne of f, there exit a mall r > 0 and Ξ > 0 uch that for any x Λ and any point y, z W r (x), we have φ (y) φ (z) Ξ d(y, z). We will tudy now the conecutive um of the function φ. Given a prehitory C, a poitive number ε < min{diamλ/2, ε 0 }, and the correponding et Λ(C, ε), we will prove that the conecutive um of two point in Λ(C, ε) are the ame, up to a contant independent of the length of C. Propoition 4. Let f : P 2 P 2 holomorphic, with Axiom A and uch that C f Λ = for a baic et Λ of untable index 1. Let alo a prehitory C of a point x in Λ, with repect to f. If m := n(c), C = (x, x 1,..., x m ) and y i an arbitrary point in Λ(C, ε), with the correponding 12

13 prehitory (y, y 1,..., y m ) ε-hadowed by C, then we have: 1 Df m (y m ) C 1 Df m (x m ) < C 1, where C 1 > 1 i a contant independent of m and C. Proof. From the fact that (y,..., y m ) i an m-prehitory of y in Λ we know in particular that y m Λ, hence there exit a local table manifold through y m of ize ε. Let u take alo ˆx be any complete prehitory in Λ of x, tarting with (x, x 1,..., x m ). Set ˆx m := ˆf m (ˆx). In thi cae W u ε (ˆx m ) interect W ε (y m ) in a unique point z. It follow from the local product tructure of Λ that z belong to Λ. From the fact that y belong to Λ(C, ε) and (y,..., y m ) i it prehitory ε-hadowed by C, we know that d(f i x m, f i y m ) < ε for all i = 0, 1,..., m. Alo from the fact that z W ε (y m ) it follow that d(f i z, f i y m ) < ε for all i = 0, 1,..., m. From the lat two inequalitie we get that d(f i x m, f i z) < 2ε for all i = 0, 1,..., m. But, ince z W u ε (ˆx m ) W ε (y m ), we have that there exit contant c > 0 and γ (0, 1) uch that for all i = 0, 1,..., m, d(f i x m, f i z) < cγ m i and d(f i y m, f i z) < cγ i. (14) Now from Theorem 1, φ (y) depend Lipchitz continuouly on y Λ and, alo φ i uniformly Lipchitz continuou on local table manifold over Λ. Thi, together with ( 14), implie that there exit a contant K > 0 uch that: m m φ (y j ) φ (x j ) j=0 j=0 m m φ (y j ) φ (f m j z) j=0 j=0 + m m φ (f m j z) φ (x j ) j=0 j=0 m m K ( d(y j, f m j z) + d(f m j z, x j )) 2K c m γ j < K, j=0 where K i a contant independent of m and ε. Hence the tatement of the propoition follow immediately from the previou inequalitie. Propoition 5. Let f : P 2 P 2 holomorphic, with Axiom A and uch that C f Λ = for a baic et Λ of untable index 1. Denote χ u := up Df u. ε > 0. Λ (a) Then we have that t n(ε) t np(ε) and that t = t n, for any poitive integer n, p and any (b) For ε < ε 0, and ρ an arbitrary number in the interval (0, χ 1 u ), denote by ρ n := ε ρ n, n > 1. Then P f n (tφ n, ρ n ) = P f n (tφ n), for any t; conequently t n(ρ n ) = t n = t, n > 1. Proof. (a) Firt we make the following notation. If m i a poitive integer, denote by C n m := {(y, y n 1,..., y n m) Λ m+1, uch that f n (y n i) = y n i+1, i = 1,.., m, and y 0 = y} Let alo C n := Cm n be the et of prehitorie of finite length for f n in Λ. Now, if n, p and ε > 0 m 0 are fixed, we conider an arbitrary number t (t n(ε), t n(ε) + 1). From the definition of t n(ε), we 13 j=0 j=0 (15)

14 get that, for N large, there exit an ε-covering Γ of Λ, Γ C n with n(c) N, C Γ and: C Γ exp(s n(c) (tφ n(c))) < exp( (t n(ε) + 1)n(2p 1) up φ ) (16) Λ For every C Γ, let u divide n(c) by p, and obtain n(c) = p m(c) + k(c), where 0 k(c) < p. If C = (y, y 1 n,..., yn n(c) ), then denote by C the m(c)-prehitory of y with repect to f np given by C = (y, z np 1,..., znp m(c) ), where znp 1 := yn p,..., z np m(c) := yn pm(c). Then it i eay to ee that Λ(C, ε) Λ(C, ε), for all C Γ. Denote by Γ the collection of all the prehitorie C aociated by the above procedure to the prehitorie C from Γ. We calculate now the conecutive um On the other hand S n(c) φ n(c) = φ n(y) φ n(y n m(c)p ) + φ n(y n m(c)p 1 ) φ n(y n n(c) ) = log Df n(pm(c)+1) (y m(c)p n nk(c) ) + log Df (y n(c) n ). S m(c) φ np(c ) = φ np(y) φ np(z np m(c) ) Thee lat two relation how that = φ (y n m(c)p ) + φ (fy n m(c)p ) φ (y) + φ (fy) φ (f np 1 y) = log Df np(m(c)+1) (y n m(c)p ). S n(c) φ n(c) = S m(c) φ np(c ) + log Df n (y) + log Df nk(c) (y n(c) n np ) log Df (y). Uing that k(c) < p and the lat equality, we obtain that S n(c) φ n(c) S m(c) φ np(c ) n(p 1) up Λ φ + log Df nk(c) (y n(c) n ) n(2p 1) up φ Λ Therefore inf{ exp(s m(c) (tφ np(c ))), Γ C np ε cover Λ} C Γ [ exp(s n(c) (tφ n(c)))] exp(tn(2p 1) up φ ) < 1. C Γ Λ (17) The lat inequality follow ince t < t n(ε) + 1 and from the way we choe Γ in the begining of the proof. But from the definition of P np, we obtain then that t t np(ε). However ince t wa taken arbitrarily in the finite interval (t n(ε), t n(ε) + 1), it follow that t n(ε) t np(ε). The inequality t (ε) t n(ε) implie that t t n, n 1. We want to prove now the oppoite inequality, i.e t t n (actually the ame proof how more generally, that P f n (tφ n) = np f (tφ )). Indeed, let u conider an arbitrary t > t n, for a fixed integer n. For a given ε > 0, let ε n > 0 atifying the following condition: for any y, z with d(y, z) < ε n we have d(f j y, f j z) < ε, 0 j n, and alo P f (tφ n, ε n n ) < 0. Hence for all m large, there exit an (m, ε n )-cover Γ n m of Λ (i.e Γ n m i a collection of m-prehitorie C with repect to f n, o that Λ = Λ(C, ε n )), atifying: C Γ n m e S m(tφ n )(C ) < 1. Now, out of every C we will form a prehitory C with repect to f in C Γ n m 14

15 the canonical way, i.e if C = (y, y n,..., y nm ), then C = (y, f n 1 y n,..., y n,..., f(y nm ), y nm ). Alo, from the condition atified by ε n, we ee that Λ(C, ε n ) Λ(C, ε); o, if Γ nm denote the collection of prehitorie C of length nm (with repect to f) obtained a above from the prehitorie C of Γ n m, we obtain that Γ nm i an (nm, ε) cover of Λ. Moreover, a found above, Snm(tφ )(C) = Sm(tφ n)(c ) + log Df (y) log Df n (y). Thee fact imply that e S nm(tφ )(C) < M n, where C Γ nm M n i a contant depending only on n. Therefore if we let m (and keep n fixed), we ee that P f (tφ, ε) 0 t t (ε). But 0 < ε < ε 0 wa arbitrary and t wa taken arbitrarily larger than t n, hence t n t. Thi prove the equality t = t n, n 1. (b) Firt from the proof of Propoition 4 we know that for all m 1, and prehitory (x, x 1,..., x m ) 1 of x in Λ, C 1 (ε) Df m (y m ) Df m(x C m) 1 (ε), for (y, y 1,..., y m ) an m-prehitory of y, ε-hadowed by (x, x 1,..., x m ). The proof of Propoition 4 implie alo that C 1 (ε) C 2 ε, 0 < ε < ε 0, for ome contant C 2 > 0. Let u conider now the ituation for f n for ome fixed n 1. Conider (x, x n,..., x np ) a p-prehitory of x in Λ (with repect to f n ), and let (y, y n,..., y np ) be another p-prehitory in Λ which i ρ n -hadowed by (x, x n,..., x np ). Then, if d(y np, x np ) < ρ n < ερ n, we get that d(f j (y np ), f j (x np )) < ε, 0 j n, and imilarly we obtain that d(f j (y np ), f j (x np )) < ε, 0 j np. Therefore the np-prehitory with repect to f, (y, y 1,..., y np ) i ε-hadowed by (x, x 1,..., x np ). prehitorie of f n : So we can apply Propoition 4 in thi cae to obtain imilar inequalitie for 1 C 1 (ε) np Df (y np ) Df np (x np ) C 1(ε), (18) for any p 1. Next, take C an arbitrary p-prehitory in Λ, with repect to f n, for n fixed. If ε i an arbitrary number in the interval (0, ρ n ), we ee that the et Λ(C, ρ n ) can be covered with at mot ( ρnc 1(ε) ε ) 4 et of the form Λ(C, ε ), where C are p-prehitorie with repect to f n. Thu, recalling the definition of P f (tφ n, ρ n n ), P f (tφ n, ε ) and inequality (18), we conclude that: n P f n (tφ n, ρ n ) = P f n (tφ n, ε ) = P f n (tφ n) The lat equality above follow from the fact that P f n (tφ n, ε ) P f n (tφ n) when ε 0. Hence, recalling alo the concluion of part (a), we get t n(ρ n ) = t n = t, n > 1. 2 Etimate from above and below for the table dimenion in the general holomorphic cae uing the invere preure of iterate up ω Λ Given a map f and a baic et Λ a in Propoition 4, define λ := inf Df (ω) and χ := ω Λ Df (ω). Remark that λ > 0 ince we aumed that Λ C f =. For every poitive integer n and mall poitive number ε, let t n(ε) (repectively t n) be the unique zero of the function t P f n (tφ n, ε) (repectively t P f n (tφ n)), where φ n(y) := log Df n (y), y Λ. 15

16 Theorem 2. Let f : P 2 P 2 be a holomorphic non-degenerate map with Axiom A and Λ a baic et of f with untable index 1. Aume alo that the critical et of f, C f doe not interect Λ. (a) Then for every x Λ, we have δ (x) t n(ρ n ) = t, where ρ n > 0 are mall number of the form ερ n, n 1, where χ u := up ε < min{ε 0, r 0 }. Λ Df u, ρ > 0 i an arbitrary number maller than χ 1 u, and (b) For all poitive number ε < ε 0, and η > 0, we get δ (x) + η t n(ε), where n n(ε, η) and n(ε, η) i a poitive integer atifying n(ε, η) > get δ (x) + ε t n(ε), for n ( 1 ε ) log 1 ε η log χ 1. In particular, if η = ε mall enough, we Proof. (a) According to Propoition 5, we have t n(ρ n ) = t. From the Theorem of [5], recalled alo in the Introduction, we have that δ (x) t. Hence δ (x) t n(ρ n ), n > 1. (b) We prove now the inequality δ (x) + η t n(ε) for ε > 0 mall enough (to be determined next), η > 0 mall, and n n(ε, η). Firt let u notice that, from definition, δ (x) 2. Let u take an arbitrary t with δ (x) < t < 3. Recall alo that ε 0 ha been introduced earlier a a poitive contant o that we can apply the Mean Value Inequality for f on ball of radiu ε 0 (inf Λ Df ) 1, and alo uch that f i injective on ball of radiu ε 0 (inf Λ Df ) 1 centered on Λ. Conider now N 0 (ε) to be the mallet cardinality of a covering of Λ with ball of radiu ε. Then if β = dim B (Λ) denote the upper box dimenion of Λ, and β 0 < β < β 1, we have that ( 1 ε )β 0 < N 0 (ε) < ( 1 ε )β 1, for ε > 0 mall enough. With ε > 0 and η > 0 fixed, conider n(ε, η) be the mallet poitive integer n uch that N 0 (ε) χ nη < 1 (19) Thi implie then n(ε, η) > 4 log 1 ε. η log 1 χ In the equel we conider ε with 0 < ε < min{ε 1 /2, r, d(λ, C f )/4}. We hall prove that, for uch an ε and η > 0, the inequality t + η > t n(ε) hold for n n(ε, η). Define now a contant 0 < α < 1 which depend only on f and on Λ, uch that for all x Λ and 0 < r << diamλ, we have that W r (y ) interect W u r (ẑ ) for all point y, z B(x, αr ) and all prehitorie ẑ ˆΛ of z. The exitence of uch a contant follow from the tranverality of table and untable manifold. Next let u cover the compact et Λ with a finite number of ball B(y 1, αε/4),..., B(y, αε/4) which are centered at point of Λ. Let u chooe one uch ball and denote it interection with Λ by Y. We will how now that there exit a poitive integer m uch that all local untable manifold W u ε (ŷ) interect the et f m (W ), for all prehitorie ŷ ˆΛ of all point y Y, where we recall that W := W r (x) Λ. Indeed, from the tranitivity of f on Λ, there exit a poitive integer m and a point z Y Λ uch that f m (z) B(x, αε/2) Λ. Take now a complete prehitory ŷ ˆΛ of an arbitrary point y from Y. From the fact that Y i contained in a ball of radiu αε/4, we can conclude that 16

17 Wε/2 (z) W ε/2 u (ŷ) and denote thi interection (which i a point) by ξ. From the local product tructure ξ belong to Λ. We have alo that f m (ξ) Wε (f m z) Λ. Take now f m ξ to be the prehitory in Λ of f m ξ given by (f m ξ, f m 1 ξ,..., ξ, ξ 1,...), where ˆξ := (ξ, ξ 1,...) i the prehitory of ξ ε/2-hadowed by ŷ; uch a prehitory of ξ exit ince ξ Wε/2 u (ŷ). So, we get that there exit a local untable manifold Wε/2 u ( f m ξ) which interect Wε/2 (x) in a point ζ; again from the local product tructure, ζ Λ and ince ζ W ε/2 (x), we obtain that ζ W. If we conider ζ m the m-th preimage of ζ obtained from the fact that ζ W u ε/2 ( f m ξ), we will have d(ζ m, ξ) < ε/2. Combining with the fact that ˆξ correpond to a prehitory of ξ ε/2-hadowed by ŷ, it follow that ζ m W u ε (ŷ) f m W. We may denote the point ζ m alo by ζ m (ŷ) when we want to emphaize it dependence on ŷ. Therefore, we proved that the et f m W interect all untable manifold W u ε (ŷ) for all prehitorie ŷ ˆΛ of point y from Y. From the fact that ζ W u ε/2 ( f m ξ), it follow that d(ζ m, ξ) < ε/2, d(fζ m, fξ) < ε/2,..., d(ζ, f m ξ) < ε/2. But ξ Λ and Λ i f- invariant, hence d(ζ, Λ) < ε/2,..., d(ζ m, Λ) < ε/2 (20) Let u denote by J m the et of thee point ζ m (ŷ) obtained for all the prehitorie ŷ of point y Y. Relation (20), together with the fact that ζ Λ W u ε/2 ( f m ξ) imply that ζ m (ŷ) Λ, therefore J m Λ. The relation in ( 20) imply alo that f m i injective on a neighbourhood of J m, ince ε < d(λ, C f )/4 and f j (J m ) C f =, j = 0,..., m. And, from our contruction, f m (J m ) W. But from above f m i injective on a neighbourhood of J m and it i bi-lipchitz on that neighbourhood, hence HD(J m ) HD(W ) = δ (x). Recall alo that t > δ (x), o t > HD(J m ). Thi mean that for any γ, 0 < γ < ε, there exit an open cover of J m with ball, U = (U i ) i I, uch that diamu i < γ and for a fixed n, n n(ε, η). (diamu i ) t < ε t+1 λ 4n χ n, (21) i I Let u chooe now an arbitrary i I and aume that Card(U i J m ) > 1. Let u denote by Y i the et of point y of Y which have ome prehitory ŷ with W u ε (ŷ) J m U i ; denote by F i the et of prehitorie ŷ ˆΛ with thi property. For each point z U i J m, there exit then a point y Y i and a prehitory ŷ ˆΛ uch that z W u ε (ŷ), and actually z = ζ m (ŷ). Therefore z ha a prehitory ẑ given by that procedure, i.e which i ε-hadowed by ŷ; thi prehitory may alo be denoted by ẑ (ŷ) if we want to emphaize it dependence on ŷ. Let alo F i := {ẑ (ŷ), ŷ F i }. Let u now take a prehitory ẑ F i. Since ε wa aumed ufficiently mall, we can define local branche of f 1 on ball of radiu ε. Let u denote by f 1 the branch of f 1 defined on B(z, ε) uch that f 1 (z ) = z 1. It may happen that the U i increae. In cae diamf 1 U i < ε, define afterward the invere iterate f 2 uch diameter of f 1 that f 2 (z ) = z 2, etc. Let u denote by n i(ẑ ) the larget integer n which i a multiple of n and for which diamf k (U i ) < ε, 0 k n, where ẑ = ẑ (ŷ) for ome ŷ F i ˆΛ a above. We do thi for all the point of U i J m and denote by n i the larget integer n i (ẑ ) for all z U i J m and 17

18 all prehitorie ẑ from F i. Obviouly we cannot tretch the open et U i in backward time forever, while keeping the diameter of it invere iterate maller than ε, hence n i i finite. Alo, n i, n i (ẑ ) are multiple of n, o they can be written a n i = nm i, n i (ẑ ) = nm i (ẑ ). In addition, for a point z U i J m and a prehitory ẑ F i, we will define alo the integer n i(ẑ ) a the mallet integer (not necearily a multiple of n) uch that diamf n i(ẑ ) U i > ε. We remark that the definition imply the inequalitie n i (ẑ ) n i (ẑ ) n i (ẑ ) + n, for any point z J m U i and any prehitory ẑ F i. We hall cover now the et Y i with et of type Λ(C, ε), where C C n (i.e C are prehitorie with repect to f n ). In order to do thi, take an arbitrary z 1 2 U i J m and a prehitory ẑ = ẑ (ŷ) F i, which correpond to ome complete (infinite) prehitory C = ŷ F i; by 1 2 U i we undertand the ball with the ame center a U i and with half it radiu. Then conider the m i (ẑ )- prehitory C of y (prehitory with repect to f n ), coming from the prehitory C, i.e we have C = (y, y n,..., y nmi (ẑ )). Recall that z W u ε/2 (ŷ). From the definition of n i(ẑ ) we ee immediately that U i P 2 (C, ε), and alo y Λ(C, ε). Recall that C i an m i (ẑ )- prehitory with repect to f n. Hence, ince N 0 (ε) i the mallet cardinality of a cover of Λ with ball of radiu ε, and ince n i = nm i i the larget integer of the form n i (ẑ ), we can cover the et Y i with at mot N 0 (ε) m i et of the form Λ(C, ε), where C are prehitorie for f n of length n(c ), with n(c ) m i. We will denote by Γ i the et of prehitorie C ued for the lat covering. So we have Y i Λ(C, ε), C Γ i and Γ i C n, n(c ) m i, C Γ i. Thi contruction can be done for every i I and, for each uch i, we have CardΓ i N 0 (ε) m i. But we proved that, for all ŷ ˆΛ, the local untable manifold W u ε (ŷ) interect J m ; on the other hand J m U i. In concluion, Y Y i, hence Y Λ(C, ε). Uing thi cover of i I i I i I C Γ i Y with et Λ(C, ε), C C n, we will etimate M f (0, (t + η)φ n, Y, N, ε) for ome large integer N n choen o that n(c ) N, C i I Γ i : M f (0, (t + η)φ n, Y, N, ε) exp(s n n(c ) (t + η)φ n(c )) i I C Γ i Let u invetigate now what i the relation between diamu i and exp(s n(c ) (t+η)φ n(c )), C Γ i. From the definition of n i (ẑ ) we know that it repreent the larget integer n, multiple of n, uch that diamf k (U i ) < ε, 0 k n, with f k being the invere branch determined by ẑ. Alo, n i (ẑ ) repreent the mallet integer (not necearily multiple of n) uch that diamf n i(ẑ ) U i > ε, where the invere branche f k were defined along the prehitory ẑ = ẑ (C). We conider now what happen to U i when taking invere iterate. Let z be another point in 1 2 U i Λ, and ζ the interection between Wr (z ) and the untable manifold Wr u (ẑ ); from the local product tructure ζ Λ. Then, ince U i i a ball, we get diamf n i(ẑ ) (Wr (z ) U i ) = contant Df n i(ẑ ) (z n i (ẑ ) 1 diamu i, and diamf n i(ẑ ) (Wr (z ) U i ) = contant Df n i(ẑ ) (ζ ni (ẑ )) 1 diamu i, due to the bounded ditortion property from Propoition 4. But ince ζ Wr u (ẑ ) and ˆζ i the prehitory of ζ following ẑ, we ee that the ditance d(z j, ζ j) decreae exponentially when 18

19 j increae; thu due to the fact that Df (z) depend Lipchitz continuouly on z (Theorem 1), we get that Df n i(ẑ ) (ζ ni (ẑ )) and Df n i(ẑ ) (z n i (ẑ )) are the ame up to a contant independent of z. Therefore up to a contant factor independent of n, C, we will obtain for every i I that: diamu i > ε exp(s n i (ẑ ) φ (C )) ε exp(s m i (ẑ ) φ n(c ))λ n, (22) where we conidered firt the n i (ẑ )-prehitory C := (y, y 1,..., y ni (ẑ )), (prehitory with repect to f, induced by the full prehitory C := ŷ), and then the m i (ẑ )-prehitory C := (y, y n,..., y nmi (ẑ )), (prehitory with repect to f n, induced by the ame complete prehitory C). We ued alo in ( 22) the fact that n i (ẑ ) n i (ẑ ) + n. We make alo the obervation that, from the definition of n(c ), for any two prehitorie C, C Γ i, we have exp(s n(c ) φ n(c )) i the ame a exp(s n( C) φ n( C)), up to a factor of le than χ n. Thu in the etimate below, we may a well ue for C a prehitory where n i i attained (called maximal n i - prehitory), i.e uch that n i (ẑ ) = n i = nm i. Therefore by employing alo ( 22) and the fact that CardΓ i N 0 (ε) m i, we can etimate M f n (0, (t + η)φ n, Y, N, ε) a follow: M f n (0, (t + η)φ n, Y, N, ε) i I i I ε t η (diamu i ) t exp(s m i (ẑ ) φ n(c )) η λ n(t+η) C Γ i [N 0 (ε) mi exp(s m i (ẑ ) φ n(c )) η ]ε t η (diamu i ) t λ n(t+η) χ nη (23) i I [N 0 (ε) χ nη ] m i χ nη ε t η (diamu i ) t λ n(t+η) where we ued in the econd inequality a maximal n i -prehitory, i I. In the above equence of inequalitie, we ued alo that 0 < η < 1, 0 < t < 3. But n i = nm i, o ( 23) implie that M f n (0, (t + η)φ n, Y, N, ε) ε t 1 i I ε t 1 λ 4n (diamu i ) t [N 0 (ε)χ ηn χ n i I ] m i λ 4n (diamu i ) t [N 0 (ε)χ ηn χ n ] m i (24) But from (19) and ince n n(ε, η), we ee that N 0 (ε)χ ηn < 1. From the way of chooing the cover U in ( 21), we have alo i I(diamU i ) t < ε t+1 λ 4n χ n. In concluion inequality ( 24) become M f n (0, (t + η)φ n, Y, N, ε) < 1 (25) Since γ and conequently diamu i, i I can be taken a mall a we wih, we ee that n(c ) can alo be made arbitrarily large, for C i I Γ i. Therefore if γ 0, N can be taken arbitrarily large, and ( 25) implie that M f n (0, (t+η)φ n, Y, ε) = 0. Thu one can conclude that P f n ((t+η)φ n, Y, ε) 0, for 0 < η < 1 and n n(ε, η). But let u alo remember that Y wa jut the interection between 19