DIMENSION ESTIMATES FOR NON-CONFORMAL REPELLERS AND CONTINUITY OF SUB-ADDITIVE TOPOLOGICAL PRESSURE

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1 DIMENSION ESTIMATES FOR NON-CONFORMAL REPELLERS AND CONTINUITY OF SUB-ADDITIVE TOPOLOGICAL PRESSURE YONGLUO CAO, YAKOV PESIN, AND YUN ZHAO Abstract. Given a non-conformal repeller Λ of a C +γ map, we study the Hausdorff dimension of the repeller and continuity of the sub-and supperadditive topological pressure of sub-additive singular valued potentials. Such a potential always possesses an equilibrium state. We then use an extended a substantially modified version of Katok s approximating argument, to construct a compact invariant set on which the corresponding dynamical quantities (such as Lyapunov exponents and metric entropy) are close to that of the equilibrium measure. This allows us to establish continuity of the subadditive topological pressure and obtain a sharp lower bound of the Hausdorff dimension of the repeller. The latter is given by the zero of the super-additive topological pressure.. Introduction Dimension is an important characteristic of invariant sets and measures of dynamical systems, (see the books [5, 6, 6, 39, 4] where the role of dimension in the theory of dynamical systems is well explained). In this regard, different versions of dimension have been put forward to characterize various dynamical phenomena. However, computing these fractal invariants is usually a challenging problem, because they depend on the microscopic structure of the set. For repellers of conformal expanding maps Bowen [0] and Ruelle [42] found that their Hausdorff dimension is a solution of an equation involving topological pressure. More precisely, if Λ is an isolated compact invariant set of a C +γ conformal expanding map f and f Λ is topologically mixing, then the Hausdorff dimension of Λ is given by the unique root s of the following equation known as Bowen s equation: P (f Λ, s log D x f ) = 0, (.) where P (f Λ, ) denotes the topological pressure. Moreover, the equilibrium measure corresponding to the potential function s log D x f has dimension s and hence, is the measure of maximal dimension. In this conformal setting, other dimension characteristics such as lower and upper box dimensions are equal to s and the topological pressure P (f Λ, log D x f ) Mathematics Subject classification: 37C45, 37D35, 37H5 Date: July 7, 208. Key words and phrases. expanding map, repeller, topological pressure, non-uniform hyperbolicity theory, dimension. The first author is partially supported by NSFC (7737, ), the second author is partially supported by NSF grant DMS , the third author is partially supported by NSFC (790274).

2 is continuous with respect to f in the C topology, hence, so is the Hausdorff dimension. Furthermore, one can also establish analytic dependence of the Hausdorff dimension on the maps with some specific classes of invariant sets called dynamically defined sets, see Ruelle s work on hyperbolic Julia sets [42], which in turn was inspired by Bowen s work on the limits sets for quasi-fuchsian groups [0]. In the case of a basic set Λ for an Axiom A diffeomorphism f on a compact surface M, the dependence of the Hausdorff dimension under small perturbations of f was shown to be continuous by McCluskey and Manning [32] and to be C by Mañé [3] (cf. also [26]). Further, Pollicott [40] showed the analytic dependence of the Hausdorff dimension of the basic set for real analytic Smale horseshoe maps. See also [29, 46, 49] for related works. The case of non-conformal repellers, which we consider in this paper, is drastically different. In this case the Hausdorff dimension of the repeller may not be equal to its (lower or upper) box dimension and there may not be any invariant measure of maximal dimension (see [5]). The study of dimension in this case is a substantially more complicated problem and to approach it different notions of topological pressure have been introduced, which allow one to obtain some upper bounds on the dimension. In [3] Barreira proved that formula (.) holds in a variety of settings. In [8], Falconer defined the topological pressure for sub-additive potentials and obtained the variational principle under a bounded distortion condition. He also proved that the zero of the topological pressure of sub-additive singular valued potentials gives an upper bound of the Hausdorff dimension of repellers. In [52], Zhang introduced a new version of Bowen s equation which involves the limit of a sequence of topological pressures for singular valued potentials, and proved that the unique solution of this equation is an upper bound of the Hausdorff dimension of repellers. Finally, in [2], using thermodynamic formalism for sub-additive potentials developed in [], the authors showed that the zero of the topological pressure of sub-additive singular valued potentials Φ f (t) = ϕ t (, f n ) n (see precise definition in Section 3.) gives an upper bound of the Hausdorff dimension of repellers, and furthermore, that the upper bounds obtained in the previous works [2, 8, 52] are all equal. We refer the reader to [3] and [7] for a detailed description of the recent progress in dimension theory of dynamical systems. We also note that in the case of iterated function system, the zero of topological pressure of sub-additive singular valued potentials always gives an upper bound for the dimension of the set, and sometimes gives the exact value of the dimension, see [7] and [45] for details. Recently, Feng and Shmerkin [2] have proved that the sub-additive topological pressure of singular valued potentials is continuous in the class of generic self-affine maps. Consequently, the dimension of typical self-affine sets is also continuous. This resolves a folklore open problem in fractal geometry. The motivation for this paper is two-fold. First, in view of Zhang s result [52], it is interesting to know whether there is a lower bound for the Hausdorff dimension of a non-conformal repeller that can be obtained as the zero of topological pressure. A natural way to proceed is to replace the sub-additive singular valued potentials with super-additive ones (see Section 3. the definition). However, in doing so one faces a difficult problem: it is not known whether the corresponding superadditive topological pressure defined in a usual way via separated sets satisfies the variational principle for a general topological dynamical system, although it is true for some special systems, see [2] for details. To overcome this difficulty, we define 2

3 the super-additive topological pressure via variational principle (see Section 2.4). This allows us to obtain a lower bound on the Hausdorff dimension of repellers for a general C +γ expanding map, see Theorem 3., which to the best of our knowledge, is the sharpest lower bound currently known. Second, in view of Feng and Shmerkin s result [2], it is natural to ask whether the topological pressure of sub-additive singular valued potentials is continuous with respect to the dynamics f in an appropriate topology. One of our main result shows that the map f P (f Λ, Φ f (t)) is continuous in the C topology within the class of C +γ expanding maps f, see Theorem 3.5. Further, we show that the unique root t(f, Λ) of Bowen s equation P (f Λ, Φ f (t)) = 0 is exactly the Carathéodory singular dimension of the repeller (see Section 4 for details). Thus replacing the Hausdorff dimension of a repeller in the non-conformal setting with its appropriately chosen Carathéodory singular dimension, one obtains a precise value of the dimension. Furthermore, our result on continuity of topological pressure thus implies that the Carathéodory singular dimension also varies continuously with the dynamics, see Theorem 4.. Our main innovation in obtaining lower bound on the dimension and on establishing continuity of topological pressure, which distinguishes our approach from previous ones, is based on some powerful new results in non-uniform hyperbolicity theory for non-invertible maps. Therefore, we require that f is of class of smoothness C +γ. These new results are given by Theorems 5. and 5.2 and are of independent interest in hyperbolicity theory. The first one is in the spirit of Katok s approximation argument, see [27] or [28, Supplement S.5] but we need a substantially stronger version of it. Namely, we show that in the setting of expanding maps, given an invariant ergodic measure µ, there is a compact invariant set K with dominated splitting whose expansion rates are close to the Lyapunov exponents of µ; moreover, the topological entropy of f K is close to the metric entropy of µ (see Theorem 5.). Further, a similar result holds for any sufficiently small perturbation g of f: there is a compact set K invariant under g with dominated splitting whose expansion rates are close to the Lyapunov exponents of µ; moreover, the topological entropy of g K is close to the metric entropy of µ (see Theorem 5.2). We shall apply these theorems in the proofs of our two main results: ) a lower bound for the Hausdorff dimension of the repeller (see Section 6.2.3) where we choose µ to be sufficiently close to the unique equilibrium measure for the singular valued supper-additive potential Ψ f (s); and 2) continuity of sub-additive topological pressure (see Theorem 3.4) where we choose µ to be a unique equilibrium measure for the singular valued sub-additive potential Φ f (t). Results similar to our Theorem 5. were obtained in some particular situations by ) Misiurewicz and Szlenk [36] for continuous and piecewise monotone maps of the interval; 2) Przytycki and Urbański [4] for holomorphic maps in the case of a measure with only positive Lyapunov exponent; 3) Persson and Schmeling [37] for dyadic Diophantine approximations. For C +γ maps results related to Katok s approximation construction (and hence, in some way to our Theorem 5.) were obtained by Chung [4], Yang [50], Gelfert [24, 25]. See also [33, 34, 35, 43, 44, 30] that represent works close to this topic. For general C +γ diffeomorphisms (i.e., invertible maps) Theorem 5. was shown by Avila, Crovisier, and Wilkinson in []. While their construction of compact 3

4 invariant set is based on the shadowing lemma, our approach is more geometrical and gives the desired compact set via a Cantor-like construction. The advantage of our approach in the settings of expanding maps is that it allows us to treat the case of non-invertible maps and obtain a similar result for small C perturbations thus proving Theorem 5.2. The paper is organized as follows. In Section 2, we recall various notions of topological pressure and dimension. In particularly, we define the super-additive topological pressure via variational relation and discuss some properties of superadditive topological pressure. In Section 3 we state our main results on dimension estimates and continuity of the topological pressure and dimension. In particular, given a (non-conformal) repeller, we prove that the zero of topological pressure of the super-additive singular valued potential gives the lower bound on its Hausdorff dimension and the zero of topological pressure of the sub-additive singular valued potential gives the upper bound on its upper box dimension (under some additional bounded distortion condition). In Section 4, we introduce the notion of Carathéodory singular dimension of a repeller and prove that it depends continuously on the map. In Section 5, we prove our two main technical results discussed above on construction of an invariant compact subset with dominated splitting and large topological entropy for the map f and its small perturbations. Section 6 contains the proofs of our main results. In the last section 7 we discuss continuity of topological pressure for the case of matrix cocycles over subshifts of finite type. Acknowledgements. The authors would like to thank Professor Dejun Feng and Wen Huang for their suggestions and comments. Ya. P. wants to thank Mittag- Leffler Institute and Department of Mathematics of Weizmann Institute of Science where part of this work was done for their hospitality. 2. Notations and Preliminaries 2.. Dimensions of sets and measures. We recall some notions and basic facts from dimension theory, see the books [6] and [39] for detailed introduction. Let X be a compact metric space equipped with a metric d. Given a subset Z of X, for s 0 and δ > 0, define Hδ(Z) s := inf U i s : Z U i, U i δ i i where denotes the diameter of a set. The quantity H s (Z) := lim δ 0 H s δ (Z) is called the s dimensional Hausdorff measure of Z. Define the Hausdorff dimension of Z, denoted by dim H Z, as follows: dim H Z = infs : H s (Z) = 0 = sups : H s (Z) =. Further define the lower and upper box dimensions of Z respectively by dim B Z = lim inf δ 0 log N(Z, δ) log δ and dim B Z = lim sup δ 0 log N(Z, δ), log δ where N(Z, δ) denotes the smallest number of balls of radius δ needed to cover the set Z. Clearly, dim H Z dim B Z dim B Z for each subset Z X. 4

5 If µ is a probability measure on X, then the Hausdorff dimension and the lower and upper box dimension of µ are defined respectively by dim H µ = inf dim H Y : Y X, µ(y ) =, dim B µ = lim inf dim B Y : Y X, µ(y ) δ, δ 0 dim B µ = lim inf dim B Y : Y X, µ(y ) δ. δ 0 The following inequalities are immediate dim H µ dim B µ dim B µ. Finally, we define the lower and upper pointwise dimensions of the measure µ at the point x X by d µ (x) = lim inf r 0 log µ(b(x, r)) log r and d µ (x) = lim sup r 0 where B(x, r) = y X : d(x, y) < r. In particular, if there exists a number s such that log µ(b(x, r)) lim = s r 0 log r for µ-almost every x X, then dim H µ = s, see [5]. log µ(b(x, r)), (2.) log r 2.2. Repellers for expanding maps. Let f : M M be a smooth map of a m 0 -dimensional smooth Riemannian manifold M and let Λ := Λ f be a compact f-invariant subset of M. We call Λ a repeller for f and f expanding if () there exists an open neighborhood U of Λ such that Λ = x U : f n (x) U, for all n 0; (2) there is κ > such that D x fv κ v, for all x Λ, and v T x M, where is the norm induced by the Riemannian metric on M Markov partitions and Gibbs measures. Let Λ be a repeller of a C expanding map f. Assume that f Λ is topologically transitive. Then there exists a partition P,..., P k of Λ which satisfies () P i and int (P i ) = P i ; (2) int (P i ) int (P j ) = if i j; (3) for each i the set f(p i ) is the union of some of the sets P j from the partition. Here int( ) denotes the interior of a set relative to Λ. Such a partition is called Markov (see [9] for proofs). A repeller admits a Markov partition into subsets of arbitrary small diameter, and in particular, we may assume that the restriction of f on each P i is injective. Given a Markov partition P,..., P k, a sequence i = (i 0 i... i n ) where i j k is called admissible, if f(p ij ) P ij+ for j = 0,,..., n 2; and we write i = n for the length of the sequence i. If i = (i 0 i... i n ) is an admissible sequence, we define the cylinder P i0i...i n = n j=0 f j (P ij ). (2.2) Denote by S n the collection of all admissible sequences of length n. 5

6 We call a (not necessarily invariant) Borel probability measure µ on Λ a Gibbs measure for a continuous function ϕ on Λ if there exists C > 0 such that for every n N, every admissible sequence (i 0 i... i n ), and x P i0i...i n we have C µ(p i0i...i n ) exp[ np + S n ϕ(x)] C, where P is a constant and S n ϕ(x) = n j=0 ϕ(f j x) Sub-additive and supper-additive topological pressures. See [4] and [9, 20] for more details. Consider a continuous transformation f : X X of a compact metric space X equipped with metric d. Denote by M(X, f) and M e (X, f) the set of all f-invariant and respectively, ergodic Borel probability measures on X. A sequence of continuous functions (potentials) Φ = ϕ n n is called sub-additive, if ϕ m+n ϕ n + ϕ m f n, for all m, n. Similarly, we call a sequence of continuous functions (potentials) Ψ = ψ n n super-additive if Ψ = ψ n n is sub-additive. For x, y X and n 0 define the d n -metric on X by d n (x, y) = maxd(f i (x), f i (y)) : 0 i < n. Given ε > 0 and n 0, denote by B n (x, ε) = y X : d n (x, y) < ε Bowen s ball centered at x of radius ε and length n and we call a subset E X (n, ε)-separated if d n (x, y) > ε for any two distinct points x, y E. Given a sub-additive sequence of continuous potentials Φ = ϕ n n, let P n (Φ, ε) = sup e ϕn(x) : E is an (n, ε) separated subset of X. x E The quantity P (f, Φ) = lim lim sup ε 0 n n log P n(φ, ε) (2.3) is called the sub-additive topological pressure of Φ. One can show (see []) that it satisfies the following variational principle: P (f, Φ) = sup h µ (f) + F (Φ, µ) : µ M(X, f), F (Φ, µ), (2.4) where h µ (f) is the metric entropy of f with respect to µ and F (Φ, µ) = lim ϕ n dµ. (2.5) n n Existence of the above limit can be shown by the standard sub-additive argument. Given a super-additive sequence of continuous potentials Ψ = ψ n n, we define the super-additive topological pressure of Ψ by P var (f, Ψ) := sup h µ (f) + F (Ψ, µ) : µ M(X, f). (2.6) Note that for any super-additive sequence of continuous potentials and any f- invariant measure µ we have F (Ψ, µ) = lim ψ n dµ = sup ψ n dµ. n n n Similarly to the definition of the sub-additive topological pressure one can define the notion of super-additive topological pressure P (f, Ψ) using (n, ε)-separated sets. 6

7 Remark 2.. It is natural to ask whether the variational principle for P (f, Ψ) holds, i.e., P (f, Ψ) = sup h µ (T ) + F (Ψ, µ) : µ M(X, f). Given a continuous function ϕ, set ϕ n = n i=0 ϕ f i. The above definitions of sub/super-additive topological pressures recover the standard notion of topological pressure of a single continuous potential (see [47] for details), which we denote by P (f, ϕ). The following result establishes some useful relations between the three notions of the pressure. Proposition 2.. Let Ψ = ψ n n be a super-additive sequence of continuous potentials on X. Then P var (f, Ψ) = lim P (f, ψ n n n ) = lim n n P (f n, ψ n ). Remark 2.2. The same results hold for sub-additive topological pressure if the entropy map is upper semi-continuous, see [2]. However, we do not need this condition in the case of super-additive topological pressure defined by variational relation. 3. Main Results: bounds on dimensions and continuity of pressure Unless otherwise stated, throughout this section we assume that f : M M is a C +γ expanding map with a repeller Λ such that f Λ is topologically transitive. In this section we establish a lower bound for the Hausdorff dimension as well as an upper bound for the upper box dimension of Λ by using super-additive and sub-additive topological pressures respectively. 3.. Singular valued potentials. Given x Λ and n, consider the differentiable operator D x f n : T x M T f n (x)m and denote its singular values in the decreasing order by For s [0, m 0 ], set and α (x, f n ) α 2 (x, f n ) α m0 (x, f n ). (3.) [s] ψ s (x, f n ) := ψ s (D x f n ) := log α i (x, f n ) + (s [s]) log α [s]+ (x, f n ) (3.2) ϕ t (x, f n ) := ϕ t (D x f n ) := i= m 0 i=m 0 [t]+ log α i (x, f n )+(t [t]) log α m0 [t](x, f n ) (3.3) for t [0, m 0 ]. Since f is smooth, the functions x α i (x, f n ), x ψ s (x, f n ) and x ϕ t (x, f n ) are continuous. It is easy to see that for all n, l N It follows that the sequences of functions ψ s (x, f n+l ) ψ s (x, f n ) + ψ s (f n (x), f l ), ϕ t (x, f n+l ) ϕ t (x, f n ) + ϕ t (f n (x), f l ). Ψ f (s) := ψ s (, f n ) n and Φ f (t) := ϕ t (, f n ) n (3.4) are respectively, super-additive and sub-additive. We call them respectively supperadditive and sub-additive singular valued potentials. 7

8 We consider the super-additive and respectively sub-additive pressure functions given by P sup (s) := P var (f, Ψ f (s)) and P sub (t) := P (f, Φ f (t)), (3.5) where P var (f, Ψ f (s)) is the supper-additive topological pressure of the potential Ψ f (s) and P (f, Φ f (t)) is the sub-additive topological pressure of the potential Φ f (t). It is obvious from the definition of super-and sub-additive topological pressures that P sup (s) and P sub (t) are continuous and strictly decreasing in s, respectively, in t Lower bound for the Hausdorff dimension of repellers. For repellers of C expanding maps the following result from [2] provides an upper bound for the Hausdorff dimension of the repeller. Proposition 3.. Let Λ be a repeller for a C expanding map f. Then the zero of the sub-additive pressure function P sub (t) gives an upper bound for the Hausdorff dimension of Λ. In view of this result it is natural to ask whether Given a C expanding map f of an m 0 -dimensional smooth Riemannian manifold M with a repeller Λ, is it true that dim H Λ s, where s is the unique root of the equation P sup (s) = 0? We give here an affirmative answer to this question assuming that f is C +γ for some γ > 0. The reason we need higher regularity of the map is that we will utilize some results from non-uniform hyperbolicity theory. Theorem 3.. Let Λ be a repeller for a C +γ expanding map f : M M. Then dim H Λ s, where s is the unique root of the equation P sup (s) = 0. As an immediate corollary of this theorem we obtain the following result that gives lower bound for the Hausdorff dimension of the repeller as the zero of the topological pressure of the potential ψ s (, f), given by (3.2). Corollary 3.. We have that dim Λ s where s is the unique root of the equation P (f, ψ s (, f)) = Upper bound for the box dimension of repellers. In view of Proposition 3., we shall show here that for expanding maps of regularity higher than C and under some additional assumptions on singular values, the zero of sub-additive pressure function gives the upper bound for the box dimension of Λ. Theorem 3.2. Assume that there is C > 0 such that for every i m 0, n N and every x, y P i0i...i n, Then where P i0i i n P sub (t) = 0. C α i(x, f n ) α i (y, f n ) C. dim B Λ t, is a cylinder (see (2.2)) and t is the unique root of the equation 8

9 3.4. Continuity of sub-additive topological pressure. Let f : M M be a C +γ map of an m 0 -dimensional smooth Riemannian manifold M with metric d, which admits a repeller Λ f. Let be the norm on the tangent space T M that is induced by the Riemannian metric on M. Let h be a C map that is sufficiently C close to f. It is well known that h admits a repeller Λ h such that (Λ f, f) and (Λ h, h) are topologically conjugate that is there is a homeomorphism π : Λ f Λ h which is close to the identity map and which commutes f and h, i.e., π f = h π. Recall that Φ h (t) is the sub-additive singular valued potential for h for each 0 t m 0 (see (3.4)). Our first result establishes upper semi-continuity of sub-additive topological pressure of singular valued potential with respect to small C perturbations. We stress that our result only requires f to be C. Theorem 3.3. Let f : M M be a C expanding map of an m 0 -dimensional smooth Riemannian manifold M, and Λ f a repeller for f. Then for each 0 t m 0 the map f P (f Λf, Φ f (t)) is upper semi-continuous, i.e., if h is a C map that is C close to f, then lim sup P (h Λh, Φ h (t)) P (f Λf, Φ f (t)). h f We now establish lower semi-continuity of sub-additive topological pressure of singular valued potential. Note that for this we require f and its small perturbations to be of class C +γ for some γ > 0. Theorem 3.4. Let f : M M be a C +γ expanding map of an m 0 -dimensional smooth Riemannian manifold M, and Λ f a repeller for f. Then for each 0 t m 0 the map f P (f Λf, Φ f (t)) is lower semi-continuous, i.e., if h is a C +γ map that is C close to f, then lim inf P (h Λ h, Φ h (t)) P (f Λf, Φ f (t)) h f Combining Theorems 3.3 and 3.4, we have the following result. Theorem 3.5. Let f : M M be a C +γ expanding map of a m 0 -dimensional smooth Riemannian manifold M, and Λ f a repeller for f. Then for each 0 t m 0 the map f P (f Λf, Φ f (t)) is continuous, i.e., if h is a C +γ map that is C close to f, then lim h f P (h Λ h, Φ h (t)) = P (f Λf, Φ f (t)). 4. Carathéodory singular dimension In this section we discuss an approach to the notion of sub-additive topological pressure which is more general than the one presented in Section 2.4 and which is based on the Carathéodory construction described in [39]. This more general approach allows one to introduce sub-additive topological pressure for arbitrary subsets of the repeller (not necessarily compact or invariant). Let Λ be a repeller for an expanding map f and let Φ = ϕ n n be a subadditive sequence of continuous functions on Λ. Given a subset Z Λ and s R, let m(z, Φ, s, r) := lim inf exp ( sn i + sup ϕ ni (y) ), (4.) N i 9 y B ni (x i,r)

10 where the infimum is taken over all collections B ni (x i, r) of Bowen s balls with x i Λ, n i N that cover Z. It is easy to show that there is a jump-up value P Z (f, Φ, r) = infs : m(z, ϕ, s, r) = 0 = sups : m(z, Φ, s, r) = +. The quantity P Z (f, Φ) = lim inf P Z(f, Φ, r) (4.2) r 0 is called the topological pressure of Φ on the subset Z. Since the map f is expanding, the entropy map µ h µ (f) is upper semicontinuous and the topological entropy is finite. This implies that () P Λ (f, Φ) = P (f Λ, Φ) i.e., the topological pressure on the whole repeller given by (4.2) is the same as the topological pressure defined by (2.3) (see [, Proposition 4.4]); (2) P Λ (f, Φ, r) = P (f Λ, Φ) for all small r > 0. We consider the case when Φ is the sub-additive singular valued potential, i.e., Φ = ϕ α (, f n ) n. Let P (α) = P (f Λ, Φ) and let α 0 be the zero of Bowen s equation P (α) = 0. It follows from (4.) that m(z, Φ, P (α 0 ), r) = lim inf exp ( sup ϕ α0 (y, f ni ) ), N i y B ni (x i,r) where the infimum is taken over all collections B ni (x i, r) of Bowen s balls with x i Λ, n i N that cover Z. This motivates us to introduce the following notion: for every Z Λ let m(z, α) := lim inf exp ( sup ϕ α (y, f ni ) ), N i y B ni (x i,r) where the infimum is taken over all collections B ni (x i, r) of Bowen s balls with x i Λ, n i N that cover Z. It is easy to see that there is a jump-up value dim C (Z) := infα : m(z, α) = 0 = supα : m(z, α) = +, (4.3) which we call the Carathéodory singular dimension of Z. We shall show that for Z = Λ the Carathéodory singular dimension of Z is exactly the zero of Bowen s equation P (α) = 0 and that it varies continuously with f. Theorem 4.. Let f : M M be a C +γ expanding map with repeller Λ f. Assume that f Λ f is topologically transitive. Then () dim C Λ f = α 0 where α 0 is the unique root of Bowen s equation P (α) = 0; (2) if h is a C +γ map h that is C close to f, then dim C (Λ h ) varies continuously with h. 5. Approximating Lyapunov exponents of expanding maps In this section we show that given an ergodic measure µ on a repeller Λ for a C +γ expanding map f, its Lyapunov exponents can be well approximated by Lyapunov exponents on a compact invariant subset which carries sufficiently large topological entropy. This statement will serve as a main technical tool in proving our main theorems. We stress that the requirement that the map is of class of smoothness C +γ is crucial as it allows us to utilize some powerful results of nonuniform hyperbolicity theory. 0

11 We first recall the definition of dominated splitting. Consider a C +γ diffeomorphism of a compact smooth manifold M of dimension m 0 and let Λ M be a compact invariant set. We say that Λ admits a dominated splitting if there is continuous invariant splitting T Λ M = E F and constants C > 0, λ (0, ) such that for each x Λ, n N, 0 u E(x), and 0 v F (x) D x f n (u) u Cλ n D xf n (v). v We write E F if F dominates E. Further, given 0 < l m 0, we call a continuous invariant splitting T Λ M = E E l dominated if there are numbers λ < < λ l such that for every j l, exp(n(λ j ε)) D x f n Ei exp(n(λ j + ε)). (5.) In particular, E E l. We shall also use the term λ j -dominated when we want to stress dependence of the numbers λ j. In what follows M is a m 0 - dimensional smooth Riemannian manifold. Theorem 5.. Let f be a C +γ expanding map of M which admits a repeller Λ, and let µ be an ergodic measure on Λ with h µ (f) > 0. Then for any ε > 0 there exists an f-invariant compact subset Q ε Λ such that the following statements hold: () h top (f Qε ) h µ (f) ε; (2) there is a λ j (µ)-dominated splitting T x M = E E 2 E l over Q ε where λ (µ) < < λ l (µ) are distinct Lyapunov exponents of f with respect to the measure µ. Consider a C +γ map h : M M that is sufficiently C close to f. Then h is expanding, has a repeller Λ h, and the maps f Λ f and h Λ h are topologically conjugate. We shall show that there is a compact invariant subset Q ε (h) of Λ h of large topological entropy such that the Lyapunov exponents of µ and of any h-invariant ergodic measure ν with support in Q ε (h) are close. Theorem 5.2. Let (f, µ) be the same as in Theorem 5.. Given a sufficiently small ε > 0, there is δ > 0 such that for any C +γ map h : M M that is δ-c close to f there exists a compact invariant subset Q ε (f) Λ h with the following properties: () h top (h Qε(h)) h µ (f) ε; (2) there is a λ j (µ)-dominated splitting T x M = E E 2 E l over Q ε (h) where λ (µ) < < λ l (µ) are distinct Lyapunov exponents of f with respect to the measure µ. (3) for any h-invariant ergodic measure ν with support in Q ε (h) and for each j m 0, there is k = k(j) such that λ k (µ) λ j (ν) ε. We begin with the proof of Theorem 5. and we split it into few steps. 5.. Preliminaries on Lyapunov exponents. Let f : ( M, µ) ( M, µ) denote the inverse limit of f : (M, µ) (M, µ) and let π i : M M be the projection defined by π i (x n n Z ) = x i for any x n n Z M. For each x = x n M, f( x) = f(x n ) n Z, and the inverse of f is given by f (x n ) = y n, where y n = x n.

12 Let Λ = x = x n M : x n Λ, for all n. It is easy to see that f( Λ) = Λ, and there exists an f-invariant ergodic measure µ supported on Λ such that (π 0 ) µ = µ. It is standard to check that with respect to the cocycle D x0 f n n, for µ-almost every x = x n Λ, the Lyapunov exponents are the same as those for the system f : (M, µ) (M, µ). The following result expresses Oseledets Multiplicative Ergodic Theorem for the system f : ( M, µ) ( M, µ). Proposition 5.. There exists a full µ-measure Borel set Λ such that for every x = x n there is an invariant splitting of the tangent space T x0 M T x0 M = E ( x) E 2 ( x) E r( x) ( x) (5.2) and numbers 0 < λ ( x) < λ 2 ( x) < < λ r( x) ( x) < + and m i ( x) (i =, 2,..., r( x)) such that dim E i ( x) = m i ( x) with r( x) i= m i( x) = m 0 and lim n ± n log T 0 n ( x)v = λ i ( x) for each i and for all 0 v E i ( x), where D x0 f n, if n > 0, T0 n ( x) = Id, if n = 0, (D xn f n ), if n < 0. Moreover, for any i j, 2,, r( x) the angle between E i ( f( x)) and E j ( f( x)) satisfies that lim n ± n log (E i( f n ( x)), E j ( f n ( x))) = 0. The splitting (5.2) is called the Oseledets splitting. Since µ is ergodic, the numbers r( x), λ i ( x) and m i ( x) for all i are in fact constants for µ-almost every point x. Hence, in the following we will denote them simply by r, λ i and m i respectively. For each x there exists a sequence of norms x,n + n= such that the following facts hold (see [8] for details): for all sufficiently small ε > 0, () for each i r, all v E i ( f n x), and n, l Z (2) for all v T xn M, e lλi l ε v x,n D xn f l v x,n+l e lλi+ l ε v x,n ; where A( x, ε) satisfies for all n Z and x. In particular, for all v E i ( x) we have Given k, consider the regular set m0 v v x,n A( x, ε)e ε n 2 v, A( f n ( x), ε) e ε n 2 A( x, ε) e λi ε v x,0 Dfv x, e λi+ε v x,0 ; e λi ε v x,0 Df v x, e λi+ε v x,0. Λ k = x : A( x, ε) e εk. 2

13 The standard results of non-uniform hyperbolicity theory (see [8]) show that Λ Λ 2 Λ k Λ k+ and for every k one has f ± (Λ k ) Λ k+ and µ(λ k ) as k. Hence, for every δ > 0, there exists K N such that µ(λ k ) > δ for every k K. We would like to point out that f is uniformly hyperbolic on each fixed set Λ k, and hence the angle between different E i ( x) is uniformly bounded away from zero. We now recall some properties of Lyapunov charts ϕ x : x Λ for f, see [8] for the proofs. Proposition 5.2. There exists an f-invariant Borel set Λ 0 Λ with full µ-measure such that for any ε > 0 there exist ) a Borel function l : Λ 0 [, + ) satisfying l( f ± ( x) e ε l( x) for all x Λ 0 ; and 2) a collection of embedding ϕ x : R(l( x) ) M such that () ϕ x (0) = x 0 and R mi ( x) := (D 0 ϕ x ) (E i ( x)) (i =,..., r) are mutually orthogonal in R m0 ; (2) For each n Z, let H ṋ x = ϕ f n ( x) f n ϕ x be the connecting map between the chart at x and the chart at f n ( x). Then for each n Z, i r, and v R mi ( x) we have e nλi n ε v D 0 H ṋ x (v) enλi+ n ε v, where is the usual Euclidean norm on R m0 ; (3) let Lip(g) denote the Lipschitz constant of the function g, then Lip(H x D 0 H x ) ε, Lip(H x D 0 H x ) ε and Lip(DH x ) l( x), Lip(DH x ) l( x); (4) For every v, v R(l( x) ), one has K 0 d(ϕ x(v), ϕ x (v )) v v l( x)d(ϕ x (v), ϕ x (v )) for some universal constant K 0 > Extending Oseledets splitting. Given x Λ k and m N, we can extend the invariant splitting (5.2) and the Lyapunov metric in the tangent space to a neighborhood B(π 0 f i ( x), (re 2ε e ε(k+i) ) α ) of π 0 f i ( x) for i = 0,,..., m, where r = Km 0 and K is the Hölder constant of the differentiable operator Df. To see this for every z B(π 0 f i ( x), (re 2ε e ε(k+i) ) α ) and v T z M we translate the vector v to a corresponding vector v T π0 f i ( x) M and we set v z = v x, i. We thus obtain a splitting of the tangent space at z T z M = E (z) E r (z), by translating the splitting at the point π 0 f i ( x) T π0 f i ( x) M = E (π 0 f i ( x)) E r (π 0 f i ( x)). Let fx i denote the corresponding inverse branch of the map f B(xi,δ 0) where δ 0 is chosen such that the map f B(x,δ0) is invertible. Then we have the following results. Lemma 5.. Let x Λ k and m N +. Assume that for each i =, 2,..., m, 3

14 () f π 0 f i ( x) (y) B(π 0 f i ( x), (re 2ε e ε(k+i+) ) α ); (2) y B(π 0 f i ( x), (re 2ε e ε(k+i) ) α ); (3) f(y) B(π 0 f i+ ( x), (re 2ε e ε(k+i ) ) α ). Then for every v E j (y) (j =,..., r) we have where y = f π 0 f i ( x) (y). e λj 2ε v y D y fv f(y) e λj+2ε v y, e λj 2ε v y D y f π 0 f i ( x) v y eλj+2ε v y, Proof of the lemma. For every v E i (y), D y f(v) f(y) = D y f(v) x, i+ D π0 f i ( x) f(v) x, i+ + D y f(v) D π0 f i ( x) f(v) x, i+ D π0 f i ( x) f(v) x, i+ + e εk e εi D y f(v) D π0 f i ( x) f(v) e λj+ε v x, i + e εk e εi K[d(π 0 f i ( x), y)] α v e λj+2ε v y. (5.3) Similarly, one can show that D y fv f(y) e λj 2ε v y. The second estimate can be proven in a similar fashion. Let x Λ k and m N +. Assume that Conditions ()-(3) in Lemma 5. hold for i =, 2,..., m. We define two cones as follows: V i (, ξ) = (v, v 2 ) : v E E i, v 2 E i+ E r, v 2 < ξ v, and U i (, ξ) = (w, w 2 ) : w E E i, w 2 E i+ E r, w < ξ w 2, where = f (y), y, f(y) and ξ > 0 is a small number. The cones have π 0 f i ( x) the following properties. Lemma 5.2. The cones are invariant in the following sense: for some λ > 0, D y f π 0 f i ( x) V i(y, ξ) V i (y, λξ) and D y fu i (y, ξ) U i (f(y), λξ) Moreover, for every v U i (y, ξ), and for every v V i (y, ξ), where y = f π 0 f i ( x) (y). Proof of the lemma. Let D y fv f(y) e (λi+ 3ε) v y D y f π 0 f i ( x) v y e (λi+3ε) v y, ( ) D π0 f i ( x) f = C 0 0 C 22 and D y f = Using an argument similar to the one in (5.3), we have that ( ) D D 2. D 2 D 22 C D f(y) e (i )ε e kε m 0 K((re 2ε e ε(k+i) ) α ) α e ε 4

15 and that C 22 D 22 f(y) e ε. Note that D 2 f(y) e ε and D 2 f(y) e ε. Therefore, for every v = (v, v 2 ) U i (y, ξ), setting D y f(v) = (v, v 2), we obtain that v f(y) = D v + D 2 v 2 f(y) e λi+2ε v y + e ε v 2 y and (ξe λi+2ε + e ε ) v 2 y v 2 f(y) = D 2 v + D 22 v 2 f(y) e λi+ 2ε v 2 y e ε v y (e λi+ 2ε ξe ε ) v 2 y. Thus, v f(y) < λξ v 2 f(y) for some λ > 0. Hence, D y fu i (y, ξ) U i (f(y), λξ). Moreover, Df(v) f(y) = (v, v 2) f(y) = v 2 f(y) (e λi+ 2ε ξe ε ) v 2 y e λi+ 3ε v y. This proves the first inequality and the second one can be proven in a similar fashion. Assume that x Λ k, π 0 ( x) = x, and f m ( x) Λ k. Let fx m denote the corresponding inverse branch of the map f B(f m (x),δ) f B(x,δ), where δ is chosen such that the map f B(x,δ) is invertible for each x Λ. We have the following corollaries of Lemmas 5. and 5.2. Corollary 5.. Assume that x Λ k, π 0 ( x) = x, and f m ( x) Λ k. Then for every y B(f m (x), (re 2ε e εk ) α ), fx m (y) B(x, (re 2ε e ε(k+m) ) α ), and every v E i (y), v E i (fx m (y)) we have where y = fx m (y). e m(λi+2ε) v y D y f m x v y e m(λi 2ε) v y, e m(λi+ 3ε) v y D y f m v y e m(λi++3ε) v y, Corollary 5.2. Assume that x Λ k, π 0 ( x) = x, and f m ( x) Λ k. Then for every y B(f m (x), (re 2ε e εk ) α ) and fx m (y) B(x, (re 2ε e ε(k+m) ) α ) we have D y fx m V i (y, ξ) V i (y, λ m ξ), D y f m U i (y, ξ) U i (y, λ m ξ) and for each v V i (y, ξ), w U i (fx m (y), ξ) we have where y = fx m (y). D y fx m v y e m(λi+3ε) v y, D y f m w y e m(λi+ 3ε) w y, 5.3. Constructing a compact invariant set with dominated splitting. The aim of this section is to construct a compact invariant subset of Λ on which the topological entropy of f is close to h µ (f) and the tangent space over this subset admits a dominated splitting with rates given by Lyapunov exponents of µ. To achieve this we will produce a sufficiently large number of points that have distinct orbits of certain length. 5

16 By Katok s entropy formula (see [27]), for each δ (0, ) we have h µ (f) = lim lim inf log N(µ, n, ε, δ) = lim lim sup log N(µ, n, ε, δ), ε 0 n n ε 0 n n where N(µ, n, ε, δ) denotes the minimal number of Bowen s balls B n (x, ε) that are needed to cover a set of measure at least δ. Fix a number δ (0, ) and a small ε > 0. Then there exists ε > 0 such that for every ε ε one can find a number N satisfying: given a set A of measure > 2δ and a number n N, any (n, ε)-separated set E A of maximal cardinality satisfies Card E exp(n(h µ (f) ε)). (5.4) Let us start by choosing some ε (0, λ /3). There is a compact regular set Λ K Λ f with µ(λ K ) > δ and such that for every x = x n Λ K, every y B(π 0 ( x), (re 2ε e εk ) α ), and every i we have D y f i π 0( x) y e i(λ 3ε), (5.5) where y = f i π (y). Set 0( x) ρ = (re 2ε e εk ) α. (5.6) Consider a cover of π 0 (Λ K ) by balls B(x, ρ/4),..., B(x j, ρ/4) with centers x i π 0 (Λ K ), i =,..., j. Let P = P,..., P l, l j, be a finite measurable partition of M such that P(x i ) B(x i, ρ/4) for i =, 2,..., j. Here P(x) denote the element of partition P that contains x. Such a partition does exist, e.g., we may take the first j elements of P as follows: k P = B(x, ρ/4),..., P k = B(x k, ρ/4) \ P i for 2 k j. We have the following result. Lemma 5.3. Given numbers ε > 0 and δ > 0 and a positive measure set Λ K as above, there exist a positive integer N 2 = N 2 (ε, δ) and a compact subset Λ 2 Λ K (possibly depending on ε and δ) such that µ(λ 2 ) > 2δ and for every x Λ 2 and n N 2 we have π 0 f k ( x) P(π 0 ( x)) and f k ( x) Λ K i= for some number k [n, n + εn]. Proof of the lemma. The partition P = P,..., P l of M induces a partition P = P,..., P l of M given by P i = x : π 0 ( x) P i. Let τ = min i l ε, µ(λ K P i )/4. From Birkhoff s ergodic theorem we derive that for each i =,..., l and µ-almost every x we have lim n n Card k 0,..., n : f k ( x) Λ K P i = µ(λ K P i ). Using Egorov s theorem, we conclude that there exists a compact set Λ 2 of µmeasure at least 2δ such that the above convergence is uniform on Λ 2 for each i =,..., l. Hence, there is a number N 2 > 0 such that for every x Λ 2, every n N 2, and all i =,..., l we have Card k 0,..., n : f k ( x) Λ K P i µ(λ K P i )n τ 2 n. 6

17 Assume that N 2 is chosen large enough that min ( µ(λ K P i ) 3τ)N 2 ε >. Thus i l for every x Λ 2, every i =,..., l, and every n N 2 we have k n,..., n( + ε) : f k ( x) Λ K P i Card = Card k 0,..., n( + ε) : f k ( x) Λ K P i Card k 0,..., n : f k ( x) Λ K P i µ(λ K P i )n( + ε) n( + ε)τ 2 µ(λ K P i )n nτ 2 = nε( µ(λ K P i ) τ 2 ) 2nτ 2 nε( µ(λ K P i ) 3τ) >. Clearly, f k (π 0 ( x)) P i if f k ( x) P i. In particular, this is true for the index i with P i = P(π 0 ( x)). This gives the proof of Lemma. Choose n maxn, N 2 such that e n(λ 3ε) e εk m 0 < 8 (5.7) (recall that m 0 = dim M) and a maximal (n, ρ)-separated subset E π 0 (Λ 2 ). It is easy to see that x E B n(x, ρ) has µ-measure at least 2δ. We partition the set E into the set F k, n k < n( + ε)), defined by F k = x E : f k (x) P(x) that is, having the same return time k to their partition element. Let m be the index satisfying Card F m = max Card F k. n k<n+εn Since Card E = Card F k, n k<n+εn we obtain that (εn)card F m Card E. Observing that εn < e εn, we find using (5.4) that Card F m Card E e n(hµ(f) 2ε). εn Choose a point x i such that the corresponding element P(x i ) has the property that Card(F m P(x i )) is maximal. We then have Card(F m P(x i )) j Card F m j en(hµ(f) 2ε). Recall that exactly after m iterations each point x F m P(x i ) returns to P(x i ), and hence to B(x i, ρ/4). Recall also that for each such x there is x with π 0 ( x) = x so that f m ( x) Λ K. Given x F m P(x i ), let U x = fx m (B(x i, ρ/2)). Notice that f m (x) B(x i, ρ/4) B(f m (x), ρ/2) and by (5.6), B(f m (x), ρ/2) B(f m (x), (re 2ε e εk ) α ). Consequently, by (5.5), for every z B(x i, ρ/4) we have D z f m x f m x (z) e m(λ 3ε). 7

18 Therefore, Hence, by (5.7), D z f m x e m(λ 3ε) e εk m 0. diam U x = diam f m x (B(x i, ρ/2)) < e m(λ 3ε) e εk m 0 ρ < 8 ρ. Thus U x B(x, 8 ρ) and U x B(x i, ρ/2). For every two distinct points x, y F m P(x i ), there exists U x = fx m (B(x i, ρ/2)) and U y = fy m (B(x i, ρ/2)) so that U x B(x i, ρ/2), U y B(x i, ρ/2) and U x U y =. Otherwise, suppose that there exists z U x U y. Then d m (x, z) ρ/2 and d m (y, z) ρ/2, which contradicts to d m (x, y) d n (x, y) > ρ, since x, y F m are (n, ρ/2)-separated. For every x, y F m P(x i ), note that fy m (U x ) U y. Therefore, we can consider the following maps f m y (U x ) f m y U x f m x For every z U x, i =,..., r we have D z f m U i (z, ξ) U(f m (z), λ m ξ), D z f m y Given a point F m P(x i ), consider the map f m : f m y (U x ) f m B(f m (x), ρ). f m V i (z, ξ) V (f m (z), λ m ξ). y (U x ). We have for every z f m y (U x ) and every i =,..., r, D z f 2m U i (z, ξ) U(f 2m (z), λ 2m ξ), Let R ε,0 = B(x i, ρ/2) and for all l 0, R ε,l+ = x F m P(x i) D z f m y V i (z, ξ) V (f m (z), λ m ξ). f m x (R ε,l ). They form a family of nested non-empty compact sets and hence, define a nonempty compact set R ε = l R ε,l that is f m -invariant and also fx m -invariant for all x F m. We also have for every z R ε,l, i =,..., r, and every point F m, D z f lm U i (z, ξ) U(f lm (z), λ lm ξ), D z f m V i (z, ξ) V (f m (z), λ m ξ). Therefore, f m Rε is uniformly expanding and it is topologically conjugate to the one-sided full shift over an alphabet with Card F m P(x i ) symbols. This implies h top (f m Rε ) = log(card F m P(x i )). Let Q ε = R ε f(r ε ) f m (R ε ). We wish to show that Q ε is the desired compact set. Clearly, Q ε is f-invariant, and we have that h top (f Qε ) = m log(cardf m P(x i )). Using the fact that ( + ε)n > m n, we obtain h top (f Qε ) m log j + n m (h µ(f) 2ε) h µ (f) 3ε. 8

19 We now show existence of a λ j -dominated splitting over Q ε satisfying (5.). Let l = Card F m P(x i ). For every z R ε there is a unique sequence (y y 2... y n... ) Σ + l (here Σ+ l is the one sided full shift over l symbols) such that Hence, for every v T z M, Also, for all l 0 z = (fy m fy m 2 n>0 fy m n (B(x i, ρ/2))). D z fy m v f m y (z) e m(λ 3ε) v z. D z f lm U i (z, ξ) U i (f lm (z), λ lm ξ). Finally, D z fy m (z)v i (z, ξ) V i (fy m (z), λ m ξ). Therefore, there exists a continuous splitting on R ε, and for every v E i (z), T z M = E (z) E 2 (z) E r (z), e m(λi 3ε) v z D z f m v f m (z) e m(λi+3ε) v z. To see this note that for every z R ε, there is a unique sequence (σ σ 2... σ n... ) Σ + l such that z = Note that fσ m n fσ m 2 result it suffices to set ( E i (z) = (fσ m n>0 f m σ 2 fσ m n (B(x i, ρ/2))). fσ m (z) R ε for every n. To obtain the desired l 0 Df lm U i (f m σ ( l 0 D(f m σ ) fσ m l (z), ξ) ) fσ m l )V i (f lm (z), ξ) Constructing compact invariant sets with dominated splitting for small perturbations. We present a proof of Theorem 5.2. Following the construction of R ε, we may find a subset R ε (h) of Λ h that we will briefly recall. Start with the same F m P(x i ) as in the previous section. For every x F m P(x i ), let diamux h = diamh m x (B(x i, ρ/2)). One has that diam U h x < 8 ρ. Moreover, one has that Ux h B(x, 8 ρ) and U x h B(x i, ρ/2). For every two distinct points x, y F m P(x i ) we have that Ux h Uy h = and h m y (U x ) U y. Consider the following maps h m y (U h x ) h m y U h x h m x B(h m (x), ρ). For z U h x, i =,..., r the cones have the following properties: D z h m U i (z, ξ) U(h m (z), λ m ξ), D z h m V i (z, ξ) V (h m (z), λ m ξ). 9 y y

20 Let Rε,0 h = B(x i, ρ/2) and for all l 0, Rε,l+ h = x F m P(x i) h m x (Rε,l). h These sets form a family of nested non-empty compact sets, yielding a non-empty compact set R ε (h) = l 0 R h ε,l which is h m -invariant and also h m x -invariant for all x F m P(x i ). Note that h m Rε(h) and f m Rε are topologically conjugate, since both of them are topologically conjugate to the full shift over l symbols with l = Card F m P(x i ). We also have for every z Rε,l h, there is a unique sequence (σ σ 2... σ n... ) Σ + l such that z = n>0 h m σ and for every i =, 2,..., r one has h m σ 2 h m σ n (B(x i, ρ/2)) D z h lm U i (z, ξ) U(h lm (z), λ lm ξ) and D z h m σ V i (z, ξ) V (h m σ (z), λ m ξ). Hence, h m Rε(h) is uniformly expanding, and the set Q ε (h) = R ε (h) h(r ε (h)) h m (R ε (h)) is h-invariant. In addition, we have that h top (h Qε(h)) = h top (f Qε(f)) h µ (f) 3ε and the first statement of the theorem follows. observe that for every l > 0 we have To prove the second statement Dh lm (z)u i (z, ξ) U i (h lm (z), λ lm ξ), D z h m σ V i (z, ξ) V i (h m σ (z), λ m ξ). Hence, there exists a continuous splitting on R ε (h) so that for every v E i (z), T z M = E (z) E 2 (z) E r (z) e m(λi 4ε) v z D z h m v hm (z) e m(λi+4ε) v z. (5.8) Thus, the second statement follows immediately. The last statement of the theorem follows immediately from (5.8). 6. Proofs of Main Results 6.. Proof of Proposition 2.. Fix a positive integer m. Since F (Ψ, µ) = sup m ψm dµ, for every µ M(X, f) we have h µ (f) + ψ m dµ h µ (f) + F (Ψ, µ) P var (f, Ψ), m The variational principle for the topological pressure of a single continuous potential (see (2.4)) yields that It follows that P (f, ψ m m ) P var(f, Ψ). lim sup n P (f, ψ n n ) P var(f, Ψ). 20

21 On the other hand, for each µ M(X, f) we have that ) h µ (f) + F (Ψ, µ) = lim ψ n dµ n ( h µ (f) + n lim inf n P (f, ψ n n ). This yields that P var (f, Ψ) = sup h µ (f) + F (Ψ, µ) : µ M(X, f) lim inf P (f, ψ n n n ) and completes the proof of the first equality. To prove the second equality choose µ M(X, f) and note that µ M(X, f k ) for every k N. By the variational principle for the topological pressure (see (2.4)), for every µ M(X, f), k P (f k, ψ k ) k It follows that lim inf k ( h µ (f k ) + k P (f k, ψ k ) h µ (f) + lim k k Since µ is any measure in M(X, f), we have that lim inf k k P (f k, ψ k ) sup ) ψ k, dµ = h µ (f) + k ψ k dµ. ψ k dµ = h µ (f) + F (Ψ, µ). h µ (f) + F (Ψ, µ) : µ M(X, f) = P var (f, Ψ). For any k N and µ M(X, f k ) the measure ν := k k i=0 f µ i is f-invariant and h ν (f) = k h µ(f k ). Since ψ nk (x) n is super-additive with respect to f k, we have ψ nk dµ ψ k dµ. lim n n ψ nk dµ = sup n n For each 0 i k, the super-additivity of ψ n (x) n with respect to f implies that ψ nk (x) df µ i ψ k i (x) df µ i + ψ (n )k (f k i x) df µ i + ψ i (f nk i x) df µ i 2m + ψ (n )k (f k x) dµ = 2m + ψ (n )k (x) dµ, where m = max 0 i k ψ i. Summing over i from 0 to k, we obtain that k ψ nk (x) dν 2mk + k ψ (n )k (x) dµ. Dividing both sides by nk and letting n, we find that kf (Ψ, ν) ψ k dµ. This implies that P var (f, Ψ) h ν (f) + F (Ψ, ν) k (h µ(f k ) + ψ k dµ). Since µ M(X, f k ) can be chosen arbitrary, this yields that P var (f, Ψ) k P (f k, Ψ k ) and since k can be chosen arbitrarily, we obtain that P var (f, Ψ) lim sup k P (f k, ψ k ) k 2

22 and the second equality follows Proof of Theorem 3.. We split the proof of the theorem into few steps Dimension estimates under bounded distortion property. First, we prove the theorem in the case when the singular value of D x f has bounded distortion property on cylinders. Lemma 6.. Assume that there exists C > 0 such that for every i m 0, n N, and every x, y P i0i...i n, C α i(x, f n ) α i (y, f n ) C. Then dim H Λ f s, where s is the unique root of Bowen s equation P (f Λ, ψ s (, f)) = 0. Proof of the lemma. Since P (f Λ, ψ s (, f)) = 0 and ψ s (, f) is a Hölder continuous function on Λ, there exists a Gibbs measure µ such that ( K exp n i=0 ) ψ s (f i (x), f) µ ( ) ( P i0i i n K exp n i=0 ) ψ s (f i (x), f) for some constant K > 0 and every x P i0i...i n. Let m satisfy m s < m + and Q = i = (i 0 i i n ) : α m+ ((D x f n ) ) r, for all x P i0i i n ; but α m+ ((D y f n ) ) > r for some y P i0i i n. Here α i (D x f) denotes the i-th singular value of D x f. Therefore, we have br < α m+ ((D x f n ) ) r for all x P i0i...i n, where b = C min x Λ D x f. Let B be a ball of radius r and B a ball of radius 2r. Put Q = i Q P i B. Hence, for i Q we have P i B contains a parallelepiped of sides aα m0 ((D x f n ) ), aα m0 ((D x f n ) ),, aα ((D x f n ) ) with centers in B. It follows that a m0 (α m+ ((D x f n ) )) m0 m α m ((D x f n ) ) α ((D x f n ) ) vol m0 (P i B). Therefore, Note that i Q a 2 (α m+ ((D x f n ) )) m0 m α m ((D x f n ) ) α ((D x f n ) ) vol m0 ( B) 2 m0 a 3 r m0. (br) m0 s i Q a 2 (α m+ ((D x f n ) )) s m α ((D x f n ) ) i Q a 2 (α m+ ((D x f n ) )) s m+m0 s α ((D x f n ) ). 22

23 Hence, i Q a 2 (α m+ ((D x f n ) )) s m α ((D x f n ) ) a 4 r s. On the other hand, using supper-additivity of ψ s, one has µ(b) µ(p i0i...i n ) (i 0i...i n ) Q K K a 5 r s. (i 0i...i n ) Q exp ( n ) ψ s (f i (x), f) i=0 (i 0i...i n ) Q (α m+ ((D x f n ) )) s m α ((D x f n ) ) This implies that dim H µ s and hence, dim H Λ dim H µ s. Observe that an f 2k -invariant measure µ must be f 2k+ -invariant. This together with the super-additivity of ψ s (, f n ) n yields that for any f 2k -invariant measure µ, Hence, 2 P (f 2k+ k+ 2 k+ (h µ(f 2 k+ ) + 2, ψ s (, f 2k+ )) ψ s (x, f 2k ) dµ) = 2 k (h µ(f 2k ) + ψ s (x, f 2k )dµ). 2 P (f 2k+ k+ By Proposition 2., we have, ψ s (, f 2k+ )) 2 k P (f 2k, ψ s (, f 2k )). (6.) P sup (s) = lim k 2 k P (f 2 k, ψ s (, f 2k )) = P var (f Λ, ψ s (, f n )). (6.2) We shall show that under the same requirements as in the above lemma, one can obtain sharper dimension estimates. Lemma 6.2. Assume that there exists C > 0 such that for every i m 0, n N, and every x, y P i0i...i n, C α i(x, f n ) α i (y, f n ) C. Then dim H Λ s where s is the unique root of Bowen s equation P sup (s) = 0. Proof of the lemma. Note that for each n N, the set Λ is also a repeller for f 2n. Using Lemma 6., for every n N, one has dim H Λ s n where s n is the unique root of the equation P (f 2n Λ, ψ s (, f 2n )) = 0. By (6.), we have that s n s n+ and hence, there is a limit s := lim n s n. We have that dim H Λ s. It now follows from (6.2) that P sup (s ) = 0. 23