On the hybrid control of metric entropy for dominated splittings

Let \begin{document}$f$\end{document} be a \begin{document}$C^1$\end{document} diffeomorphism on a compact Riemannian manifold without boundary and \begin{document}$\mu$\end{document} an ergodic \begin{document}$f$\end{document} -invariant measure whose Oseledets splitting admits domination. We give a hybrid estimate from above for the metric entropy of \begin{document}$\mu$\end{document} in terms of Lyapunov exponents and volume growth. Furthermore, for any \begin{document}$C^1$\end{document} diffeomorphism away from tangencies, its topological entropy is bounded by the volume growth.


1.
Introduction. In dynamical systems, the notion of entropy plays a crucial role in measuring the complexity of evolutions. A system with positive entropy is considered to be chaotic. In general it is a difficult task to calculate the value of entropy due to its technical definitions. Thus estimates of entropy in various settings interest us frequently.
If f is further a C 1+α diffeomorphism, in 1980, Przytycki [15] gave another estimate from above of h µ (f ) in terms of the volume growth: where (D x f ) ∧i is the map on the i-th exterior algebra of the tangent space T x M induced by D x f . Together with the well known variational principle, (2) in fact gives rise to an upper bound of the topological entropy of f : where (D x f ) ∧ = max 1≤i≤dim M (D x f ) ∧i is the map on the exterior algebra of the tangent space T x M induced by D x f . Besides, Newhouse [12] has extended (3) to all C 1+α maps and Kozlovski [8] established the inverse part of (3) for all C ∞ maps.
In the present paper, we plan to give a hybrid estimate of the metric entropy for systems with dominated splittings, replacing the Hölder condition of the derivative in the estimate of the volume growth. Here, a dominated splitting Noting that the domination on any f -invariant set can be naturally extended to its closure, we consider dominated splittings over the support supp(µ) with respect to every µ ∈ M erg (M, f ).
Usually, there is no necessary connection between Lyapunov exponent and volume growth except when µ is just the volume measure Leb. When µ is Leb and ergodic, by the concavity of log t, one has Therefore, for a C 1 generic f in Diff 1 Leb (M ) which denotes the space of volumepreserving C 1 diffeomorphisms on M , by the domination property along orbits of Leb-almost all points in Theorem 1 of [3] and the ergodicity statement in Theorem A of [2], the Oseledets splitting of Leb is dominated, which combing with the Pesin entropy formula in the domination situation [18], implies Some results in the setting of C 1 diffeomorphisms with domination corresponding to the C 1+α Pesin theory have been obtained, for example, the stable manifold theorem [1] and Pesin entropy formula [18]. However, in some cases, it also may happen that the results are not parallel: in [10], it is proved that the entropy map is upper semi-continuous in the setting of C 1 diffeomorphisms with domination but not upper semi-continuous in the setting of C 1+α . Concerning the hybrid control of metric entropy, assuming a partially hyperbolic splitting T M = E cs ⊕ E u over the whole manifold M , Saghin [17] showed that the metric entropy with respect to an ergodic measure µ is bounded from above by the volume growth on E u plus the maximum λ + E cs between zero and the maximal Lyapunov exponent λ E cs on E cs multiplied by dim E cs . Here we further obtain the hybrid control of metric entropy from the patially hyperbolic splitting over M to the dominated splitting over the support.
The upper bound in Theorem 1.2 consists of two parts: the Lyapunov exponent and the volume growth. To deduce the term on Lyapunov exponent, we don't use the methods in the proof of Ruelle inequality and Pesin entropy formula but directly estimate the average maximal expanding rate on the E-bundle. For the estimate on volume growth, we use the domination to obtain invariant C 1 plaques with uniform size, which inherit the dynamical properties of tangent bundles (i.e., derivatives). Theorem 1.1 is applicable for any C 1 diffeomorphism f away from tangencies and any µ ∈ M erg (M, f ) ( [11,4,20]): the Oseledets bundle with negative (positive) exponents and the Oseledets bundle with nonnegative (nonpositive) exponents are dominated by Proposition 3.4 of [11], hence the term on Lyapunov exponent can be taken trivially in Theorem 1.1. Together with the variational principle, we actually get the C 1+α theorem of Przytycki [15] in the setting away from tangencies. Corollary 1.3. Let f be a C 1 diffeomorphism on a compact Riemannian manifold M without boundary and away from tangencies, then be its Oseledets splitting, which exists for µ-a.e., x ∈ M . We first prove The- where the minimal norm m(A) for a linear map A is defined by inf v =1 Av . By Oseledets theorem, for any η > 0, For any sufficiently small ε > 0, we can take N large such that µ(Q N,η ) > 1 − ε/4. Let Λ = Q N,η in the following discussions.
Note that E + = F 1 ⊕F 2 . By the domination T supp(µ) M = E − ⊕ < E + , we are able to construct continuous families of C 1 plaques: {W * x : x ∈ supp(µ)}, * ∈ {E − , E + } by [6]. Precisely, there is some β > 0 such that denotes the angle between E 1 and E 2 , v ∧ w is the wedge product of two vectors v, w. Then for any γ ∈ (0, β/4) (the value of γ will be specified later), there exist 0 < r 1 < r 0 such that for any x ∈ supp(µ), there exist C 1 submanifolds W * x with the following properties: (i) almost tangency: T x W * x = * (x) and T y W * x lies in a cone of width γ of * (x) for y ∈ W * x (x, r 0 ), where V (z, ρ) denotes the ball on a submanifold V centered at z with radius ρ; . For studying dynamics in neighborhoods, we take C 1 local foliations F x on B(x, r 0 ) for every x ∈ supp(µ), with leaves whose tangent bundles lie in a cone of width γ of W E + x (Note that we don't require invariance of F, because invariant foliations may not be C 1 smooth [5]). Given a foliation F and a point y in the domain, we denote by F(y) the leaf through y. We have that F x is uniformly absolutely continuous: there exist continuous functions, In the above, by the compactness of supp(µ), we can choose a constant a 0 > 1 such that a −1 0 < p x , q x < a 0 for any x ∈ supp(µ), and for any ball of radius ρ inside F x (or W E − x ) with respect to the induced metric in F x (or W E − x ) has l-volume between a −1 0 ρ l and a 0 ρ l , where l = dim E + (or dim E − ). Moreover, for any ρ 1 , ρ 2 ∈ (0, r 0 ), any submanifold V containing y ∈ W E − x (x, ρ 1 ) with tangent bundles in γ-cone of E + (x), it holds that the ball B V (y, ρ 2 ) in V with center y and raduis ρ 2 is contained in B(x, ρ 1 + ρ 2 ).
To estimate local dynamical growth, we adopt the concept of Bowen ball. For each x ∈ M , n ∈ N, ρ > 0, define n-step Bowen ball by B n (x, ρ, f ) = {y ∈ M : By the invariance property (ii) we could also define Bowen ball along W * x for any x ∈ supp(µ) and ρ ≤ r 1 by W * . As g is C 1 , we could take 0 < γ 1 < γ 2 , 0 < r 1 < r 0 small enough and find some b 0 ∈ (0, r 1 ) such that for any d(y, z) ≤ b 0 , any unit vectors v j ) ≤ γ 1 , 1 ≤ i 1 = i 2 ≤ i, k = 1, 2, 1 ≤ j ≤ i, the following holds: We let the γ from the construction of the local invariant plaques and the local foliations be in (0, γ 1 ). Observing µ may be not ergodic for g, we give a lemma to deal with this problem.

XUFENG GUO, GANG LIAO, WENXIANG SUN AND DAWEI YANG
Next we estimate the growth of Bowen balls along F : , g), there exists a connected subset R x (y, n, ρ, g) ⊂ F x (y) containing x and satisfying (i) R x (y, n, ρ, g) ⊂ B n (x, 2ρ, g); Proof. First, take R 0 as a ball in F x (y) centered at y with radius ρ. By domination, g(R 0 ) is a submanifold with tangent bundles in γ-cone of E + (g(x)). By induction, we suppose R i has been defined, 0 ≤ i < n − 1. If g i (x) ∈ Λ = Q N,η , then g has expanding rate ≥ e −2N η on R i which implies we can choose a ball R i+1 in g(R i ) centered at g i+1 (y) whose radius is ρe −2N η times the radius of R i ; otherwise, choose R i+1 as the ball in g(R i ) centered at g i+1 (y) whose radius is ρe −N a1 times the radius of R i . Continue the process until R n−1 is defined. As the frequency in Λ is no less than 1 − 2ε 1/2 , we deduce the radius of R n−1 is at least ρe −2(1−2ε 1/2 )nN η−2ε 1/2 nN a1 . Let R x (y, n, ρ, g) = g −(n−1) (R n−1 ). Then by domination, for any i ∈ [0, n), g i (R x (y, n, ρ, g)) belongs to R i whose tangent bundles lie in γ-cone of E + (g i (x)). Hence, g i (R x (y, n, ρ, g)) ⊂ B(g i (x), 2ρ) and we have vol dim E + (g (n−1) (R x (y, n, ρ, g)))