Dominated Splitting and Pesins Entropy Formula

arXiv:1004.3441v2 [math.DS] 24 Aug 2010 Dominated Splitting and Pesin’s Entropy Formula Wenxiang Sun ∗ LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China E-mail: sunwx@math.pku.edu.cn Xueting Tian † School of Mathematical Sciences, Peking University, Beijing 100871, China E-mail: txt@pku.edu.cn Abstract Let M be a compact manifold and f : M →M be a C1 diffeomor- phism on M. If µ is an f-invariant probability measure which is abso- lutely continuous relative to Lebesgue measure and for µ a. e. x ∈M, there is a dominated splitting Torb(x)M = E ⊕F on its orbit orb(x), then we give an estimation through Lyapunov characteristic exponents from below in Pesin’s entropy formula, i.e., the metric entropy hµ(f) satisfies hµ(f) ≥ Z χ(x)dµ, where χ(x) = Pdim F (x) i=1 λi(x) and λ1(x) ≥λ2(x) ≥· · · ≥λdim M(x) are the Lyapunov exponents at x with respect to µ. Consequently, by us- ing a dichotomy for generic volume-preserving diffeomorphism we show that Pesin’s entropy formula holds for generic volume-preserving dif- feomorphisms, which generalizes a result of Tahzibi [12] in dimension 2. ∗Sun is supported by National Natural Science Foundation ( # 10671006, # 10831003) and National Basic Research Program of China(973 Program)(# 2006CB805903) † Tian is the corresponding author. Key words and phrases: metric entropy, Lyapunov exponents, Pesin’s entropy formula, dominated splitting AMS Review: 37A05, 37A35, 37D25, 37D30 1 1 Introduction To estimate metric entropy through Lyapunov exponents is an important topic in differential ergodic theory. In 1977 Ruelle [11] got from above an estimate of metric entropy of an invariant measure, and Pesin[10] in 1978 got from below an estimation of metric entropy of an invariant measure absolutely continuous relative to Lebesgue measure and thus got a so called Pesin’ entropy formula. Pesin’s proof is based on the stable manifold theorem. In 1980 Ma˜n´e [7] gave another ingenious and very simple proof without using the theory of stable manifolds. In 1985 Ledrappier and Young[4] generalized the formula to all SRB measures, not necessarily absolutely continuous relative to Lebesgue measure. There are also more generalizations[5, 6]. Pesin’s entropy formula by Pesin and by Ma˜n´e and by others assumes that not only the differentiability of the given dynamics is of class C1 but also that the first derivative satisfies an α-H¨older condition for some α > 0. It is interest- ing to investigate Pesin’s entropy formula under the weaker C1 differentiability hypothesis plus some additional condition, for example, dominated splitting. The aim of this paper is to prove that Pesin’s entropy formula remains true for invariant probability measure absolutely continuous relative to Lebesgue measure in the C1 diffeomorphisms with dominated splitting. In the proof of [7], the combination of the graph transform method (Lemma 3 there) and the distortion property deduced from the H¨older condition of the derivative play important roles. The domination assumption in our C1 diffeomorphism helps us to overcome much trouble. Our proof follows Ma˜n´e without using the theory of stable manifolds, as noted by Katok that it seems that Ma˜n´e’s proof can also be extended to the more general framework. Tahzibi showed in [12] that there is a residual subset R in C1 volume- preserving surface diffeomorphisms such that every system in R satisfies Pesin’s entropy formula. As an consequence our main Theorem 2.2 and a result of Bochi and Viana[2], we generalize the result of Tahzibi into any di- mensional case. 2 Results Before stating our main results we need to introduce the concept of dominated splitting. Denote the minimal norm of a linear map A by m(A) = ∥A−1∥−1. 2 Definition 2.1. Let f : M →M be a C1 diffeomorphism on a compact Remainnian manifold. (1). (Dominated splitting at one point) Let x ∈M and Torb(x)M = E ⊕F be a Df−invariant splitting on orb(x). Torb(x)M = E ⊕F is called to be N(x)-dominated at x, if there exists a constant N(x) ∈Z+ such that ∥Df N(x)|E(fj(x))∥ m(Df N(x)|F (fj(x))) ≤1 2, ∀j ∈Z. (2). (Dominated splitting on an invariant set) Let ∆be an f-invariant set and T∆M = E ⊕F be a Df−invariant splitting on ∆. We call T∆M = E ⊕F to be a N-dominated splitting, if there exists a constant N ∈Z+ such that ∥Df N|E(y)∥ m(Df N|F (y)) ≤1 2, ∀y ∈∆. For a Borel measurable map f : M →M on a compact metric space M and an f−invariant measure µ, we denote by hµ(f) the metric entropy. Now we state our results as follows. Theorem 2.2. Let f : M →M be a C1 diffeomorphism on a compact Re- mainnian manifold. Let f preserve an invariant probability measure µ which is absolutely continuous relative to Lebesgue measure. For µ a.e. x ∈M, denote by λ1(x) ≥λ2(x) ≥· · · ≥λdim M(x) the Lyapunov exponents at x. Let m(·) : M →N be an f-invariant measurable function. If for µ a. e. x ∈M, there is a m(x)-dominated splitting: Torb(x)M = Eorb(x) ⊕Forb(x), then hµ(f) ≥ Z χ(x)dµ, where χ(x) = Pdim F (x) i=1 λi(x). In particu

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