Dominated splitting and Pesin's entropy formula

Wenxiang Sun; Xueting Tian

doi:10.3934/dcds.2012.32.1421

Article Contents

2012, Volume 32, Issue 4: 1421-1434. Doi: 10.3934/dcds.2012.32.1421

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Dominated splitting and Pesin's entropy formula

Wenxiang Sun^1, and
Xueting Tian^2,

1.
LMAM, School of Mathematical Sciences, Peking University, Beijing 100871
2.
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190

Received: May 2010

Revised: September 2011

Published: April 2012

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Abstract

Let $M$ be a compact manifold and $f:\,M\rightarrow M$ be a $C^1$ diffeomorphism on $M$. If $\mu$ is an $f$-invariant probability measure which is absolutely continuous relative to Lebesgue measure and for $\mu$ $a.\,\,e.\,\,x\in M,$ there is a dominated splitting $T_{orb(x)}M=E\oplus F$ 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 $h_\mu(f)$ satisfies $$h_{\mu}(f)\geq\int \chi(x)d\mu,$$ where $\chi(x)=\sum_{i=1}^{dim\,F(x)}\lambda_i(x)$ and $\lambda_1(x)\geq\lambda_2(x)\geq\cdots\geq\lambda_{dim\,M}(x)$ are the Lyapunov exponents at $x$ with respect to $\mu.$
Consequently, we obtain that Pesin's entropy formula always holds for (1) volume-preserving Anosov diffeomorphisms, (2) volume-preserving partially hyperbolic diffeomorphisms with one-dimensional center bundle, (3) volume-preserving diffeomorphisms far away from homoclinic tangency, and (4) generic volume-preserving diffeomorphisms.

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Mathematics Subject Classification: 37A05, 37A35, 37D25, 37D30.

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