Dominated Splitting and Pesin's Entropy Formula

Abstract

Let $M$ be a compact manifold and $f:M\to M$ 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 $$h_{\mu }(f)\ge \int \chi (x)d\mu ,$$ where $\chi (x)={\sum }_{i=1}^{dimF(x)}{\lambda }_i(x)$ and ${\lambda }_1(x)\ge {\lambda }_2(x)\ge ...\ge {\lambda }_{dimM}(x)$ are the Lyapunov exponents at $x$ with respect to $\mu .$ Consequently, by using a dichotomy for generic volume-preserving diffeomorphism we show that Pesin's entropy formula holds for generic volume-preserving diffeomorphisms, which generalizes a result of Tahzibi in dimension 2.

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