Dominated Splitting and Pesin's Entropy Formula

Let $M$ be a compact manifold and $f:\,M\to M$ be a $C^1$ diffeomorphism on $M$. If $μ$ is an $f$-invariant probability measure which is absolutely continuous relative to Lebesgue measure and for $μ$ $a.\,\,e.\,\,x\in M,$ 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 $h_μ(f)$ satisfies $h_μ(f)\geq\int χ(x)dμ,$ where $χ(x)=\sum_{i=1}^{dim\,F(x)}λ_i(x)$ and $λ_1(x)\geqλ_2(x)\geq...\geqλ_{dim\,M}(x)$ are the Lyapunov exponents at $x$ with respect to $μ.$ 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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