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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 absolutelycontinuous 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 belowin 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$\lambda_1(x)\geq\lambda_2(x)\geq...\geq\lambda_{dim\,M}(x)$ are the Lyapunovexponents at $x$ with respect to $\mu.$ Consequently, by using a dichotomy forgeneric volume-preserving diffeomorphism we show that Pesin's entropy formulaholds for generic volume-preserving diffeomorphisms, which generalizes a resultof Tahzibi in dimension 2.

Dominated Splitting and Pesin's Entropy Formula