Dominated Splitting and Pesin's Entropy Formula

TL;DR

This paper establishes a lower bound for metric entropy in terms of Lyapunov exponents for certain invariant measures under $C^1$ diffeomorphisms with dominated splitting, and proves Pesin's entropy formula for generic volume-preserving cases.

Contribution

It provides a new estimation of entropy via Lyapunov exponents under dominated splitting and extends Pesin's entropy formula to generic volume-preserving diffeomorphisms.

Findings

Lower bound for metric entropy using Lyapunov exponents

Pesin's entropy formula holds for generic volume-preserving diffeomorphisms

Generalization of Tahzibi's result to higher dimensions

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 $$h_{\mu}(f)\geq\int \chi(x)d\mu,$$ 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 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.