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arXiv · 2112.11149

On positive Lyapunov exponents and SRB measures for partially hyperbolic systems

Reza Mohammadpour
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Abstract

In this paper we consider $C^{1}$ diffeomorphisms on compact Riemannian manifolds admitting a dominated splitting $E^{cs} \oplus E^{cu}$. First, we prove that the smallest Lyapunov exponent along $E^{cu}$, computed with respect to the Lebesgue measure, is computable using observable measures. Then we show that if the Lyapunov exponents along $E^{cu}$ are positive Lebesgue almost everywhere and $E^{cu}$ admits a finest 1-dominated splitting on the support of an ergodic observable measure then $f$ is non-uniformly expanding along $E^{cu}$. As a byproduct, every $C^{1+\alpha}$ diffeomorphism exhibiting a dominated splitting $E^{s} \oplus E^{cu}$ where $E^{cu}$ fulfills the previous assumptions admits an SRB measure.

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BibTeXRIS

Reza Mohammadpour. 2021-12-21. On positive Lyapunov exponents and SRB measures for partially hyperbolic systems. https://arxiv.org/abs/2112.11149

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