On positive Lyapunov exponents and SRB measures for partially hyperbolic systems

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 $C^{1+α}$ diffeomorphism exhibiting a dominated splitting $E^{s} \oplus E^{cu}$ where $E^{cu}$ fulfills the previous assumptions admits an SRB measure.