Hyperbolic SRB measures and the leaf condition

Abstract

Let $M$ be a Riemannian, boundaryless, and compact manifold, with $\dim M\geq 2$ and let $f$ be a $C^{1+}$ diffeomorphism. We show that there is a hyperbolic SRB measure if and only if there exists an unstable leaf with a subset of positive leaf volume of hyperbolic points which return to some Pesin set with positive frequency. This answers a question of Pesin.