Hyperbolic SRB Measures and the Leaf Condition

Let M be a Riemannian, boundaryless, and compact manifold, with dim M >= 2 and let f be a C1+ 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.

Automated summary

In a Riemannian, boundaryless, and compact manifold with dim M >= 2, the existence of a hyperbolic SRB measure is linked to the presence of an unstable leaf containing hyperbolic points that return to a Pesin set with positive frequency. This observation provides an answer to a question raised by Pesin.