# Hyperbolic SRB measures and the leaf condition

## TL;DR

This paper establishes a criterion for the existence of hyperbolic SRB measures on compact manifolds, linking it to the presence of unstable leaves with positive leaf volume of hyperbolic points that frequently return to Pesin sets.

## Contribution

It provides a necessary and sufficient condition for hyperbolic SRB measures in terms of unstable leaves and their recurrence properties, answering a question posed by Pesin.

## Key findings

- Hyperbolic SRB measures exist if and only if certain unstable leaves have positive leaf volume of hyperbolic points.
- Unstable leaves with positive leaf volume of hyperbolic points are key to the existence of SRB measures.
- The result characterizes SRB measures via leaf recurrence and hyperbolic behavior.

## 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.