Srb Measures For Almost Axiom A Diffeomorphisms

TL;DR

This paper establishes conditions under which an Almost Axiom A diffeomorphism admits a probability SRB measure, focusing on hyperbolic sets with positive measure and sufficiently long stable and unstable leaves.

Contribution

It proves the existence of SRB measures for Almost Axiom A diffeomorphisms under specific hyperbolic and measure conditions, extending understanding of statistical properties in such systems.

Findings

Existence of SRB measure under given conditions

Hyperbolic set with positive measure implies SRB measure

Long stable and unstable leaves are crucial for measure existence

Abstract

We consider a diffeomorphism f of a compact manifold M which is Almost Axiom A, i.e. f is hyperbolic in a neighborhood of some compact f-invariant set, except in some singular set of neutral points. We prove that if there exists some f-invariant set of hyperbolic points with positive unstable-Lebesgue measure such that for every point in this set the stable and unstable leaves are "long enough", then f admits a probability SRB measure.