Unique SRB measures and transitivity for Anosov diffeomorphisms

TL;DR

This paper proves the uniqueness of SRB measures for $C^2$ Anosov diffeomorphisms and shows that all $C^1$ Anosov diffeomorphisms on compact manifolds are transitive, highlighting their ergodic properties.

Contribution

It establishes the uniqueness of SRB measures for $C^2$ Anosov diffeomorphisms and demonstrates transitivity for all $C^1$ Anosov diffeomorphisms on compact manifolds.

Findings

Unique SRB measures for $C^2$ Anosov diffeomorphisms

Basins cover Lebesgue almost every point

All $C^1$ Anosov diffeomorphisms are transitive

Abstract

We prove that every $C^2$ Anosov diffeomorphism in a compact and connected Riemannian manifold has a unique SRB and physical probability measure, whose basin of attraction covers Lebesgue almost every point in the manifold. Then, we use structural stability of Anosov diffeomorphisms to deduce that all $C^1$ Anosov diffeomorphisms on compact and connected Riemannian manifolds are transitive.