On The Uniqueness of SRB Measures for Endomorphisms

TL;DR

This paper demonstrates that weak hyperbolic transitivity guarantees the uniqueness of hyperbolic SRB measures, leading to ergodicity and stability results for certain dynamical systems.

Contribution

It improves existing results by establishing the link between weak hyperbolic transitivity and the uniqueness of SRB measures, with implications for ergodicity and stability.

Findings

Weak hyperbolic transitivity implies unique hyperbolic SRB measures.

Ergodicity is established in conservative systems under these conditions.

Conditions for stable and statistical stability in $C^2$-regular maps are identified.

Abstract

In this paper we improve the results of \cite{MT} and show that a weak hyperbolic transitivity implies the uniqueness of hyperbolic SRB measures. As an important corollary, it arises the ergodicity of the system in a conservative setting. It also arises the condition which implies the stable ergodicity as well as the statistical stability for a general $C^2$-regular map.