Open sets of partially hyperbolic skew products having a unique SRB measure

TL;DR

This paper establishes the existence of open sets of dissipative, partially hyperbolic skew product systems with a unique SRB measure, by classifying and ruling out certain $u$-Gibbs measures through a measure rigidity approach.

Contribution

It introduces a measure rigidity framework for $u$-Gibbs measures in partially hyperbolic skew products, leading to the identification of open sets with unique SRB measures.

Findings

Existence of $C^2$-open sets with unique SRB measures

Classification of $u$-Gibbs measures near a Berger-Carrasco example

Ruling out multiple $u$-Gibbs measures in the neighborhood

Abstract

In this paper we obtain $C^2$-open sets of dissipative, partially hyperbolic skew products having a unique SRB measure with full support and full basin. These partially hyperbolic systems have a two dimensional center bundle which presents both expansion and contraction but does not admit any further dominated splitting of the center. These systems are non conservative perturbations of an example introduced by Berger-Carrasco. To prove the existence of SRB measures for these perturbations, we obtain a measure rigidity result for $u$-Gibbs measures for partially hyperbolic skew products. This is an adaptation of a measure rigidity result by A. Brown and F. Rodriguez Hertz for stationary measures of random product of surface diffeomorphisms. In particular, we classify all the possible $u$-Gibbs measures that may appear in a neighborhood of the example. Using this classification, and ruling out some of the possibilities, we obtain open sets of systems, in a neighborhood of the example, having a unique $u$-Gibbs measure which is SRB.