Uniqueness of $u$-Gibbs measures for hyperbolic skew products on $\mathbb{T}^4$

TL;DR

This paper proves that for a class of hyperbolic skew products on the 4-torus, the $u$-Gibbs measure is unique and SRB, leading to minimality of the strong unstable foliation, with results dense and open in certain topologies.

Contribution

It establishes the generic uniqueness and SRB property of $u$-Gibbs measures for hyperbolic skew products on $ ext{T}^4$.

Findings

$u$-Gibbs measures are unique and SRB in a dense and open set.

The strong unstable foliation is minimal under these conditions.

Results apply to systems with strong and weak unstable directions.

Abstract

We study the $u$-Gibbs measures of a certain class of uniformly hyperbolic skew products on $\mathbb{T}^4$. These systems have a strong unstable and a weak unstable directions. We show that $C^r$-dense and $C^2$-open in this set every $u$-Gibbs measure is SRB, in particular, there is only one such measure. As an application of this, we can obtain the minimality of the strong unstable foliation.