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

Sylvain Crovisier, Davi Obata, Mauricio Poletti

Published 2022-09-19Version 1

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.