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

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.

18 pages, 4 figures

References in corpus (2)

• Geometric properties of partially hyperbolic measures and applications to measure rigidity
• Rigidity of $\text{U}$-Gibbs measures near conservative Anosov diffeomorphisms on ${\mathbb{T}}^3$