Uniqueness of u-Gibbs Measures for Hyperbolic Skew Products on \(\mathbb {T}^4\)

We study the u-Gibbs measures of a certain class of uniformly hyperbolic skew products on \(\mathbb {T}^4\) ${\mathbb{T}}^4$ . These systems have a strong unstable and a weak unstable direction. Among such skew products, we show the existence of a subset which is \(C^r\) $C^r$ -dense and \(C^2\) $C^2$ -open for which every u-Gibbs measure is SRB. In particular, there is only one such measure. As an application, we obtain the minimality of the strong unstable foliation.