I have been reading the famous [paper of Alves, Bonatti, and Viana](https://link.springer.com/article/10.1007/s002220000057) where they proved that there is an SRB measure for partially hyperbolic systems. Since I am new to this field, I have some basic questions.


1)All results were proved for $C^2$ diff, but I saw at some talks that they mentioned their results for $C^{1+\varepsilon}.$ Is it true that their results work for $C^{1+\varepsilon}.$ I am aware that the limsup condition of their result can be replaced by the liminf condition (Alves, Luzzatto. etc), but I want to know whether all their results work for $C^{1+\varepsilon}$ or not.


2)Can one give an example that a Gibbs-u state is not a hyperbolic SRB measure?



SRB measure and Gibbs u-state
The results are true for $C^{1+}$. [This paper](https://arxiv.org/pdf/1403.2937.pdf) that you mentioned uses $C^{1+}$ regularity. Actually, in this business the main thing that usually comes from $C^{1+}$ condition is control of distortion along an unstable disc.


For you second question, consider the following. Let $f$ be a $C^{1+}$ Anosov diffeomorphism of $\mathbb{T}^2$. We know that such system has only one $u$-Gibbs measure that is the SRB measure $\mu\_f$.


Now, let $g$ be the north/South Pole on $S^1$ and suppose that the hyperbolicity of $g$ is weaker then the hyperbolicity of $f$, so that $f\times g$ is a partially hyperbolic diffeomorphism of $\mathbb{T}^3$. Let $p\_1$ be the North Pole (repeller for $g$) and $p\_2$ be the South Pole (attractor). The measure $\mu\_f \times \delta\_{p\_1}$ is $u$-Gibbs but it is not SRB. However the measure $\mu\_f \times \delta\_{p\_2}$ is $u$-Gibbs and SRB.


