# SRB measure and Gibbs u-state

I have been reading the famous paper of Alves, Bonatti, and Viana 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?

The results are true for $$C^{1+}$$. This paper 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.

• Thank you very much for the answer. A quick question: what do you mean by hyperbolicity of $g$ is weaker then the hyperbolicity of $f$?