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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?

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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.

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  • $\begingroup$ Thank you very much for the answer. A quick question: what do you mean by hyperbolicity of $g$ is weaker then the hyperbolicity of $f$? $\endgroup$
    – Adam
    Commented Mar 3, 2022 at 10:20
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    $\begingroup$ To talk about u-Gibbs you need to have a strong unstable direction, so you can have a splitting of the form E^s + E^c + E^u for example. In the example I mentioned, you want g to give the dynamics in the center direction. For the system f x g to really be partially hyperbolic, you want ||Df|_E^s|| < m(Dg) \leq ||Dg|| <|| Df|_E^u||, where m(Dg) is the conorm. That is what I meant by Hyperbolicity of g be weaker than Hyperbolicity of f. $\endgroup$ Commented Mar 4, 2022 at 22:10
  • $\begingroup$ Thank you very much. $\endgroup$
    – Adam
    Commented Mar 6, 2022 at 16:56

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