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?