# SRB measures for partially hyperbolic systems whose central direction is weakly expanding

### José F. Alves

Universidade do Porto, Portugal### Carla L. Dias

Instituto Politécnico de Portalegre, Portugal### Stefano Luzzatto

Abdus Salam International Centre for Theoretical Physics, Trieste, Italy### Vilton Pinheiro

Universidade Federal da Bahia, Salvador, Brazil

## Abstract

We consider partially hyperbolic $C^{1+}$ diffeomorphisms of compact Riemannian manifolds of arbitrary dimension which admit a partially hyperbolic tangent bundle decomposition $E^s \otimes E^{cu}$. Assuming the existence of a set of positive Lebesgue measure on which $f$ satisfies a weak nonuniform expansivity assumption in the centre unstable direction, we prove that there exists at most a finite number of transitive attractors each of which supports an SRB measure. As part of our argument, we prove that each attractor admits a Gibbs–Markov–Young geometric structure with integrable return times. We also characterize in this setting SRB measures which are liftable to Gibbs–Markov–Young structures.

## Cite this article

José F. Alves, Carla L. Dias, Stefano Luzzatto, Vilton Pinheiro, SRB measures for partially hyperbolic systems whose central direction is weakly expanding. J. Eur. Math. Soc. 19 (2017), no. 10, pp. 2911–2946

DOI 10.4171/JEMS/731