JournalsjemsVol. 19, No. 10pp. 2911–2946

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

We consider partially hyperbolic C1+C^{1+} diffeomorphisms of compact Riemannian manifolds of arbitrary dimension which admit a partially hyperbolic tangent bundle decomposition EsEcuE^s \otimes E^{cu}. Assuming the existence of a set of positive Lebesgue measure on which ff 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