Journal of the European Mathematical Society
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Published online: 2017-09-18
SRB measures for partially hyperbolic systems whose central direction is weakly expanding
José F. Alves[1], Carla L. Dias[2], Stefano Luzzatto[3] and Vilton Pinheiro (1) Universidade do Porto, Portugal(2) Instituto Politécnico de Portalegre, Portugal
(3) Abdus Salam International Centre for Theoretical Physics, Trieste, Italy
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.
Keywords: SRB measures, Lyapunov exponents, Nonuniform expansion, GMY structures
Alves José, Dias Carla, Luzzatto Stefano, Pinheiro Vilton: SRB measures for partially hyperbolic systems whose central direction is weakly expanding. J. Eur. Math. Soc. 19 (2017), 2911-2946. doi: 10.4171/JEMS/731