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

2014/03/12 by Alves, Jose F., Dias, C. L., Luzzatto, S. +1 · 3 citations
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1403.2937

Abstract

We consider partially hyperbolic \( C¹⁺ \) diffeomorphisms of compact Riemannian manifolds of arbitrary dimension which admit a partially hyperbolic tangent bundle decomposition \( E^s⊕ E^cu \). 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.

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