Open sets of partially hyperbolic skew products having a unique SRB measure

Abstract

In this paper we obtain $C^2$-open sets of dissipative, partially hyperbolic skew products having a unique SRB measure with full support and full basin. These partially hyperbolic systems have a two dimensional center bundle which presents both expansion and contraction but does not admit any further dominated splitting of the center. These systems are non conservative perturbations of an example introduced by Berger-Carrasco. To prove the existence of SRB measures for these perturbations, we obtain a measure rigidity result for $u$-Gibbs measures for partially hyperbolic skew products. This is an adaptation of a measure rigidity result by A. Brown and F. Rodriguez Hertz for stationary measures of random product of surface diffeomorphisms. In particular, we classify all the possible $u$-Gibbs measures that may appear in a neighborhood of the example. Using this classification, and ruling out some of the possibilities, we obtain open sets of systems, in a neighborhood of the example, having a unique $u$-Gibbs measure which is SRB.

Final version. To appear in Advances in Mathematics. 78 pages, 2 figures

References in corpus (5)

• Lyapunov exponents for random perturbations of some area-preserving maps including the standard map
• The geometric approach for constructing Sinai-Ruelle-Bowen measures
• Smoothness of stable holonomies inside center-stable manifolds and the $C^2$ hypothesis in Pugh-Shub and Ledrappier-Young theory
• Stationary measures and orbit closures of uniformly expanding random dynamical systems on surfaces
• Measure rigidity for random dynamics on surfaces with positive entropy