Robust ergodic properties in partially hyperbolic dynamics

Abstract

We study ergodic properties of partially hyperbolic systems whose central direction is mostly contracting. Earlier work of Bonatti and Viana (2000) about existence and finitude of physical measures is extended to the case of local diffeomorphisms. Moreover, we prove that such systems constitute a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C squared"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">C^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-open set in which statistical stability is a dense property. In contrast, <italic>all</italic> mostly contracting systems are shown to be stable under small random perturbations.