Robust transitivity and almost robust ergodicity

Abstract

In this paper, we prove a relation between robust transitivity and robust ergodicity for conservative diffeomorphisms. In dimension two, robustly transitive diffeomorphisms are robustly ergodic. In higher dimensions, we work with partially hyperbolic diffeomorphisms with non-zero Lyapunov exponents. We define almost robust ergodicity and prove that in the three-dimensional case robustly transitive diffeomorphisms are approximated by almost robustly ergodic ones. In higher dimensions under the condition of the central Lyapunov exponents being of the same sign, we prove that robustly transitive diffeomorphisms are robustly ergodic.