Stable Ergodicity for Expanding on Average Partially Hyperbolic Diffeomorphisms with Two-Dimensional Center

Abstract

We study a property called expansion on average (EA) defined for any partially hyperbolic diffeomorphism satisfying an unstable center bunching condition. Under a mild set of assumptions, we show that the EA property implies $C^2$-stable ergodicity with respect to volume. Our theorem applies to any conservative, $u$-center bunched, robustly dynamically coherent, partially hyperbolic $C^2$-diffeomorphism with two-dimensional center and one-dimensional unstable bundle, whose action on $E^c$ is close to conservative. We do not assume that the center leaves are compact or that the center foliation has a fibration structure. Our mechanism for ergodicity also does not use accessibility or the existence of a dominated splitting in $E^c$. The EA property is inspired by a condition with the same name defined for IID random dynamical systems on surfaces, and our proofs extend ideas from that setting. The EA property is systematically studied in very recent work of Carrasco-Obata-Saghin-Yang, where they obtain continuity of center Lyapunov exponents, strong positive recurrence, and existence of finitely many SRB measures. With input from our result, they obtain uniqueness of SRB measures for dissipative perturbations of certain EA diffeomorphisms.