Lyapunov stability and the Bernoulli property

Abstract

This paper addresses, and was motivated by, two open questions concerning the Bernoulli property for partially hyperbolic systems with some controlled behavior along the center direction. Namely, we address a question due to A. Wilkinson concerning the Bernoullicity of accessible, center-bunched volume preserving C 1+αpartially hyperbolic diffeomorphisms with Lyapunov stable center and a question due to F. Hertz, J. Hertz and R. Ures [18] concerning the Bernoulli property for jointly integrable volume preserving perturbations of ergodic linear automorphisms of T N .We prove that for a volume-preserving C 1+α -partially hyperbolic f with dim(E c ) = 1, accessibility and bi-Lyapunov stability of the center direction imply that either the center foliation is atomic, or it is C ∞ and f is Bernoulli.For the context of perturbations of linear ergodic automorphisms of T N we prove that for f ∈ P H 2 m (T N ), C ∞ close to a linear ergodic automorphism A : T N → T N , with dim(E c ) = 2, then either i) F c is weakly-atomic; or ii) f is Bernoulli.The main novelty of our proofs is the use of a recent technique introduced in [27], which consists in using a system of leafwise invariant metrics to recover informations on the disintegration of ergodic invariant measures.

Equilibrium states for self‐products of flows and the mixing properties of rank 1 geodesic flows

“…We also establish the Bernoulli property for the measure of maximal entropy. Studying the Kolmogorov and Bernoulli properties, and conditions under which 𝐾 implies Bernoulli, is an active area in smooth ergodic theory, with recent references including [24,29,36,37].…”

Equilibrium states for self-products of flows and the mixing properties of rank 1 geodesic flows

“…We also establish the Bernoulli property for the measure of maximal entropy. Studying the Kolmogorov and Bernoulli properties, and conditions under which K implies Bernoulli, is an active area in smooth ergodic theory, with recent references including [25,31,20,32].…”