On the abundance of non-zero central Lyapunov exponents, physical measures and stable ergodicity for partially hyperbolic dynamics

TL;DR

This paper demonstrates that certain Anosov flows can be perturbed to produce diffeomorphisms with positive central Lyapunov exponents and unique physical measures, leading to stable ergodicity without requiring a preserved smooth measure.

Contribution

It introduces a perturbative method to achieve positive central Lyapunov exponents and stable ergodicity for maps close to Anosov flows, independent of measure preservation.

Findings

Time-1 maps of specific Anosov flows can be approximated by stably ergodic diffeomorphisms.

These diffeomorphisms have positive central Lyapunov exponents Lebesgue almost everywhere.

The approach does not depend on preserving a smooth measure.

Abstract

We show that the time-1 map of an Anosov flow, whose strong-unstable foliation is $C^2$ smooth and minimal, is $C^2$ close to a diffeomorphism having positive central Lyapunov exponent Lebesgue almost everywhere and a unique physical measure with full basin, which is $C^r$ stably ergodic. Our method is perturbative and does not rely on preservation of a smooth measure.