Perturbation of the time-1 map of a generic volume-preserving $3$-dimensional Anosov flow

TL;DR

This paper studies the perturbations of time-1 maps of generic volume-preserving Anosov flows on 3D manifolds, showing exponential convergence of measures and various dynamical properties, with notable applications to stability and approximation questions.

Contribution

It demonstrates new stability and mixing properties of perturbed Anosov flow maps and provides counterexamples to longstanding conjectures in dynamical systems.

Findings

Exponential convergence of pushforward measures to a full support limit measure.

Topological mixing and unique physical measure for the perturbed maps.

Existence of stable transitivity without periodic points in certain diffeomorphisms.

Abstract

Let $s > 1$ be a large integer, and let $f$ be a diffeomorphism sufficiently close in the $C^{s}$-topology to the time-1 map of a $C^{s}$ generic volume-preserving Anosov flow on a $3$-dimensional compact manifold. We show that for any probability measure $\mu$ with smooth density, $f^n_* \mu$ converges exponentially fast to a common limit measure with full support. As corollaries, we show the following: $f$ is topologically mixing; $f$ has a unique physical measure with basin of full Lebesgue measure, which is also the unique u-Gibbs state; if $f$ is volume preserving, then $f$ is exponentially mixing with respect to the volume form. As applications, we give a class of time-1 maps of transitive Anosov flows non-approximable in $C^{s}$ by Axiom A maps, giving negative answer to a question of Palis-Pugh (1974); the first example of a $C^{s}$-stably transitive time-1 map of Anosov flow, a question mentioned in Bonatti-Guelman (2010), Rodriguez Hertz (2010); as well as the first example of a $C^{s}$-stably transitive diffeomorphism without periodic points.