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

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.