# Stability of mixing and rapid mixing for hyperbolic flows

### Abstract

We obtain general results on the stability of mixing and rapid mixing (superpolynomial decay of correlations) for hyperbolic flows. Amongst $C^r$ Axiom A flows, $r\ge2$, we show that there is a $C^2$-open, $C^r$-dense set of flows for which each nontrivial hyperbolic basic set is rapid mixing. This is the first general result on the stability of rapid mixing (or even mixing) for Axiom A flows that holds in a $C^r$, as opposed to Hölder, topology.

## Authors

Michael Field

Department of Mathematics
University of Houston
Houston, TX 77004
United States

Ian Melbourne

Department of Mathematics
University of Surrey
Guildford, Surrey
GU2 7XH
United Kingdom

Andrei Török

Department of Mathematics
University of Houston
Houston, TX 77004
United States
and
Institute of Mathematics of the Romanian Academy
014700 Bucharest
Romania