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.