Stability of mixing and rapid mixing for hyperbolic flows

Annals of Mathematics 166 (2007) 269-291

Michael Field, Ian Melbourne and Andrew Török

Abstract We obtain general results on the stability of mixing and rapid mixing (superpolynomial decay of correlations) for hyperbolic flows. Amongst Cr Axiom A flows, $r\ge2$, we show that there is a C2-open, Cr-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 Cr, as opposed to Hölder, topology.

Postscript file or pdf file
Previous preprints from May 2003 and July 2004 established results on stability of mixing and rapid mixing respectively, under the additional hypothesis
r \ge 2dim M - 1
where M is the ambient manifold. The current preprint removes this restriction and hence supercedes the previous preprints.