Stability of mixing and rapid mixing for hyperbolic flows

  AUTHORS:
        Michael Field, Ian Melbourne, Andrew T\"or\"ok

        (MF & AT at U. of Houston, IM at U. of Surrey)

  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 $C2$-open,
   $C^r$-dense set of flows for which each nontrivial hyperbolic basic set
   is rapid mixing.

   Our approach relies on estimating the discrepancy from an arithmetic
   progression of the periods of closed orbits shadowing a periodic orbit
   of the flow. To obtain the rapid decay, we combine this with a result of
   Dolgopyat.