Open sets of exponentially mixing Anosov flows

Abstract

We prove that an Anosov flow with \mathcal C^1 stable bundle mixes exponentially whenever the stable and unstable bundles are not jointly integrable. This allows us to show that if a flow is sufficiently close to a volume-preserving Anosov flow and dim \mathbb E_s = 1 , dim \mathbb E_u \geq 2 then the flow mixes exponentially whenever the stable and unstable bundles are not jointly integrable. This implies the existence of non-empty open sets of exponentially mixing Anosov flows. As part of the proof of this result we show that \mathcal C^{1+} uniformly expanding suspension semiflows (in any dimension) mix exponentially when the return time is not cohomologous to a piecewise constant.