Open Sets of Exponentially Mixing Anosov Flows

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 $\dim {\mathbb{E}}_s=1$, $\dim {\mathbb{E}}_u\ge 2$ 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 in not cohomologous to a piecewise constant.