We prove that an Anosov flow with 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 Es = 1, dim Eu ≥ 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 C 1+ uniformly-expanding suspension semiflows (in any dimension) mix exponentially when the return time in not cohomologous to a piecewise constant.Date: 20th November 2017. 2010 Mathematics Subject Classification. Primary: 37A25; Secondary: 37C30. With pleasure we thank Matias Delgadino, Stefano Luzzatto, Ian Melbourne, Masato Tsujii and Sina Türeli for stimulating discussions. We also thank Viviane Baladi, François Ledrappier and the anonymous referee for highlighting an issue in a previous version of this paper. We are grateful to the ESI (Vienna) for hospitality during the event "Mixing Flows and Averaging Methods" where this work was initiated. OB was partially supported by CNRS. KW was partially supported by DFG (CRC/TRR 191). 4 Stoyanov [34] obtained results similar to Dolgopyat [17] for Axiom A flows but, among other assumptions, required that local stable and unstable laminations are Lipschitz.

We obtain Margulis-type asymptotic estimates for the number of free homotopy classes of closed geodesics on certain manifolds without conjugate points. Our results cover all compact surfaces of genus at least 2 without conjugate points.

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We prove that for closed surfaces M with Riemannian metrics without conjugate points and genus ≥ 2 the geodesic flow on the unit tangent bundle T 1 M has a unique measure of maximal entropy. Furthermore, this measure is fully supported on T 1 M and the flow is mixing with respect to this measure. We formulate conditions under which this result extends to higher dimensions.

We give new sufficient conditions for the integrability and unique integrability of continuous tangent sub-bundles on manifolds of arbitrary dimension, generalizing Frobenius' classical Theorem for C 1 sub-bundles. Using these conditions we derive new criteria for uniqueness of solutions to ODE's and PDE's and for the integrability of invariant bundles in dynamical systems. In particular we give a novel proof of the Stable Manifold Theorem and prove some integrability results for dynamically defined dominated splittings.