No abstract

We consider the skew product F : (x, u) → (f (x), u + τ (x)), where the base map f : T 1 → T 1 is piecewise C 2 , covering and uniformly expanding, and the fibre map τ : T 1 → R is piecewise C 2 . We show the dichotomy that either this system mixes exponentially or τ is cohomologous (via a Lipschitz function) to a piecewise constant.

We prove sharp results on polynomial decay of correlations for nonuniformly hyperbolic flows. Applications include intermittent solenoidal flows and various Lorentz gas models including the infinite horizon Lorentz gas.

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.

Abstract. For any dimension d ≥ 3 we construct C 1 -open subsets of the space of C 3 vector fields such that the flow associated to each vector field is Axiom A and exhibits a non-trivial attractor which mixes exponentially with respect to the unique SRB measure.