Open sets of exponentially mixing Anosov flows

Abstract

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.

Rates of mixing for non-Markov infinite measure semiflows

“…In both references, the map G is one-dimensional. Higher-dimensional C 2 maps are treated in [6] but under very restrictive smoothness assumptions on the boundaries of the partition elements, and this also extends to C 1+α maps [16]. Hence Theorem 7.1 can be shown to hold for semiflows over C 1+α piecewise expanding maps in arbitrary dimensions, subject to this restriction on the partition elements.…”

“…Decay of correlations is a delicate phenomenon for continuous time dynamical systems. Exponential decay of correlations has been established for certain classes of Anosov flows [16,19,30,48], and the techniques have been extended to various (non)uniformly hyperbolic flows [3,4,6,7,8,15]. Nevertheless, the class of flows for which exponential decay has been established is very restricted.…”

On the rate of mixing of circle extensions of Anosov maps

“…There is however a major issue there: unlike in the flow case, our roof function τ does not come naturally as a return time function and is not constant on stable leaves. Projecting τ on unstable leaves would produce a merely Hölder function, destroying all options to use [7]. On the other hand, Liverani as shown in [10] that by working directly with Anisotropic norms, one can avoid this situation.…”

“…See also [7] for new results in higher dimensions. We don't know if Tsujii's recent technique can be used to prove exponential mixing in our context, and this should be pursued elsewhere.…”

Decay of correlations for certain isometric extensions of Anosov flows

“…Perhaps the most notable result was due to Dolgopyat, who showed in [11] that smooth Anosov flows with C 1 stable and unstable foliations were exponentially mixing, via a quantitative non-integrability estimate. Recently, [9]-following related work by [3][4][5]-established exponential mixing for Anosov flows in the absence of any regularity hypotheses on the unstable foliation.…”