Abstract
AbstractWe establish exponential decay of correlations of all orders for locally G-accessible isometric extensions of transitive Anosov flows, under the assumption that the strong stable and strong unstable distributions of the base Anosov flow are $C^1$ . This is accomplished by translating accessibility properties of the extension into local non-integrability estimates measured by infinitesimal transitivity groups used by Dolgopyat, from which we obtain contraction properties for a class of ‘twisted’ symbolic transfer operators.
Highlights
One of the strongest characteristics of chaotic behaviour in dynamical systems is the exponential decay of correlations, or exponential mixing; in addition to being of intrinsic interest, this is typically accompanied by other strong statistical properties for regular observables
Hyperbolicity and the joint structure of the strong stable and strong unstable foliations are among the most well-understood mechanisms driving chaotic behaviour in both discrete-time and continuous-time dynamical systems
The main portion of our argument, obtaining quantitative non-integrability estimates for the extension in terms of the infinitesimal transitivity group described in Definition 4.3, only requires that the strong stable distribution is C1
Summary
One of the strongest characteristics of chaotic behaviour in dynamical systems is the exponential decay of correlations, or exponential mixing; in addition to being of intrinsic interest, this is typically accompanied by other strong statistical properties for regular observables. The main portion of our argument, obtaining quantitative non-integrability estimates for the extension in terms of the infinitesimal transitivity group described in Definition 4.3, only requires that the strong stable distribution is C1. Theorem A is analogous to a result of Dolgopyat, who showed in [12] that accessible compact group extensions of discrete-time expanding dynamical systems are exponentially mixing, and whose techniques we make heavy use of in our proof. One of the most striking applications of quantitative mixing results in this setting has been the recent work of Kahn and Markovic, using the exponential mixing of frame flows on hyperbolic manifolds in their resolution of the surface subgroup conjecture in [14]. If the frame flow ft is locally accessible, it enjoys exponential decay of correlations of all orders
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