Rapid mixing for compact group extensions of hyperbolic flows

Pollicott, Mark; Zhang, Daofei

doi:10.1090/tran/9424

Rapid mixing for compact group extensions of hyperbolic flows
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by Mark Pollicott and Daofei Zhang;

Trans. Amer. Math. Soc. 378 (2025), 5011-5056

DOI: https://doi.org/10.1090/tran/9424

Published electronically: March 25, 2025

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Abstract:

In this article, we give explicit conditions for compact group extensions of hyperbolic flows (including geodesic flows on negatively curved manifolds) to exhibit quantifiable rates of mixing (or decay of correlations) with respect to the natural probability measures, which are locally the product of a Gibbs measure for a Hölder potential and the Haar measure. More precisely, we show that the mixing rate with respect to Hölder functions will be faster than any given polynomial (i.e., rapid mixing). We also give error estimates on the equidistribution of the holonomy around closed orbits. In particular, these results apply to some frame flows for manifolds with negative sectional curvatures.

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Rapid mixing for compact group extensions of hyperbolic flows
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Abstract: