Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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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

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.
References
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Bibliographic Information
  • Mark Pollicott
  • Affiliation: Department of Mathematics, University of Warwick, Coventry CV4 7AL, United Kingdom
  • MR Author ID: 140805
  • ORCID: 0000-0002-0206-2200
  • Email: masdbl@warwick.ac.uk
  • Daofei Zhang
  • Affiliation: School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, People’s Republic of China; and Department of Mathematics, University of Warwick, Coventry CV4 7AL, United Kingdom
  • MR Author ID: 1505707
  • ORCID: 0009-0000-3234-3698
  • Email: Daofei.Zhang@guet.edu.cn
  • Received by editor(s): June 23, 2024
  • Received by editor(s) in revised form: December 31, 2024
  • Published electronically: March 25, 2025
  • Additional Notes: Daofei Zhang is the corresponding author
    The authors were partly supported by ERC-Advanced Grant 833802-Resonances.
  • © Copyright 2025 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 378 (2025), 5011-5056
  • MSC (2020): Primary 37A25, 37C30, 37D40; Secondary 37D20, 37D30
  • DOI: https://doi.org/10.1090/tran/9424