Rapid mixing for compact group extensions of hyperbolic flows
HTML articles powered by AMS MathViewer
- 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
- HTML | PDF | Request permission
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
- David Applebaum, Probability on compact Lie groups, Probability Theory and Stochastic Modelling, vol. 70, Springer, Cham, 2014. With a foreword by Herbert Heyer. MR 3243650, DOI 10.1007/978-3-319-07842-7
- Viviane Baladi, Positive transfer operators and decay of correlations, Advanced Series in Nonlinear Dynamics, vol. 16, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. MR 1793194, DOI 10.1142/9789812813633
- Rufus Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math. 95 (1973), 429–460. MR 339281, DOI 10.2307/2373793
- Rufus Bowen and David Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29 (1975), no. 3, 181–202. MR 380889, DOI 10.1007/BF01389848
- Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Second revised edition, Lecture Notes in Mathematics, vol. 470, Springer-Verlag, Berlin, 2008. With a preface by David Ruelle; Edited by Jean-René Chazottes. MR 2423393, DOI 10.1007/978-3-540-77695-6
- M. I. Brin, The topology of group extensions of $C$-systems, Mat. Zametki 18 (1975), no. 3, 453–465 (Russian). MR 394764
- M. I. Brin, Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature, Funkcional. Anal. i Priložen. 9 (1975), no. 1, 9–19 (Russian). MR 370660
- M. Brin and M. Gromov, On the ergodicity of frame flows, Invent. Math. 60 (1980), no. 1, 1–7. MR 582702, DOI 10.1007/BF01389897
- M. Brin, Ergodic theory of frame flows, Ergodic theory and dynamical systems, II (College Park, Md., 1979/1980) Progr. Math., vol. 21, Birkhäuser, Boston, MA, 1982, pp. 163–183. MR 670078
- M. Brin and H. Karcher, Frame flows on manifolds with pinched negative curvature, Compositio Math. 52 (1984), no. 3, 275–297. MR 756723
- Yann Bugeaud and Michel Laurent, On exponents of homogeneous and inhomogeneous Diophantine approximation, Mosc. Math. J. 5 (2005), no. 4, 747–766, 972. MR 2266457, DOI 10.17323/1609-4514-2005-5-4-747-766
- Keith Burns and Mark Pollicott, Stable ergodicity and frame flows, Geom. Dedicata 98 (2003), 189–210. MR 1988429, DOI 10.1023/A:1024057924334
- Mihajlo Cekić, Thibault Lefeuvre, Andrei Moroianu, and Uwe Semmelmann, On the ergodicity of the frame flow on even-dimensional manifolds, Invent. Math. 238 (2024), no. 3, 1067–1110. MR 4824734, DOI 10.1007/s00222-024-01297-7
- M. Cekić and T. Lefeuvre, Semiclassical analysis on principal bundles, arXiv:2405.14846.
- Dmitry Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math. (2) 147 (1998), no. 2, 357–390. MR 1626749, DOI 10.2307/121012
- Dmitry Dolgopyat, Prevalence of rapid mixing in hyperbolic flows, Ergodic Theory Dynam. Systems 18 (1998), no. 5, 1097–1114. MR 1653299, DOI 10.1017/S0143385798117431
- Dmitry Dolgopyat, On mixing properties of compact group extensions of hyperbolic systems, Israel J. Math. 130 (2002), 157–205. MR 1919377, DOI 10.1007/BF02764076
- Michael Field, Ian Melbourne, Matthew Nicol, and Andrei Török, Statistical properties of compact group extensions of hyperbolic flows and their time one maps, Discrete Contin. Dyn. Syst. 12 (2005), no. 1, 79–96. MR 2121250, DOI 10.3934/dcds.2005.12.79
- Todd Fisher and Boris Hasselblatt, Hyperbolic flows, Zurich Lectures in Advanced Mathematics, EMS Publishing House, Berlin, [2019] ©2019. MR 3972204, DOI 10.4171/200
- Jeremy Kahn and Vladimir Markovic, Immersing almost geodesic surfaces in a closed hyperbolic three manifold, Ann. of Math. (2) 175 (2012), no. 3, 1127–1190. MR 2912704, DOI 10.4007/annals.2012.175.3.4
- William Parry and Mark Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187-188 (1990), 268 (English, with French summary). MR 1085356
- Joseph F. Plante, Anosov flows, Amer. J. Math. 94 (1972), 729–754. MR 377930, DOI 10.2307/2373755
- Mark Pollicott, On the rate of mixing of Axiom A flows, Invent. Math. 81 (1985), no. 3, 413–426. MR 807065, DOI 10.1007/BF01388579
- Mark Pollicott and Richard Sharp, Error terms for closed orbits of hyperbolic flows, Ergodic Theory Dynam. Systems 21 (2001), no. 2, 545–562. MR 1827118, DOI 10.1017/S0143385701001274
- Mark Pollicott and Richard Sharp, Periodic orbits and holonomy for hyperbolic flows, Geometric and probabilistic structures in dynamics, Contemp. Math., vol. 469, Amer. Math. Soc., Providence, RI, 2008, pp. 289–302. MR 2478476, DOI 10.1090/conm/469/09172
- M. Ratner, Markov partitions for Anosov flows on $n$-dimensional manifolds, Israel J. Math. 15 (1973), 92–114. MR 339282, DOI 10.1007/BF02771776
- Peter Sarnak and Masato Wakayama, Equidistribution of holonomy about closed geodesics, Duke Math. J. 100 (1999), no. 1, 1–57. MR 1714754, DOI 10.1215/S0012-7094-99-10001-9
- Ralf Spatzier and Daniel Visscher, Equilibrium measures for certain isometric extensions of Anosov systems, Ergodic Theory Dynam. Systems 38 (2018), no. 3, 1154–1167. MR 3784258, DOI 10.1017/etds.2016.62
- Mitsuo Sugiura, Fourier series of smooth functions on compact Lie groups, Osaka Math. J. 8 (1971), 33–47. MR 294571
- Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108, DOI 10.1007/978-1-4612-5775-2
- D. Zhang, Rapid mixing and superpolynomial equidistribution for torus extensions of hyperbolic flows, arXiv:2405.11367.
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