Hostname: page-component-cb9f654ff-hqlzj Total loading time: 0 Render date: 2025-08-20T08:38:31.067Z Has data issue: false hasContentIssue false
  • English
  • Français

Exponential mixing of frame flows for convex cocompact hyperbolic manifolds

Part of: Smooth dynamical systems: general theory Ergodic theory Dynamical systems with hyperbolic behavior Lie groups

Published online by Cambridge University Press:  08 November 2021

Pratyush Sarkar  and
Dale Winter
Pratyush Sarkar
Affiliation:
Department of Mathematics, Yale University, New Haven, CT 06511, USA pratyush.sarkar@yale.edu
Dale Winter
Affiliation:
Winchester, MA 01890, USA dale.alan.winter@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

The aim of this paper is to establish exponential mixing of frame flows for convex cocompact hyperbolic manifolds of arbitrary dimension with respect to the Bowen–Margulis–Sullivan measure. Some immediate applications include an asymptotic formula for matrix coefficients with an exponential error term as well as the exponential equidistribution of holonomy of closed geodesics. The main technical result is a spectral bound on transfer operators twisted by holonomy, which we obtain by building on Dolgopyat's method.

Keywords

convex cocompact subgroupZariski dense subgrouphyperbolic manifoldframe flowmixingexponential mixingtransfer operatorDolgopyat operatorasymptotic formulamatrix coefficientequidistributionholonomyBowen–Margulis–Sullivan measure

MSC classification

Primary: 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 37A25: Ergodicity, mixing, rates of mixing 37C30: Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dynamical systems 22E40: Discrete subgroups of Lie groups

Information

Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 12 , December 2021 , pp. 2585 - 2634
Copyright
© 2021 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Babillot, M., On the mixing property for hyperbolic systems, Israel J. Math. 129 (2002), 61–76; MR 1910932.CrossRefGoogle Scholar
Benoist, Y. and Oh, H., Effective equidistribution of $S$-integral points on symmetric varieties, Ann. Inst. Fourier (Grenoble) 62 (2012), 1889–1942; MR 3025156.CrossRefGoogle Scholar
Bowen, R., Markov partitions for Axiom $A$ diffeomorphisms, Amer. J. Math. 92 (1970), 725–747; MR 0277003.CrossRefGoogle Scholar
Bowen, R., Periodic points and measures for Axiom $A$ diffeomorphisms, Trans. Amer. Math. Soc. 154 (1971), 377–397; MR 0282372.Google Scholar
Bowen, R., Symbolic dynamics for hyperbolic flows, Amer. J. Math. 95 (1973), 429–460; MR 339281.CrossRefGoogle Scholar
Bowen, R., Equilibrium states and the ergodic theory of Anosov diffeomorphisms, revised edition, Lecture Notes in Mathematics, vol. 470 (Springer, Berlin, 2008). With a preface by David Ruelle. Edited by Jean-René Chazottes; MR 2423393.CrossRefGoogle Scholar
Brin, M. I. and Pesin, J. B., Partially hyperbolic dynamical systems, Math. USSR Izv. 8 (1974), 177–218; MR 0343316.CrossRefGoogle Scholar
Brin, M., Ergodic theory of frame flows, in Ergodic theory and dynamical systems, II (College Park, Md., 1979/1980), Progress in Mathematics, vol. 21 (Birkhäuser, Boston, 1982), 163–183; MR 670078.CrossRefGoogle Scholar
Chernov, N., Invariant measures for hyperbolic dynamical systems, in Handbook of dynamical systems, vol. 1A (North-Holland, Amsterdam, 2002), 321–407; MR 1928521.CrossRefGoogle Scholar
Das, T., Fishman, L., Simmons, D. and Urbański, M., Extremality and dynamically defined measures, part II: Measures from conformal dynamical systems, Ergodic Theory Dynam. Systems 41 (2021), 2311–2348.CrossRefGoogle Scholar
Dolgopyat, D., On decay of correlations in Anosov flows, Ann. of Math. (2) 147 (1998), 357–390; MR 1626749.CrossRefGoogle Scholar
Dolgopyat, D., On mixing properties of compact group extensions of hyperbolic systems, Israel J. Math. 130 (2002), 157–205; MR 1919377.CrossRefGoogle Scholar
Duke, W., Rudnick, Z. and Sarnak, P., Density of integer points on affine homogeneous varieties, Duke Math. J. 71 (1993), 143–179; MR 1230289.CrossRefGoogle Scholar
Eskin, A. and McMullen, C., Mixing, counting, and equidistribution in Lie groups, Duke Math. J. 71 (1993), 181–209; MR 1230290.CrossRefGoogle Scholar
Edwards, S. C. and Oh, H., Spectral gap and exponential mixing on geometrically finite hyperbolic manifolds, Duke Math. J., to appear. Preprint (2021), arXiv:2001.03377.Google Scholar
Gorodnik, A. and Spatzier, R., Exponential mixing of nilmanifold automorphisms, J. Anal. Math. 123 (2014), 355–396; MR 3233585.CrossRefGoogle Scholar
Kaimanovich, V. A., Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds, Ann. Inst. H. Poincaré Phys. Théor. 53 (1990), 361–393; MR 1096098.Google Scholar
Kato, T., Perturbation theory for linear operators, Classics in Mathematics (Springer, Berlin, 1995), Reprint of the 1980 edition; MR 1335452.CrossRefGoogle Scholar
Kelmer, D. and Oh, H., Shrinking targets for geodesic flow on geometrically finite hyperbolic manifolds, J. Mod. Dyn., to appear. Preprint (2021), arXiv:1812.05251.Google Scholar
Kleinbock, D. Y. and Margulis, G. A., Bounded orbits of nonquasiunipotent flows on homogeneous spaces, in Sinaĭ's Moscow Seminar on Dynamical Systems, American Mathematical Society Translations: Series 2, vol. 171 (American Mathematical Society, Providence, RI, 1996), 141–172; MR 1359098.CrossRefGoogle Scholar
Lasota, A. and Yorke, J. A., On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481–488; MR 335758.CrossRefGoogle Scholar
Mangum, B. S., Incompressible surfaces and pseudo-Anosov flows, Topology Appl. 87 (1998), 29–51; MR 1626072.CrossRefGoogle Scholar
Margulis, G. A., On some aspects of the theory of Anosov systems, Springer Monographs in Mathematics (Springer, Berlin, 2004); MR 2035655.CrossRefGoogle Scholar
Margulis, G., Mohammadi, A. and Oh, H., Closed geodesics and holonomies for Kleinian manifolds, Geom. Funct. Anal. 24 (2014), 1608–1636; MR 3261636.CrossRefGoogle Scholar
Mather, J. N., Characterization of Anosov diffeomorphisms, Nederl. Akad. Wetensch. Proc. Ser. A 71 = Indag. Math. 30 (1968), 479–483; MR 0248879.CrossRefGoogle Scholar
Mohammadi, A. and Oh, H., Matrix coefficients, counting and primes for orbits of geometrically finite groups, J. Eur. Math. Soc. (JEMS) 17 (2015), 837–897; MR 3336838.CrossRefGoogle Scholar
Mok, K. P., On the differential geometry of frame bundles of Riemannian manifolds, J. Reine Angew. Math. 302 (1978), 16–31; MR 511689.Google Scholar
Moore, C. C., Exponential decay of correlation coefficients for geodesic flows, in Group representations, ergodic theory, operator algebras, and mathematical physics (Berkeley, Calif., 1984), Mathematical Sciences Research Institute Publications, vol. 6 (Springer, New York, 1987), 163–181; MR 880376.CrossRefGoogle Scholar
Naud, F., Expanding maps on Cantor sets and analytic continuation of zeta functions, Ann. Sci. Éc. Norm. Supér. (4) 38 (2005), 116–153; MR 2136484.CrossRefGoogle Scholar
Oh, H. and Shah, N. A., Equidistribution and counting for orbits of geometrically finite hyperbolic groups, J. Amer. Math. Soc. 26 (2013), 511–562; MR 3011420.CrossRefGoogle Scholar
Oh, H. and Winter, D., Uniform exponential mixing and resonance free regions for convex cocompact congruence subgroups of ${\rm SL_2}(\Bbb {Z})$, J. Amer. Math. Soc. 29 (2016), 1069–1115; MR 3522610.CrossRefGoogle Scholar
Oh, H. and Winter, D., Prime number theorems and holonomies for hyperbolic rational maps, Invent. Math. 208 (2017), 401–440; MR 3639596.CrossRefGoogle Scholar
Parry, W. and Pollicott, M., Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187–188 (1990); MR 1085356.Google Scholar
Paulin, F., Pollicott, M. and Schapira, B., Equilibrium states in negative curvature, Astérisque 373 (2015); MR 3444431.Google Scholar
Patterson, S. J., The limit set of a Fuchsian group, Acta Math. 136 (1976), 241–273; MR 0450547.CrossRefGoogle Scholar
Petkov, V. and Stoyanov, L., Ruelle transfer operators with two complex parameters and applications, Discrete Contin. Dyn. Syst. 36 (2016), 6413–6451; MR 3543593.Google Scholar
Pollicott, M., On the rate of mixing of Axiom A flows, Invent. Math. 81 (1985), 413–426; MR 807065.CrossRefGoogle Scholar
Ratner, M., Markov partitions for Anosov flows on $n$-dimensional manifolds, Israel J. Math. 15 (1973), 92–114; MR 0339282.CrossRefGoogle Scholar
Ratner, M., The rate of mixing for geodesic and horocycle flows, Ergodic Theory Dynam. Systems 7 (1987), 267–288; MR 896798.CrossRefGoogle Scholar
Roblin, T., Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.) 96 (2003); MR 2057305.Google Scholar
Ruelle, D., The thermodynamic formalism for expanding maps, Comm. Math. Phys. 125 (1989), 239–262; MR 1016871.CrossRefGoogle Scholar
Sarnak, P. and Wakayama, M., Equidistribution of holonomy about closed geodesics, Duke Math. J. 100 (1999), 1–57; MR 1714754.Google Scholar
Sasaki, S., On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. (2) 10 (1958), 338–354; MR 0112152.CrossRefGoogle Scholar
Stoyanov, L., Spectra of Ruelle transfer operators for axiom A flows, Nonlinearity 24 (2011), 1089–1120; MR 2776112.CrossRefGoogle Scholar
Sullivan, D., The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171–202; MR 556586.CrossRefGoogle Scholar
Sullivan, D., Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math. 153 (1984), 259–277; MR 766265.CrossRefGoogle Scholar
Warner, G., Harmonic analysis on semi-simple Lie groups. I, Grundlehren der mathematischen Wissenschaften, vol. 188 (Springer, Berlin, 1972); MR 0498999.Google Scholar
Winter, D., Mixing of frame flow for rank one locally symmetric spaces and measure classification, Israel J. Math. 210 (2015), 467–507; MR 3430281.CrossRefGoogle Scholar