Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-02T21:58:37.587Z Has data issue: false hasContentIssue false
  • English
  • Français

Excursions to the cusps for geometrically finite hyperbolic orbifolds and equidistribution of closed geodesics in regular covers

Part of: Dynamical systems with hyperbolic behavior Ergodic theory

Published online by Cambridge University Press:  15 November 2021

RON MOR Open the ORCID record for RON MOR [Opens in a new window]
RON MOR*
Affiliation:
The Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel
*
(e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a finitary criterion for the convergence of measures on non-elementary geometrically finite hyperbolic orbifolds to the unique measure of maximal entropy. We give an entropy criterion controlling escape of mass to the cusps of the orbifold. Using this criterion, we prove new results on the distribution of collections of closed geodesics on such an orbifold, and as a corollary, we prove the equidistribution of closed geodesics up to a certain length in amenable regular covers of geometrically finite orbifolds.

Keywords

hyperbolic orbifoldsgeometrically finiteentropyclosed geodesicsequidistribution

MSC classification

Primary: 37A17: Homogeneous flows 37A35: Entropy and other invariants, isomorphism, classification 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 12 , December 2022 , pp. 3745 - 3791
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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.)

References

Adler, R. L., Konheim, A. G. and McAndrew, M. H.. Topological entropy. Trans. Amer. Math. Soc. 114 (1965), 309–319.CrossRefGoogle Scholar
Apanasov, B. N.. Discrete Groups in Space and Uniformization Problems (Mathematics and Its Applications (Soviet Series), 40). Kluwer Academic Publishers Group, Dordrecht, 1991. Translated and revised from the 1983 Russian original.Google Scholar
Apanasov, B. N.. Conformal Geometry of Discrete Groups and Manifolds (De Gruyter Expositions in Mathematics, 32). Walter de Gruyter & Co., Berlin, 2000.CrossRefGoogle Scholar
Beardon, A. F.. The exponent of convergence of Poincaré series. Proc. Lond. Math. Soc. (3) 18 (1968), 461–483.CrossRefGoogle Scholar
Bowditch, B. H.. Geometrical finiteness for hyperbolic groups. J. Funct. Anal. 113(2) (1993), 245–317.CrossRefGoogle Scholar
Bowen, R.. Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153 (1971), 401–414.CrossRefGoogle Scholar
Brooks, R.. The bottom of the spectrum of a Riemannian covering. J. Reine Angew. Math. 357 (1985), 101–114.Google Scholar
Dinaburg, E. I.. A correlation between topological entropy and metric entropy. Dokl. Akad. Nauk SSSR 190 (1970), 19–22.Google Scholar
Duke, W.. Hyperbolic distribution problems and half-integral weight Maass forms. Invent. Math. 92(1) (1988), 73–90.CrossRefGoogle Scholar
Eberlein, P.. Geodesic flows on negatively curved manifolds. I. Ann. of Math. (2) 95 (1972), 492–510.CrossRefGoogle Scholar
Einsiedler, M., Kadyrov, S. and Pohl, A.. Escape of mass and entropy for diagonal flows in real rank one situations. Israel J. Math. 210(1) (2015), 245–295.CrossRefGoogle Scholar
Einsiedler, M., Lindenstrauss, E., Michel, P. and Venkatesh, A.. The distribution of closed geodesics on the modular surface, and Duke’s theorem. Enseign. Math. (2) 58(3–4) (2012), 249–313.CrossRefGoogle Scholar
Einsiedler, M., Lindenstrauss, E. and Ward, T.. Entropy in Ergodic Theory and Toplogical Dynamics, manuscript in preparation, https://tbward0.wixsite.com/books/entropy.Google Scholar
Gehring, F. W., Martin, G. J. and Palka, B. P.. An Introduction to the Theory of Higher-Dimensional Quasiconformal Mappings (Mathematical Surveys and Monographs, 216). American Mathematical Society, Providence, RI, 2017.Google Scholar
Goodman, T. N. T.. Relating topological entropy and measure entropy. Bull. Lond. Math. Soc. 3 (1971), 176–180.CrossRefGoogle Scholar
Goodwyn, L. W.. Topological entropy bounds measure-theoretic entropy. Proc. Amer. Math. Soc. 23 (1969), 679–688.CrossRefGoogle Scholar
Handel, M. and Kitchens, B.. Metrics and entropy for non-compact spaces. Israel J. Math. 91(1–3) (1995), 253–271. With an appendix by Daniel J. Rudolph.CrossRefGoogle Scholar
Iommi, G., Riquelme, F. and Velozo, A.. Entropy in the cusp and phase transitions for geodesic flows. Israel J. Math. 225(2) (2018), 609–659.CrossRefGoogle Scholar
Kadyrov, S.. Positive entropy invariant measures on the space of lattices with escape of mass. Ergod. Th. & Dynam. Sys. 32(1) (2012), 141–157.CrossRefGoogle Scholar
Katok, A.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 51 (1980), 137–173.CrossRefGoogle Scholar
Margulis, G. A.. On Some Aspects of the Theory of Anosov Systems. With a Survey by Richard Sharp: Periodic Orbits of Hyperbolic Flows (Springer Monographs in Mathematics). Springer-Verlag, Berlin, 2004. Translated from the Russian by Valentina Vladimirovna Szulikowska.CrossRefGoogle Scholar
McMullen, C. T.. Hausdorff dimension and conformal dynamics. I. Strong convergence of Kleinian groups. J. Differential Geom. 51(3), 471–515.Google Scholar
Misiurewicz, M.. A short proof of the variational principle for a ${\mathbb{Z}}_{+}^N$ action on a compact space. Int. Conf. on Dynamical Systems in Mathematical Physics (Rennes, 1975). Astérisque, 40 (1976), 147–157.Google Scholar
Mor, R.. Geometrically finite groups, Patterson–Sullivan measures, and entropy. Master’s Thesis, The Hebrew University of Jerusalem, 2018.Google Scholar
Nicholls, P. J.. The Ergodic Theory of Discrete Groups (London Mathematical Society Lecture Note Series, 143). Cambridge University Press, Cambridge, 1989.CrossRefGoogle Scholar
Oh, H. and Pan, W.. Local mixing and invariant measures for horospherical subgroups on abelian covers. Int. Math. Res. Not. IMRN 19 (2019), 6036–6088.CrossRefGoogle Scholar
Otal, J.-P. and Peigné, M.. Principe variationnel et groupes kleiniens. Duke Math. J. 125(1) (2004), 15–44.CrossRefGoogle Scholar
Patterson, S. J.. The limit set of a Fuchsian group. Acta Math. 136(3–4) (1976), 241–273.CrossRefGoogle Scholar
Paulin, F., Pollicott, M. and Schapira, B.. Equilibrium states in negative curvature. Astérisque 373 (2015), viii + 281 pp.Google Scholar
Prokhorov, Yu. V.. Convergence of random processes and limit theorems in probability theory. Teor. Veroyatn. Primen. 1 (1956), 177–238.Google Scholar
Quint, J.-F.. An overview of Patterson–Sullivan theory. Workshop ‘The Barycenter Method’. FIM, Zurich, 2006.Google Scholar
Riquelme, F. and Velozo, A.. Escape of mass and entropy for geodesic flows. Ergod. Th. & Dynam. Sys. 39(2) (2019), 446–473.CrossRefGoogle Scholar
Roblin, T.. Ergodicité et équidistribution en courbure négative. Mém. Soc. Math. Fr. (N.S.) 95 (2003), vi + 96.Google Scholar
Roblin, T.. Un théorème de Fatou pour les densités conformes avec applications aux revêtements galoisiens en courbure négative. Israel J. Math. 147 (2005), 333–357.CrossRefGoogle Scholar
Selberg, A.. On discontinuous groups in higher-dimensional symmetric spaces. Contributions to Function Theory (International Colloquium on Function Theory, Bombay, 1960). Tata Institute of Fundamental Research, Bombay, 1960, pp. 147–164.Google Scholar
Sharp, R.. Critical exponents for groups of isometries. Geom. Dedicata 125 (2007), 63–74.CrossRefGoogle Scholar
Sullivan, D.. The density at infinity of a discrete group of hyperbolic motions. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 171–202.CrossRefGoogle Scholar
Sullivan, D.. Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math. 153(3–4) (1984), 259–277.CrossRefGoogle Scholar
Winter, D.. Mixing of frame flow for rank one locally symmetric spaces and measure classification. Israel J. Math. 210(1) (2015), 467–507.CrossRefGoogle Scholar