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Counting closed geodesics in a compact rank-one locally CAT(0) space

Part of: Measure-theoretic ergodic theory Dynamical systems with hyperbolic behavior Global differential geometry

Published online by Cambridge University Press:  27 August 2021

RUSSELL RICKS
RUSSELL RICKS*
Affiliation:
Binghamton University, Binghamton, New York, USA
*
e-mail: [email protected]
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Abstract

Let X be a compact, geodesically complete, locally CAT(0) space such that the universal cover admits a rank-one axis. Assume X is not homothetic to a metric graph with integer edge lengths. Let $P_t$ be the number of parallel classes of oriented closed geodesics of length at most t; then $\lim \nolimits _{t \to \infty } P_t / ({e^{ht}}/{ht}) = 1$ , where h is the entropy of the geodesic flow on the space $GX$ of parametrized unit-speed geodesics in X.

Keywords

classical ergodic theoryentropynon-positive curvature

MSC classification

Primary: 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 53C22: Geodesics
Secondary: 28D20: Entropy and other invariants
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 3: Anatole Katok Memorial Issue Part 2: Special Issue of Ergodic Theory and Dynamical Systems , March 2022 , pp. 1220 - 1251
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Ballmann, W.. Lectures on Spaces of Nonpositive Curvature (DMV Seminar, 25) . Birkhäuser, Basel, 1995, with an appendix by Misha Brin.CrossRefGoogle Scholar
Bridson, M. R. and Haefliger, A.. Metric Spaces of Non-Positive Curvature (Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319). Springer, Berlin, 1999.Google Scholar
Burns, K., Climenhaga, V., Fisher, T. and Thompson, D. J.. Unique equilibrium states for geodesic flows in nonpositive curvature. Geom. Funct. Anal. 28(5) (2018), 1209–1259.CrossRefGoogle Scholar
Gekhtman, I. and Yang, W.-Y.. Counting conjugacy classes in groups with contracting elements. Preprint, 2019, arXiv:1810.02969.Google Scholar
Gunesch, R.. Counting closed geodesics on rank one manifolds. Preprint, 2007, arXiv:0706.2845.Google Scholar
Knieper, G.. The uniqueness of the measure of maximal entropy for geodesic flows on rank $1$ manifolds. Ann. of Math. (2) 148(1) (1998), 291–314.CrossRefGoogle Scholar
Knieper, G.. Closed geodesics and the uniqueness of the maximal measure for rank 1 geodesic flows. Smooth Ergodic Theory and Its Applications (Seattle, WA, 1999) (Proceedings of Symposia in Pure Mathematics, 69). American Mathematical Society, Providence, RI, 2001, pp. 573–590.CrossRefGoogle Scholar
Link, G.. Equidistribution and counting of orbit points for discrete rank one isometry groups of Hadamard spaces. Tunis. J. Math. 2(4) (2020), 791–839.CrossRefGoogle Scholar
Liu, F., Wang, F. and Wu, W.. On the Patterson–Sullivan measure for geodesic flows on rank $1$ manifolds without focal points. Discrete Contin. Dyn. Syst. 40 (2020), 1517–1554.CrossRefGoogle Scholar
Margulis, G. A.. On Some Aspects of the Theory of Anosov Systems (Springer Monographs in Mathematics). Springer, Berlin, 2004, with a survey by Richard Sharp: Periodic Orbits of Hyperbolic Flows, translated from the Russian by Valentina Vladimirovna Szulikowska.CrossRefGoogle Scholar
Papasoglu, P. and Swenson, E.. Boundaries and JSJ decompositions of $\mathrm{CAT}(0)$ -groups. Geom. Funct. Anal. 19(2) (2009), 559–590.CrossRefGoogle Scholar
Ricks, R.. Flat strips, Bowen–Margulis measures, and mixing of the geodesic flow for rank one $\mathrm{CAT}(0)$ spaces. Ergod. Th. & Dynam. Sys. 37(3) (2017), 939–970.CrossRefGoogle Scholar
Ricks, R.. The unique measure of maximal entropy for a compact rank one locally $\mathrm{CAT}(0)$ space. Discrete Contin. Dyn. Syst. 41(2) (2021), 507–523.CrossRefGoogle Scholar
Roblin, T.. Ergodicité et équidistribution en courbure négative. Mém. Soc. Math. Fr. (N.S.) 95 (2003), vi + 96.Google Scholar
Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York, 1982.CrossRefGoogle Scholar