Counting closed geodesics in a compact rank-one locally CAT(0) space. (English) Zbl 1490.37045
Summary: 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 \(X\).
MSC:
| 37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |
| 53C22 | Geodesics in global differential geometry |
| 28D20 | Entropy and other invariants |
References:
| [1] | Ballmann, W.. Lectures on Spaces of Nonpositive Curvature (DMV Seminar, 25). Birkhäuser, Basel, 1995, with an appendix by Misha Brin. · Zbl 0834.53003 |
| [2] | 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. · Zbl 0988.53001 |
| [3] | 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. · Zbl 1401.37038 |
| [4] | Gekhtman, I. and Yang, W.-Y.. Counting conjugacy classes in groups with contracting elements. Preprint, 2019, arXiv:1810.02969. |
| [5] | Gunesch, R.. Counting closed geodesics on rank one manifolds. Preprint, 2007, arXiv:0706.2845. |
| [6] | 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. · Zbl 0946.53045 |
| [7] | 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. · Zbl 1010.53062 |
| [8] | 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. · Zbl 1516.20101 |
| [9] | 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. · Zbl 1501.37032 |
| [10] | 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. · Zbl 1140.37010 |
| [11] | Papasoglu, P. and Swenson, E.. Boundaries and JSJ decompositions of \(\text{CAT}(0)\) -groups. Geom. Funct. Anal.19(2) (2009), 559-590. · Zbl 1226.20038 |
| [12] | Ricks, R.. Flat strips, Bowen-Margulis measures, and mixing of the geodesic flow for rank one \(\text{CAT}(0)\) spaces. Ergod. Th. & Dynam. Sys.37(3) (2017), 939-970. · Zbl 1368.37037 |
| [13] | Ricks, R.. The unique measure of maximal entropy for a compact rank one locally \(\text{CAT}(0)\) space. Discrete Contin. Dyn. Syst.41(2) (2021), 507-523. · Zbl 1460.37034 |
| [14] | Roblin, T.. Ergodicité et équidistribution en courbure négative. Mém. Soc. Math. Fr. (N.S.)95 (2003), vi + 96. · Zbl 1056.37034 |
| [15] | Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York, 1982. · Zbl 0475.28009 |
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