Hostname: page-component-7f64f4797f-kg7gq Total loading time: 0.001 Render date: 2025-11-05T08:53:43.509Z Has data issue: false hasContentIssue false
  • English
  • Français

Counting closed geodesics on rank-one manifolds without focal points

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

Published online by Cambridge University Press:  03 November 2025

WEISHENG WU
WEISHENG WU*
Affiliation:
School of Mathematical Sciences, Xiamen University , China
*
e-mail: wuweisheng@xmu.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article, we consider a closed rank-one Riemannian manifold M without focal points. Let $P(t)$ be the set of free-homotopy classes containing a closed geodesic on M with length at most t, and $\# P(t)$ its cardinality. We obtain the following Margulis-type asymptotic estimates:

$$ \begin{align*}\lim_{t\to \infty}\#P(t)/\frac{e^{ht}}{ht}=1\end{align*} $$
where h is the topological entropy of the geodesic flow. We also show that the unique measure of maximal entropy of the geodesic flow has the Bernoulli property.

Keywords

geodesic flowmeasure of maximal entropyclosed geodesicsBernoulli property

MSC classification

Primary: 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
Secondary: 37C40: Smooth ergodic theory, invariant measures

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 28
Check for updates
Copyright
© The Author(s), 2025. 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.)

Article purchase

Temporarily unavailable

References

Alansari, N.. Ergodic properties of measures with local product structure. Preprint, 2022, arXiv:2206.02242.Google Scholar
Anosov, D. V.. Geodesic flows on closed Riemannian manifolds with negative curvature. Proc. Steklov Inst. Math. 90 (1967), 1–235.Google Scholar
Babillot, M.. On the mixing property for hyperbolic systems. Israel J. Math. 129(1) (2002), 61–76.10.1007/BF02773153CrossRefGoogle Scholar
Barreira, L. and Pesin, Ya B.. Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents (Encyclopedia of Mathematics and Its Applications, 115). Cambridge University Press, Cambridge, 2007.10.1017/CBO9781107326026CrossRefGoogle Scholar
Bowen, R.. Periodic orbits for hyperbolic flows. Amer. J. Math. 94 (1972), 1–30.10.2307/2373590CrossRefGoogle Scholar
Bowen, R.. Maximizing entropy for a hyperbolic flow. Math. Systems Theory 7 (1974), 300–303.10.1007/BF01795948CrossRefGoogle Scholar
Burns, K., Climenhaga, V., Fisher, T. and Thompson, D. J.. Unique equilibrium states for geodesic flows in nonpositive curvature. Geom. Funct. Anal. 28 (2018), 1209–1259.10.1007/s00039-018-0465-8CrossRefGoogle Scholar
Burns, K. and Katok, A.. Manifolds with non-positive curvature. Ergod. Th. & Dynam. Sys. 5 (1985), 307–317.10.1017/S0143385700002935CrossRefGoogle Scholar
Call, B. and Thompson, D. J.. Equilibrium states for self-products of flows and the mixing properties of rank 1 geodesic flows. J. Lond. Math. Soc. (2) 105(2) (2022), 794–824.10.1112/jlms.12517CrossRefGoogle Scholar
Chen, D., Kao, L. and Park, K.. Unique equilibrium states for geodesic flows over surfaces without focal points. Nonlinearity 33(3) (2020), 1118–1155.10.1088/1361-6544/ab5c06CrossRefGoogle Scholar
Chen, D., Kao, L. and Park, K.. Properties of equilibrium states for geodesic flows over manifolds without focal points. Adv. Math. 380 (2021), 107564.10.1016/j.aim.2021.107564CrossRefGoogle Scholar
Chernov, N. I. and Haskell, C.. Nonuniformly hyperbolic K-systems are Bernoulli. Ergod. Th. & Dynam. Sys. 16(1) (1996), 19–44.10.1017/S0143385700008695CrossRefGoogle Scholar
Climenhaga, V., Knieper, G. and War, K.. Uniqueness of the measure of maximal entropy for geodesic flows on certain manifolds without conjugate points. Adv. Math. 376 (2021), 107452.10.1016/j.aim.2020.107452CrossRefGoogle Scholar
Climenhaga, V., Knieper, G. and War, K.. Closed geodesics on surfaces without conjugate points, Commun. Contemp. Math. 24(6) (2022), Paper no. 2150067.10.1142/S021919972150067XCrossRefGoogle Scholar
Climenhaga, V., Pesin, Y. B. and Zelerowicz, A.. Equilibrium measures for some partially hyperbolic systems. J. Mod. Dyn. 16 (2020), 155–205.10.3934/jmd.2020006CrossRefGoogle Scholar
Croke, C. B. and Schroeder, V.. The fundamental group of compact manifolds without conjugate points. Comment. Math. Helv. 61 (1986), 161–175.10.1007/BF02621908CrossRefGoogle Scholar
Dal’Bo, F., Peigné, M. and Sambusetti, A.. On the horoboundary and the geometry of rays of negatively curved manifolds. Pacific J. Math. 259 (2012), 55–100.10.2140/pjm.2012.259.55CrossRefGoogle Scholar
Druetta, M. J. and Ferraris, C. J.. The duality condition on manifolds without focal points. Arch. Math. 36(1) (1981), 370–377.10.1007/BF01223712CrossRefGoogle Scholar
Eberlein, P.. Geodesic flow in certain manifolds without conjugate points. Trans. Amer. Math. Soc. 167 (1972), 151–170.10.1090/S0002-9947-1972-0295387-4CrossRefGoogle Scholar
Eberlein, P. and O’Neill, B.. Visibility manifolds. Pacific J. Math. 46(1) (1973), 45–109.10.2140/pjm.1973.46.45CrossRefGoogle Scholar
Fisher, T. and Hasselblatt, B.. Hyperbolic Flows (Zurich Lectures in Advanced Mathematics). EMS Publishing Houise, Berlin, 2019.10.4171/200CrossRefGoogle Scholar
Freire, A. and Mañé, R.. On the entropy of the geodesic flow in manifolds without conjugate points. Invent. Math. 69 (1982), 375–392.10.1007/BF01389360CrossRefGoogle Scholar
Gelfert, K. and Ruggiero, R.. Geodesic flows modelled by expansive flows. Proc. Edinb. Math. Soc. (2) 62 (2019), 61–95.10.1017/S0013091518000160CrossRefGoogle Scholar
Gunesch, R.. Counting closed geodesics on rank one manifolds. Preprint, 2007, arXiv:0706.2845.Google Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and Its Applications, 54). Cambridge University Press, Cambridge, 1995.10.1017/CBO9780511809187CrossRefGoogle Scholar
Knieper, G.. On the asymptotic geometry of nonpositively curved manifolds. Geom. Funct. Anal. 7 (1997), 755–782.10.1007/s000390050025CrossRefGoogle Scholar
Knieper, G.. The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds. Ann. of Math. (2) 148 (1998), 291–314.10.2307/120995CrossRefGoogle Scholar
Knieper, G.. Hyperbolic dynamics and riemannian geometry. Handbook of Dynamical Systems, Vol. 1A. Ed. B. Hasselblatt and A. Katok. North-Holland, Amsterdam, 2002, pp. 453–545.Google Scholar
Ledrappier, F.. Mesures d’équilibre d’entropie complètement positive. Astérisque 50 (1977), 251–272.Google Scholar
Ledrappier, F. and Young, L.-S.. The metric entropy of diffeomorphisms: Part II: relations between entropy, exponents and dimension. Ann. of Math. (2) 122(3) (1985), 540–574.10.2307/1971329CrossRefGoogle Scholar
Link, G. and Picaud, J.-C.. Ergodic geometry for non-elementary rank one manifolds. Discrete Contin. Dyn. Syst. Ser. A 36(11) (2016), 6257–6284.Google Scholar
Liu, F., Liu, X. and Wang, F.. On the mixing and Bernoulli properties for geodesic flows on rank 1 manifolds without focal points. Discrete Contin. Dyn. Syst. Ser. A 41(10) (2021), 4791–4804.10.3934/dcds.2021057CrossRefGoogle 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. Ser. A 40(3) (2020), 1517–1554.10.3934/dcds.2020085CrossRefGoogle Scholar
Margulis, G. A.. Certain applications of ergodic theory to the investigation of manifolds of negative curvature. Funktsional. Anal. i Prilozhen. 3(4) (1969), 89–90.Google Scholar
Margulis, G. A.. On Some Aspects of the Theory of Anosov Systems (Springer Monographs in Mathematics). Springer-Verlag, Berlin, 2004; with a survey by Richard Sharp: Periodic orbits of hyperbolic flows, translated from the Russian by Valentina Vladimirovna Szulikowska.10.1007/978-3-662-09070-1CrossRefGoogle Scholar
O’Sullivan, J. J.. Riemannian manifolds without focal points. J. Differential Geom. 11 (1976), 321–333.Google Scholar
Ornstein, D. and Weiss, B.. Geodesic flows are Bernoullian. Israel J. Math. 14 (1973), 184–198.10.1007/BF02762673CrossRefGoogle Scholar
Ornstein, D. and Weiss, B.. On the Bernoulli nature of systems with some hyperbolic structure. Ergod. Th. & Dynam. Sys. 18(2) (1998), 441–456.10.1017/S0143385798100354CrossRefGoogle Scholar
Papasoglu, P. and Swenson, E.. Boundaries and JSJ decompositions of CAT $(0)$ -groups. Geom. Funct. Anal. 19(2) (2009), 559–590.10.1007/s00039-009-0012-8CrossRefGoogle Scholar
Parry, W. and Pollicott, M.. An analogue of the prime number theorem for closed orbits of Axiom A flows. Ann. of Math. (2) 118(3) (1983), 573–591.10.2307/2006982CrossRefGoogle Scholar
Patterson, S. J.. The limit set of a Fuchsian group. Acta Math. 136 (1976), 241–273.10.1007/BF02392046CrossRefGoogle Scholar
Pesin, J. B.. Characteristic Ljapunov exponents, and smooth ergodic theory. Uspekhi Mat. Nauk 32(4(196)) (1977), 55–114.Google Scholar
Pesin, J. B.. Geodesic flows in closed Riemannian manifolds without focal points. Izv. Akad. Nauk SSSR Ser. Mat. 41(6) (1977), 1195–1228.Google Scholar
Ponce, G., Tahzibi, A. and Varão, R.. On the Bernoulli property for certain partially hyperbolic diffeomorphisms. Adv. Math. 329 (2018), 329–360.10.1016/j.aim.2018.02.019CrossRefGoogle Scholar
Ponce, G. and Varão, R.. An Introduction to the Kolmogorov–Bernoulli Equivalence (SpringerBriefs in Mathematics). Springer International Publishing, Cham, 2019.10.1007/978-3-030-27390-3CrossRefGoogle Scholar
Ratner, M.. Anosov flows with Gibbs measures are also Bernoullian. Israel J. Math. 17(4) (1974), 380–391.10.1007/BF02757140CrossRefGoogle Scholar
Ricks, R.. Counting closed geodesics in a compact rank-one locally CAT(0) space. Ergod. Th. & Dynam. Sys. 42(3) (2022), 1220–1251.10.1017/etds.2021.83CrossRefGoogle Scholar
Ruggiero, R.. Expansive geodesic flows in manifolds with no conjugate points. Ergod. Th. & Dynam. Sys. 17 (1997), 211–225.10.1017/S0143385797060963CrossRefGoogle Scholar
Ruggiero, R.. Dynamics and global geometry of manifolds without conjugate points. Ensaios Mat. 12 (2007), 1–181.Google Scholar
Sullivan, D.. The density at infinity of a discrete group of hyperbolic motions. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 171–202.10.1007/BF02684773CrossRefGoogle Scholar
Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer-Verlag, New York–Berlin, 1982.10.1007/978-1-4612-5775-2CrossRefGoogle Scholar
Watkins, J.. The higher rank rigidity theorem for manifolds with no focal points. Geom. Dedicata. 164 (2013), 319–349.10.1007/s10711-012-9776-3CrossRefGoogle Scholar
Weaver, B.. Growth rate of periodic orbits for geodesic flows over surfaces with radially symmetric focusing caps. J. Mod. Dyn. 8(2) (2014), 139–176.10.3934/jmd.2014.8.139CrossRefGoogle Scholar