Closed orbits in homology classes. (English) Zbl 0728.58026

The authors consider a smooth, transitive and weakly mixing Anosov flow on a compact manifold. By introducing an analogue of the density theorem for prime numbers, they succeed to estimate the number of closed orbits in a homology class.
Given a surjective homomorphism \(\psi\) of \(H_ 1(X,Z)\) onto an abelian group H, for each \(\alpha\in H\) and for each positive number x, the authors set \[ \Pi(x,\alpha)= \{{\mathfrak p};\text{ closed orbits with } \psi[{\mathfrak p}]=\alpha \text{ and }\ell({\mathfrak p})<x\}, \] and \[ \pi(x,\alpha) = \text{ the cardinality of }\Pi(x,\alpha) \] where [\({\mathfrak p}]\) denotes the homology class and \(\ell ({\mathfrak p})\) the least period of \({\mathfrak p}.\)
One of the main results in the present paper is concerned with an asymptotic estimate of \(\pi\) (x,\(\alpha\)) as x goes to infinity. The authors have remarked the resemblance of this problem to a number theoretic problem, and demonstrate that an analogue of the density theorem for prime numbers holds. However, the main problem is that the “Galois group” H is possibly of infinite order, such that interesting extra phenomena appear.
Reviewer: D.Savin (Montreal)


37D99 Dynamical systems with hyperbolic behavior
37A99 Ergodic theory
Full Text: DOI Numdam EuDML


[1] T. Adachi, Meromorphic extension of L-functions of Anosov flows and profinite graphs,Kumamoto J. Math.,1 (1988), 9–24. · Zbl 0668.58041
[2] T. Adachi andT. Sunada, Homology of closed geodesics in a negatively curved manifold,J. Diff. Geom.,26 (1987), 81–99. · Zbl 0618.58028
[3] T. Adachi andT. Sunada, Twisted Perron-Frobenius theorem and L-functions,J. Funct. Anal.,71 (1987), 1–46. · Zbl 0658.58034 · doi:10.1016/0022-1236(87)90014-0
[4] R. Bowen, Symbolic dynamics for hyperbolic flows,Amer. J. Math.,95 (1973), 429–459. · Zbl 0282.58009 · doi:10.2307/2373793
[5] M. Denker andW. Philipp, Approximation by Brownian motion for Gibbs measures and flows under a function,Ergod. Th. & Dynam. Syst.,4 (1984), 541–552. · Zbl 0554.60077
[6] C. Epstein, Asymptotics for closed geodesics in a homology class–finite volume case–,Duke Math. J.,55 (1987), 717–757. · Zbl 0648.58041 · doi:10.1215/S0012-7094-87-05536-0
[7] I. M. Gel’fand andI. I. Pyateckii-Shapiro, A theorem of Poincaré,Dokl. Akad. Nauk.,127 (1959), 490–493 (Russian).
[8] V. Guillemin andD. Kazhdan, Some inverse spectral results for negatively curved 2-manifolds,Topology,11 (1980), 301–312. · Zbl 0465.58027 · doi:10.1016/0040-9383(80)90015-4
[9] P. H. Halmos,Measure Theory, New York, Van Nostrand Co. Inc., 1950.
[10] A. Katsuda, Density theorem for closed orbits,Proc. of Taniguchi Symp., Springer Lect. Notes in Math., vol. 1339, 1988, to appear. · Zbl 0668.58043
[11] A. Katsuda andT. Sunada, Homology and closed geodesics in a compact Riemann surface,Amer. J. Math.,110 (1988), 145–156. · Zbl 0647.53036 · doi:10.2307/2374542
[12] S. Lang,Algebraic Number Theory, London, Addison-Wisley, 1970. · Zbl 0211.38404
[13] A. N. Livcic, Some homological properties of U-systems,Math. Zametki,10 (1971), 555–564.
[14] S. Manabe andY. Ochi, The central limit theorem for current-valued processes induced by geodesic flows,Osaka J. Math.,26 (1989), 191–205. · Zbl 0703.60019
[15] W. Parry, Bowen’s equidistribution theory and the Dirichlet density theorem,Ergod. Th. & Dynam. Syst.,4 (1984), 117–134. · Zbl 0567.58014
[16] W. Parry andM. Pollicott, An analogue of the prime number theorem for closed orbits of Axiom A flows,Ann. of Math.,118 (1983), 573–591. · Zbl 0537.58038 · doi:10.2307/2006982
[17] W. Parry andM. Pollicott, The Chebotarev theorem for Galois coverings of Axiom A flows,Ergod. Th. & Dynam. Syst.,6 (1986), 133–148. · Zbl 0626.58006
[18] R. Phillips andP. Sarnak, Geodesics in homology classes,Duke Math. J.,55 (1987), 287–297. · Zbl 0642.53050 · doi:10.1215/S0012-7094-87-05515-3
[19] J. Plante, Anosov flows,Amer. J. Math.,94 (1972), 729–758. · Zbl 0257.58007 · doi:10.2307/2373755
[20] J. Plante, Homology of closed orbits of Axiom A flows,Proc. A. M. S.,37 (1973), 297–300. · Zbl 0249.58012 · doi:10.1090/S0002-9939-1973-0310927-0
[21] M. Pollicott, On the rate of mixing of Axiom A flows,Invent. Math.,81 (1985), 413–426. · Zbl 0591.58025 · doi:10.1007/BF01388579
[22] D. Ruelle,Thermodynamic Formalism, London, Addison-Wisley, 1978.
[23] A. Selberg, Harmonic analysis and discontinuous subgroups in weakly symmetric Riemannian spaces with applications to Dirichlet series,J. Indian Math. Soc.,20 (1956), 47–87. · Zbl 0072.08201
[24] J.-P. Serre, Divisibilité de certaines fonctions arithmétiques,L’Ens. Math.,22 (1976), 227–260.
[25] S. Zelditch,Geodesics in homology classes and periods of automorphic forms, preprint, Johns Hopkins Univ., 1986.
[26] M. H. Delange, Généralisation du théorème de Ikehara,Ann. Scient. Ec. Norm. Sup., Sér. 3,71 (1954), 213–242. · Zbl 0056.33101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.