Hostname: page-component-6bf8c574d5-zc66z Total loading time: 0 Render date: 2025-03-12T02:37:07.186Z Has data issue: false hasContentIssue false
  • English
  • Français

Distribution of closed geodesics with a preassigned homology class in a negatively curved manifold

Published online by Cambridge University Press:  22 January 2016

Toshiaki Adachi
Toshiaki Adachi*
Affiliation:
Department of Mathematics, Nagoya University, Chikusa-ku Nagoya, 464Japan
*
Department of Mathematics, College of General Education, Kumamoto University, Kumamoto, 860Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let M be a compact Riemannian manifold whose geodesic flow φi : UM→UM on the unit tangent bundle is of Anosov type. In this paper we count the number of φi-closed orbits and study the distribution of prime closed geodesies in a given homology class in H1(M, Z). Here a prime closed geodesic means an (oriented) image of a φi-closed orbit by the projection p : UM → M.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 110 , June 1988 , pp. 1 - 14
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1988

References

[1] Adachi, T., Markov families for Anosov flows with an involutive action, Nagoya Math. J., 104 (1986), 55–62.CrossRefGoogle Scholar
[2] Adachi, T. and Sunada, T., Twisted Perron-Frobenius Theorem and L-functions J. Funct. Anal., 71 (1987), 1–46.CrossRefGoogle Scholar
[3] Adachi, T. and Sunada, T., Homology of closed geodesies in a negatively curved manifold, J. Diff. Geom., 26 (1987), 81–99.Google Scholar
[4] Bowen, R., Periodic orbits for hyperbolic flows, Amer. J. Math., 94 (1972), 1–30.CrossRefGoogle Scholar
[5] Bowen, R., The equidistribution of closed geodesies, Amer. J. Math., 94 (1972), 413–423.CrossRefGoogle Scholar
[6] Bowen, R., Symbolic dynamics for hyperbolic flows, Amer. J. Math., 95 (1973), 429–460.CrossRefGoogle Scholar
[7] Eberlein, P., When is a geodesic flow Anosov type? I, J. Differential Geom., 8 (1973), 437–463.Google Scholar
[8] Parry, W., Bowen’s equidistribution theory and the Dirichlet density theorem, Ergodic Theory Dynamical Systems, 4 (1984), 117–134.CrossRefGoogle Scholar
[9] Parry, W. and Pollicott, M., An analogue of the prime number theorem for closed orbits of Axiom A flows, Ann. of Math., 118 (1983), 573–591.CrossRefGoogle Scholar
[10] Parry, W. and Pollicott, M., The Chebotarev theorem for Galois coverings of Axiom A flows, preprint (1984), Warwick Univ.Google Scholar
[11] Plante, J. F., Homology of closed orbits of Anosov flows, Proc. Amer. Math. Soc, 37 (1973), 297–300.CrossRefGoogle Scholar
[12] Pollicott, M., Meromorphic extension of generalised zeta function, Invent, math., 85 (1986), 147–164.CrossRefGoogle Scholar
[13] Pollicott, M., A note on uniform distribution for primes and closed orbits, preprint.CrossRefGoogle Scholar
[14] Rees, M., Checking ergodicity for some geodesic flows with infinite Gibbs measure, Ergodic Theory Dynamical Systems, 1 (1981), 107–133.CrossRefGoogle Scholar
[15] Ruelle, D., Thermodynamic Formalism. Addison-Wesley, 1978.Google Scholar
[16] Sunada, T., Geodesic flows and geodesic random walks. Advanced Studies in Pure Math., 3 (Geometry of Geodesies and Related Topics) 1984, 47–86.CrossRefGoogle Scholar
[17] Sunada, T., Tchebotarev’s density theorem for closed geodesies in a compact locally symmetric space of negative curvature, preprint.Google Scholar