Hostname: page-component-7c8c6479df-27gpq Total loading time: 0 Render date: 2024-03-27T10:30:54.341Z Has data issue: false hasContentIssue false

Closed orbits in homology classes for Anosov flows

Published online by Cambridge University Press:  19 September 2008

Richard Sharp*
Affiliation:
Institut des Hautes Etudes Scientifiques, 35 route de Chartres, 91440 Bures-sur-Yvette, France
*
School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London E1 4NS, UK.

Abstract

We consider transitive Anosov flows φ: MM and give necessary and sufficient conditions for every homology class in H1(M,ℤ) to contain a closed φ-orbit. Under these conditions, we derive an asymptotic formula for the number of closed φ-orbits in a fixed homology class, generalizing a result of Katsuda and Sunada.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1993

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.)

References

REFERENCES

[1]Abramov, L. M.. On the entropy of a flow. Amer. Math. Soc. Transl. 49 (1996), 167170.Google Scholar
[2]Anosov, D.. Geodesic flows on closed Riemannian manifolds with negative curvature. Proc. Steklov Inst. Math. 90 (1967), 1235.Google Scholar
[3]Bowen, R.. Symbolic dynamics for hyperbolic flows. Amer. J. Math. 95 (1973), 429459.CrossRefGoogle Scholar
[4]Bowen, R. & Ruelle, D.. The ergodic theory of Axiom A flows. Invent. Math. 29 (1975), 181202.CrossRefGoogle Scholar
[5]Bruschlinsky, N.. Stetige Abbildungen und Bettische Gruppen. Math. Ann. 109 (1934), 525537.CrossRefGoogle Scholar
[6]Rham, G. de. Variétés Différentiates. Formes Courantes, Formes Harmoniques. Herman, Paris, 1955.Google Scholar
[7]Delange, H.. Généralization du Théorème de Ikehara. Ann. Sci. Ecole Norm. Sup. 17 (1954), 213242.CrossRefGoogle Scholar
[8]Epstein, C.. Asymptotics for closed geodesies in a homology class-finite volume case. Duke Math. J. 55 (1987), 717757.CrossRefGoogle Scholar
[9]Franks, J. & Williams, R. F.. Anomalous Anosov flows. Global Theory of Dynamical Systems, Proceedings, Northwestern 1979. Nitecki, Z. and Robinson, C., eds, Springer Lecture Notes 819. Springer, Berlin, Heidelberg, New York, 1980.Google Scholar
[10]Fried, D.. The geometry of cross sections to flows. Topology 21 (1982), 353371.CrossRefGoogle Scholar
[11]Katsuda, A.. Density theorem for closed orbits. Proc. Taniguchi Symp. 1988. Springer Lecture Notes 1339. Springer, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1988.Google Scholar
[12]Katsuda, A. & Sunada, T.. Homology and closed geodesies in a compact Riemann surface. Amer. J. Math. 110 (1988), 145156.CrossRefGoogle Scholar
[13]Katsuda, A. & Sunada, T.. Closed orbits in homology classes. Publ. Math. IHES 71 (1990), 532.CrossRefGoogle Scholar
[14]Lalley, S.. Closed geodesies in homology classes on surfaces of variable negative curvature. Duke Math. J. 58 (1989), 795821.CrossRefGoogle Scholar
[15]Lang, S.. Algebraic Number Theory. Addison-Wesley, Reading, MA, 1970.Google Scholar
[16]Livsic, A. N.. Homology properties of Y-systems. Math. Notes 10 (1971), 758763.Google Scholar
[17]Manning, A.. Axiom A diffeomorphisms have rational zeta functions. Bull. London Math. Soc. 3 (1971), 215220.CrossRefGoogle Scholar
[18]Marcus, B. & Tuncel, S.. Entropy at a weight-per-symbol and embeddings of Markov chains. Invent. Math. 102 (1990), 235266.CrossRefGoogle Scholar
[19]Parry, W.. Bowen's equidistribution theory and the Dirichlet density theorem. Ergod. Th. & Dynam. Sys. 4 (1984), 117134.CrossRefGoogle Scholar
[20]Parry, W. & Pollicott, M.. The Chebotarov theorem for Galois coverings of Axiom A flows. Ergod. Th. & Dynam. Sys. 6 (1986), 133148.CrossRefGoogle Scholar
[21]Parry, W. & Pollicott, M.. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187–188 (1990).Google Scholar
[22]Phillips, R. & Sarnak, P.. Geodesies in homology classes. Duke Math. J. 55 (1987), 287297.CrossRefGoogle Scholar
[23]Plante, J.. Anosov flows. Amer. J. Math. 94 (1972), 729754.CrossRefGoogle Scholar
[24]Pollicott, M.. A complex Ruelle-Perron-Frobenius theorem and two counter examples. Ergod. Th. & Dyam. Sys. 4 (1984), 135146.CrossRefGoogle Scholar
[25]Pollicott, M.. Homology and closed geodesies in a compact negatively curved surface. Amer. J. Math. 113 (1991), 379385.CrossRefGoogle Scholar
[26]Ruelle, D.. Generalized zeta functions for Axiom A basic sets. Bull. Amer. Math. Soc. 82 (1976), 153156.CrossRefGoogle Scholar
[27]Ruelle, D.. Thermodynamic Formalism. Addison-Wesley, Reading, MA, 1978.Google Scholar
[28]Schwartzman, S.. Asymptotic Cycles. Ann. of Math. 66 (1957), 270284.CrossRefGoogle Scholar
[29]Sharp, R.. Prime orbits theorems with multi-dimensional constraints for Axiom A flows. Preprint, 1990.Google Scholar
[30]Sinai, Ya. G.. Gibbs measures in ergodic theory. Russian Math. Surveys 27(3) (1972), 2164.CrossRefGoogle Scholar
[31]Walters, P.. An Introduction to Ergodic Theory. Springer Graduate Texts in Mathematics 79. Springer, Berlin, Heidelberg, New York, 1982.Google Scholar