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Closed orbits in homology classes for Anosov flows

  • Richard Sharp (a1)

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.

Corresponding author
School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London E1 4NS, UK.
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[1]Abramov L. M.. On the entropy of a flow. Amer. Math. Soc. Transl. 49 (1996), 167170.
[2]Anosov D.. Geodesic flows on closed Riemannian manifolds with negative curvature. Proc. Steklov Inst. Math. 90 (1967), 1235.
[3]Bowen R.. Symbolic dynamics for hyperbolic flows. Amer. J. Math. 95 (1973), 429459.
[4]Bowen R. & Ruelle D.. The ergodic theory of Axiom A flows. Invent. Math. 29 (1975), 181202.
[5]Bruschlinsky N.. Stetige Abbildungen und Bettische Gruppen. Math. Ann. 109 (1934), 525537.
[6]Rham G. de. Variétés Différentiates. Formes Courantes, Formes Harmoniques. Herman, Paris, 1955.
[7]Delange H.. Généralization du Théorème de Ikehara. Ann. Sci. Ecole Norm. Sup. 17 (1954), 213242.
[8]Epstein C.. Asymptotics for closed geodesies in a homology class-finite volume case. Duke Math. J. 55 (1987), 717757.
[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.
[10]Fried D.. The geometry of cross sections to flows. Topology 21 (1982), 353371.
[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.
[12]Katsuda A. & Sunada T.. Homology and closed geodesies in a compact Riemann surface. Amer. J. Math. 110 (1988), 145156.
[13]Katsuda A. & Sunada T.. Closed orbits in homology classes. Publ. Math. IHES 71 (1990), 532.
[14]Lalley S.. Closed geodesies in homology classes on surfaces of variable negative curvature. Duke Math. J. 58 (1989), 795821.
[15]Lang S.. Algebraic Number Theory. Addison-Wesley, Reading, MA, 1970.
[16]Livsic A. N.. Homology properties of Y-systems. Math. Notes 10 (1971), 758763.
[17]Manning A.. Axiom A diffeomorphisms have rational zeta functions. Bull. London Math. Soc. 3 (1971), 215220.
[18]Marcus B. & Tuncel S.. Entropy at a weight-per-symbol and embeddings of Markov chains. Invent. Math. 102 (1990), 235266.
[19]Parry W.. Bowen's equidistribution theory and the Dirichlet density theorem. Ergod. Th. & Dynam. Sys. 4 (1984), 117134.
[20]Parry W. & Pollicott M.. The Chebotarov theorem for Galois coverings of Axiom A flows. Ergod. Th. & Dynam. Sys. 6 (1986), 133148.
[21]Parry W. & Pollicott M.. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187–188 (1990).
[22]Phillips R. & Sarnak P.. Geodesies in homology classes. Duke Math. J. 55 (1987), 287297.
[23]Plante J.. Anosov flows. Amer. J. Math. 94 (1972), 729754.
[24]Pollicott M.. A complex Ruelle-Perron-Frobenius theorem and two counter examples. Ergod. Th. & Dyam. Sys. 4 (1984), 135146.
[25]Pollicott M.. Homology and closed geodesies in a compact negatively curved surface. Amer. J. Math. 113 (1991), 379385.
[26]Ruelle D.. Generalized zeta functions for Axiom A basic sets. Bull. Amer. Math. Soc. 82 (1976), 153156.
[27]Ruelle D.. Thermodynamic Formalism. Addison-Wesley, Reading, MA, 1978.
[28]Schwartzman S.. Asymptotic Cycles. Ann. of Math. 66 (1957), 270284.
[29]Sharp R.. Prime orbits theorems with multi-dimensional constraints for Axiom A flows. Preprint, 1990.
[30]Sinai Ya. G.. Gibbs measures in ergodic theory. Russian Math. Surveys 27(3) (1972), 2164.
[31]Walters P.. An Introduction to Ergodic Theory. Springer Graduate Texts in Mathematics 79. Springer, Berlin, Heidelberg, New York, 1982.
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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
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