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Closed orbits in homology classes for Anosov flows

  • Richard Sharp (a1)
Abstract
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.

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School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London E1 4NS, UK.
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[3] R. Bowen . Symbolic dynamics for hyperbolic flows. Amer. J. Math. 95 (1973), 429459.

[4] R. Bowen & D. Ruelle . The ergodic theory of Axiom A flows. Invent. Math. 29 (1975), 181202.

[5] N. Bruschlinsky . Stetige Abbildungen und Bettische Gruppen. Math. Ann. 109 (1934), 525537.

[8] C. Epstein . Asymptotics for closed geodesies in a homology class-finite volume case. Duke Math. J. 55 (1987), 717757.

[10] D. Fried . The geometry of cross sections to flows. Topology 21 (1982), 353371.

[12] A. Katsuda & T. Sunada . Homology and closed geodesies in a compact Riemann surface. Amer. J. Math. 110 (1988), 145156.

[13] A. Katsuda & T. Sunada . Closed orbits in homology classes. Publ. Math. IHES 71 (1990), 532.

[14] S. Lalley . Closed geodesies in homology classes on surfaces of variable negative curvature. Duke Math. J. 58 (1989), 795821.

[17] A. Manning . Axiom A diffeomorphisms have rational zeta functions. Bull. London Math. Soc. 3 (1971), 215220.

[18] B. Marcus & S. Tuncel . Entropy at a weight-per-symbol and embeddings of Markov chains. Invent. Math. 102 (1990), 235266.

[22] R. Phillips & P. Sarnak . Geodesies in homology classes. Duke Math. J. 55 (1987), 287297.

[23] J. Plante . Anosov flows. Amer. J. Math. 94 (1972), 729754.

[25] M. Pollicott . Homology and closed geodesies in a compact negatively curved surface. Amer. J. Math. 113 (1991), 379385.

[26] D. Ruelle . Generalized zeta functions for Axiom A basic sets. Bull. Amer. Math. Soc. 82 (1976), 153156.

[28] S. Schwartzman . Asymptotic Cycles. Ann. of Math. 66 (1957), 270284.

[30] Ya. G. Sinai . Gibbs measures in ergodic theory. Russian Math. Surveys 27(3) (1972), 2164.

[31] P. Walters . 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
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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