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Closed orbits in homology classes for Anosov flows. (English) Zbl 0783.58059
We consider transitive Anosov flows \(\varphi:M \to M\) and give necessary and sufficient conditions for every homology class in \(H_ 1(M,\mathbb{Z})\) to contain a closed \(\varphi\)-orbit. Under these conditions, we derive an asymptotic formula for the number of closed \(\varphi\)-orbits in a fixed homology class, generalizing a result of Katsuda and Sunada.
Reviewer: R.Sharp

MSC:
37D99 Dynamical systems with hyperbolic behavior
37C10 Dynamics induced by flows and semiflows
37A99 Ergodic theory
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[1] DOI: 10.1215/S0012-7094-87-05515-3 · Zbl 0642.53050 · doi:10.1215/S0012-7094-87-05515-3
[2] DOI: 10.2307/2373755 · Zbl 0257.58007 · doi:10.2307/2373755
[3] Parry, Ergod. Th. & Dynam. Sys. 6 pp 133– (1986)
[4] Franks, Global Theory of Dynamical Systems, Proceedings 819 (1979) · Zbl 0452.58013
[5] DOI: 10.1215/S0012-7094-87-05536-0 · Zbl 0648.58041 · doi:10.1215/S0012-7094-87-05536-0
[6] Delange, Ann. Sci. Ecole Norm. Sup. 17 pp 213– (1954)
[7] Rham, Variétés Différentiates. Formes Courantes, Formes Harmoniques (1955)
[8] DOI: 10.1007/BF01449153 · Zbl 0008.37302 · doi:10.1007/BF01449153
[9] DOI: 10.1007/BF01389848 · Zbl 0311.58010 · doi:10.1007/BF01389848
[10] DOI: 10.2307/2373793 · Zbl 0282.58009 · doi:10.2307/2373793
[11] Anosov, Proc. Steklov Inst. Math. 90 pp 1– (1967)
[12] Abramov, Amer. Math. Soc. Transl. 49 pp 167– (1996) · Zbl 0185.21803 · doi:10.1090/trans2/049/08
[13] Walters, Introduction to Ergodic Theory. Springer Graduate Texts in Mathematics 79 (1982) · Zbl 0475.28009 · doi:10.1007/978-1-4612-5775-2
[14] DOI: 10.1070/RM1972v027n04ABEH001383 · doi:10.1070/RM1972v027n04ABEH001383
[15] Parry, Ergod. Th. & Dynam. Sys. 4 pp 117– (1984)
[16] DOI: 10.1007/BF01233428 · Zbl 0705.28012 · doi:10.1007/BF01233428
[17] DOI: 10.1112/blms/3.2.215 · Zbl 0219.58007 · doi:10.1112/blms/3.2.215
[18] Livsic, Math. Notes 10 pp 758– (1971) · Zbl 0235.58010 · doi:10.1007/BF01109040
[19] Lang, Algebraic Number Theory (1970) · Zbl 0211.38404
[20] DOI: 10.1215/S0012-7094-89-05837-7 · Zbl 0732.53035 · doi:10.1215/S0012-7094-89-05837-7
[21] Katsuda, Publ. Math. IHES 71 pp 5– (1990) · Zbl 0728.58026 · doi:10.1007/BF02699875
[22] DOI: 10.2307/2374542 · Zbl 0647.53036 · doi:10.2307/2374542
[23] Katsuda, Proc. Taniguchi Symp. 1988. Springer Lecture Notes 1339 (1988)
[24] DOI: 10.1016/0040-9383(82)90017-9 · Zbl 0594.58041 · doi:10.1016/0040-9383(82)90017-9
[25] DOI: 10.2307/1969999 · Zbl 0207.22603 · doi:10.2307/1969999
[26] Ruelle, Thermodynamic Formalism (1978)
[27] DOI: 10.1090/S0002-9904-1976-14003-7 · Zbl 0316.58016 · doi:10.1090/S0002-9904-1976-14003-7
[28] DOI: 10.2307/2374830 · Zbl 0728.53031 · doi:10.2307/2374830
[29] Pollicott, Ergod. Th. & Dyam. Sys. 4 pp 135– (1984)
[30] Parry, Astérisque 187?188 pp none– (1990)
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