Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Homology of closed orbits of Anosov flows


Author: J. F. Plante
Journal: Proc. Amer. Math. Soc. 37 (1973), 297-300
MSC: Primary 58F15
DOI: https://doi.org/10.1090/S0002-9939-1973-0310927-0
MathSciNet review: 0310927
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that for a volume preserving Anosov flow on a compact manifold the closed orbits span the first homology (with real coefficients). The proof uses the notion of asymptotic cycles and results concerning the space of invariant measures for hyperbolic flows.


References [Enhancements On Off] (What's this?)

  • [1] D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature, Trudy Mat. Inst. Steklov. 90 (1967) = Proc Steklov Inst. Math. 90 (1967). MR 39 #3527. MR 0224110 (36:7157)
  • [2] V. I. Arnol'd, The one-dimensional cohomologies of the Lie algebra of divergencefree vector fields, and the winding numbers of dynamical systems, Funkcional Anal. i Priložen. 3 (1969), no 4, 77-78. (Russian) MR 41 #7706. MR 0263101 (41:7706)
  • [3] R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math. 94 (1972), 1-30. MR 0298700 (45:7749)
  • [4] M. W. Hirsch, C. C. Pugh and M. Shub, Invariant manifolds, Bull. Amer. Math. Soc. 76 (1970), 1015-1019. MR 0292101 (45:1188)
  • [5] J. Plante, Diffeomorphisms with invariant line bundles, Invent. Math. 13 (1971), 325-334. MR 0307263 (46:6383)
  • [6] -, Anosov flows, Amer. J. Math. 94 (1972). MR 0377930 (51:14099)
  • [7] S. Schwartzman, Asymptotic cycles, Ann. of Math. (2) 66 (1957), 270-284. MR 19, 568. MR 0088720 (19:568i)
  • [8] K. Sigmund, On the space of invariant measures for hyperbolic flows, Amer. J. Math. 94 (1972), 31-37. MR 0302866 (46:2009)
  • [9] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. MR 37 #3598. MR 0228014 (37:3598)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58F15

Retrieve articles in all journals with MSC: 58F15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0310927-0
Keywords: Anosov flow, homology, closed orbit, invariant measure, asymptotic cycle
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society