A Note on Expansiveness and Hyperbolicity for Generic Geodesic Flows

  • Mário BessaEmail author


In this short note we contribute to the generic dynamics of geodesic flows associated to metrics on compact Riemannian manifolds of dimension ≥ 2. We prove that there exists a C2-residual subset \(\mathscr{R}\) of metrics on a given compact Riemannian manifold such that if \(g\in \mathscr{R}\), then its associated geodesic flow \({\varphi ^{t}_{g}}\) is expansive if and only if the closure of the set of periodic orbits of \({\varphi ^{t}_{g}}\) is a uniformly hyperbolic set. For surfaces, we obtain a stronger statement: there exists a C2-residual \(\mathscr{R}\) such that if \(g\in \mathscr{R}\), then its associated geodesic flow \({\varphi ^{t}_{g}}\) is expansive if and only if \({\varphi ^{t}_{g}}\) is an Anosov flow.


Expansiveness Residual sets Anosov Geodesic flows 

Mathematics Subject Classification (2010)

37C20 37D40 53D25 



The author was partially supported by FCT - ‘Fundação para a Ciência e a Tecnologia’, through Centro de Matemática e Aplicações (CMA-UBI), Universidade da Beira Interior, project UID/MAT/00212/2013. The author would like to thank the reviewers for their careful reading of the manuscript and the helpful comments, CMUP for providing the necessary conditions in which this work was developed, and to Maria Joana Torres for conversations about this subject.


  1. 1.
    Abraham, R.: Bumpy metrics. In: Global analysis (Proc. Sympos. Pure Math., vol. XIV, Berkeley, Calif., 1968), pp 1–3. Amer. Math. Soc., Providence (1970)Google Scholar
  2. 2.
    Anosov, D.V.: Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov., 90. (Russian) (1967)Google Scholar
  3. 3.
    Arnaud, M.-C.: Type des Points Fixes des Difféomorphismes Symplectiques de \(\mathbb {T}^{n} \mathbb {R}^{n}\), p 48. Mém. Soc. Math. , France (1992)Google Scholar
  4. 4.
    Bessa, M.: Generic incompressible flows are topological mixing. C. R. Acad. Sci. Paris, Ser. I 346, 1169–1174 (2008)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bessa, M., Torres, M.J.: The c 0 general density theorem for geodesic flows. C. R. Acad. Sci. Paris, Ser. I 353(6), 545–549 (2015)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bessa, M., Lee, M., Wen, X.: Shadowing, expansiveness and specification for c 1-conservative systems. Acta. Math. Sci. 35B(3), 583–600 (2015)CrossRefzbMATHGoogle Scholar
  7. 7.
    Bessa, M., Dias, J.L., Torres, M.J.: On shadowing and hyperbolicity for geodesic flows on surfaces. Nonlinear Anal. Theory Methods Appl. 55, 250–263 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bowen, R., Walters, P.: Expansive one-parameter flows. J. Diff. Equat. 12, 180–193 (1972)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Contreras, G.: Geodesic flows with positive topological entropy, twist maps and hyperbolicity. Annals Math. 172, 761–808 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Contreras, G., Paternain, G.: Genericity of geodesic flows with positive topological entropy on s 2. J. Differ. Geom. 61(1), 1–49 (2002)CrossRefzbMATHGoogle Scholar
  11. 11.
    Klingenberg, W., Takens, F.: Generic properties of geodesic flows. Math. Ann. 197, 323–334 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Le Calvez, P.: PropriétéS Dynamiques Des Difféomorphismes De l’Anneau Et Du Tore. Astérisque 204, Soc. Math., France (1991)zbMATHGoogle Scholar
  13. 13.
    Mãné, R.: Expansive diffeomorphisms, Lecture Notes in Mathematics, vol. 468, pp 162–174. Springer, Berlin (1975)Google Scholar
  14. 14.
    Paternain, M.: Expansive geodesic flows on surfaces. Ergod. Th. & Dynam. Sys. 13(1), 153–165 (1993)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Pugh, C., Robinson, C.: The c 1 closing lemma, including Hamiltonians. Ergod. Th. & Dynam. Sys. 3, 261–313 (1983)CrossRefzbMATHGoogle Scholar
  16. 16.
    Rifford, L.: Closing geodesics in c 1 topology. J. Differ. Geom. 91, 361–381 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ruggiero, R.: Persistently expansive geodesic flows. Commun. Math. Phys. 140, 203–215 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ruggiero, R.: On the creation of conjugate points. Math. Z. 208, 41–55 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ruggiero, R.: On a conjecture about expansive geodesic flows. Ergod. Th. & Dynam. Sys. 16(3), 545–553 (1996)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Zehnder, E.: Homoclinic points near elliptic fixed points. Comm. Pure Appl. Math. 26, 131–182 (1973)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade da Beira InteriorCovilhãPortugal

Personalised recommendations