Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-12-05T02:44:39.786Z Has data issue: false hasContentIssue false
  • English
  • Français

Expansive geodesic flows on surfaces

Published online by Cambridge University Press:  19 September 2008

Miguel Paternain
Miguel Paternain
Affiliation:
IMPA. Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, Brazil
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove the following result: if M is a compact Riemannian surface whose geodesic flow is expansive, then M has no conjugate points. This result and the techniques of E. Ghys imply that all expansive geodesic flows of a compact surface are topologically equivalent.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 13 , Issue 1 , March 1993 , pp. 153 - 165
Copyright
Copyright © Cambridge University Press 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bowen, R. & Walters, P.. Expansive one-parameter flows. J. Diff. Eq. 12 (1972), 180–193.CrossRefGoogle Scholar
[2]Epstein, D.. Curves on 2-manifolds and isotopies. Acta Math. 115 (1966), 83–107.CrossRefGoogle Scholar
[3]Eschenburg, J. H.. Horospheres and the stable part of the geodesic flow. Math. Zeitschrift 153 (1977), 237–251.CrossRefGoogle Scholar
[4]Franks, J.. Anosov Diffeomorphisms. Proc. Symp. Pure Math. vol. 14 (1970), 61–93.CrossRefGoogle Scholar
[5]Ghys, E.. Flots d'anosov sur les 3-Varietes Fibrees en Cercles. Ergod. Th. & Dynam. Sys. 4 (1984), 67–80.CrossRefGoogle Scholar
[6]Green, L.. Surfaces without conjugate points. Trans. Amer. Math. Soc. 76 (1954), 529–546.CrossRefGoogle Scholar
[7]Hector, G. & Hirsch, U.. Introduction to the Geometry of Foliations. Part B. Aspects of Mathematics. Vieweg (1987).CrossRefGoogle Scholar
[8]Hiraide, K.. Expansive homeomorphisms of compact surfaces are pseudo-Anosov. Osaka J. Math. 27 (1990), 117–162.Google Scholar
[9]Klingenberg, W.. Riemannian manifolds with geodesic flows of Anosov type. Ann. Math. 99 (1974), 1–13.CrossRefGoogle Scholar
[10]Lewowicz, J.. Lyapunov functions and stability of geodesic flows. Springer Lecture Notes in Mathematics 1007 Springer, Berlin, 1981, 463–479.Google Scholar
[11]Lewowicz, J.. Expansive homeomorphisms of surfaces. Bol. Soc. Bras. Mat. 20 (1989), 113–133.CrossRefGoogle Scholar
[12]Mañé, R.. On a Theorem of Klingenberg in Dynamical Systems and Bifurcation Theory. Eds, Camacho, M., Pacifico, M. and Takens, F., Pitman Research Notes in Mathematics 160 Longman, Harlow, 1987, 319–345.Google Scholar
[13]Paternain, M.. Expansive flows in 3-manifolds. Thesis IMPA (1990).Google Scholar