Hostname: page-component-7b7567c8c-2n9c9 Total loading time: 0 Render date: 2026-11-15T09:25:44.188Z Has data issue: false hasContentIssue false
  • English
  • Français

Hyperbolic behaviour of geodesic flows on manifolds with no focal points

Published online by Cambridge University Press:  19 September 2008

Keith Burns
Keith Burns
Affiliation:
Department of Mathematics, Southern Illinois University, Carbondale, Illinois 62901
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

It is shown that the unit tangent bundle of a compact uniform visibility manifold with no focal points contains a subset of positive Liouville measure on which all the characteristic exponents of the geodesic flow (except in the flow direction) are non-zero. This completes Pesin's proof that the geodesic flow of such a manifold is Bernoulli.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 3 , Issue 1 , March 1983, pp. 1 - 12
Copyright
Copyright © Cambridge University Press 1983

References

REFERENCES

[1]Ballmann, W.. Einige neue Resultate über Mannigfaltigkeiten nicht positiver Krümmung. Bonner Mathematische Schriften 113.Google Scholar
[2]Ballmann, W.. Axial isometries of manifolds of non-positive curvature. Math. Annalen 259 (1982), 131–144.CrossRefGoogle Scholar
[3]Ballmann, W. & Brin, M.. On the ergodicity of geodesic flows. Ergodic Theory and Dynamical Systems. To appear.Google Scholar
[4]Dugundji, J.. Topology. Allyn and Bacon: Boston, 1966.Google Scholar
[5]Eberlein, P.. Geodesic flow in certain manifolds without conjugate points. Trans. Amer. Math. Soc. 167 (1972), 151–170.CrossRefGoogle Scholar
[6]Eberlein, P.. Geodesic flows on negatively curved manifolds I. Ann. of Math. 95 (1972), 492–510.CrossRefGoogle Scholar
[7]Eberlein, P.. Geodesic flows on negatively curved manifolds II. Trans. Amer. Math. Soc. 178 (1973), 57–82.CrossRefGoogle Scholar
[8]Eberlein, P.. When is a geodesic flow of Anosov type? I. J. Differential Geometry. 8 (1973), 437–463.Google Scholar
[9]Eberlein, P.. When is a geodesic flow of Anosov type? II. J. Differential Geometry. 8 (1973), 565–577.Google Scholar
[10]Eberlein, P. & O'Neill, B.. Visibility manifolds. Pacific J. Math. 46 (1973), 45–109.CrossRefGoogle Scholar
[11]Eschenburg, J. H.. Stabilitätsverhalten des geodätischen Flusses riemannscher Mannigfaltigkeiten. Bonner Mathematische Schriften 87.Google Scholar
[12]Eschenburg, J. H.. Horospheres and the stable part of the geodesic flow. Math. Zeitschrift. 153 (1977), 237–251.CrossRefGoogle Scholar
[13]Eschenburg, J. H. & O'Sullivan, J. J.. Growth of Jacobi fields and divergence of geodesics. Math. Zeitschrift. 150 (1976), 221–237.CrossRefGoogle Scholar
[14]Gromoll, D., Klingenberg, W. & Meyer, W.. Riemannsche Geometrie im Grossen. Springer Lecture Notes in Math. no. 55. Springer: Berlin, 1968.CrossRefGoogle Scholar
[15]Nemytskii, V. V. & Stepanov, V. V.. Qualitative theory of differential equations. Princeton University Press: Princeton, 1960.Google Scholar
[16]O'Sullivan, J. J.. Riemannian manifolds without focal points. J. Differential Geometry. 11 (1976), 321–333.CrossRefGoogle Scholar
[17]Pesin, Ja. B.. Characteristic Lyapunov exponents and smooth ergodic theory. Russian Math. Surveys. 32 (1977), 55–114. Uspekhi Mat. Nauk. 32 (1977), 55–112.CrossRefGoogle Scholar
[18]Pesin, Ja. B.. Geodesic flows on closed Riemannian manifolds without focal points. Math. USSR Izvestija. 11 (1977), 1195–1228. Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), 1252–1288.CrossRefGoogle Scholar
[19]Pesin, Ja. B.. Equations for the entropy of a geodesic flow on a compact Riemannian manifold without conjugate points. Math. Notes. 24 (1978), 796–805. Matematicheskie Zametki. 24 (1978), 553–570.CrossRefGoogle Scholar
[20]Sasaki, S.. On the differential geometry of tangent bundles of Riemannian manifolds. Tôhoku Math. J. II Ser. 10 (1958), 338–354.CrossRefGoogle Scholar