Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-12-01T02:40:36.334Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamics of the geodesic flow of a foliation

Published online by Cambridge University Press:  19 September 2008

Paweł G. Walczak
Paweł G. Walczak
Affiliation:
University of Łódź, Institute of Mathematics, Banacha 22, PL 90238 Łódź, Poland
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.

The geodesic flow of a foliated Riemannian manifold (M, F) is studied. The invariance of some smooth measure is established under some geometrical conditions on F. The Lyapunov exponents and the entropy of this flow are estimated. As an application, the non-existence of foliations with ‘short’ second fundamental tensors is obtained on compact negatively curved manifolds.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 8 , Issue 4 , December 1988 , pp. 637 - 650
Copyright
Copyright © Cambridge University Press 1988

References

REFERENCES

[A]Anosov, D. V.. Geodesic flows on closed Riemannian manifolds with negative curvature. Trudy Mat. Inst. Steklov 90 (1967), 1–235 (in Russian, English translation: Proc. Steklov Inst. Math.).Google Scholar
[BBE]Ballman, W., Brin, M. & Eberlein, P.. Structure of manifolds of non-positive curvature, I. Ann. of Math. 122 (1985), 171–203.CrossRefGoogle Scholar
[BBS]Ballman, W., Brin, M. & Spatzier, R.. Structure of manifolds of non-positive curvature, II. Ann. of Math. 122 (1985), 205–235.Google Scholar
[Br]Brito, F. G. B.. Une obstruction géométrique a l'existence de feuilletages de codimension 1 totalement géodésiques. J. Diff. Geom. 16 (1981), 675–684.Google Scholar
[E]Eberlein, P.. When is a geodesic flow of Anosov type? J Diff. Geom. 8 (1972), I 437–463, II 565–577.Google Scholar
[FM]Freire, A. & Mañe, R.. On the entropy of the geodesic flow in manifolds without conjugate points. Invent. Math. 69 (1982), 375–392.Google Scholar
[Ga]Garnett, L.. Foliations, the ergodic theorem and Brownian motion. J. Func. Anal. 51 (1983), 285–311.Google Scholar
[GLW]Ghys, E., Langevin, R. & Walczak, P. G.. Entropie géométrique des feuilletages. Ada Math. 168 (1988), 105–142.Google Scholar
[Gr]Green, L. W.. A theorem of E. Hopf. Mich. Math. J. 5 (1958), 31–34.Google Scholar
[GKM]Gromoll, D., Klingenberg, W. & Meyer, W.. Riemannsche Geometrie im Grossen, Springer Verlag: Berlin-Heidelberg-New York, 1968.CrossRefGoogle Scholar
[KN]Kobayashi, S. & Nomizu, K.. Foundations of Differential Geometry, II, Interscience Publishers: New York-London-Sydney, 1969.Google Scholar
[K]Klingenberg, W.. Lecture on Closed Geodesies, Springer Verlag: Berlin-Heidelberg-New York, 1978.Google Scholar
[M]Mañe, R.. Ergodic Theory and Differentiable Dynamics, Springer Verlag: Berlin-Heidelberg-New York, 1987.Google Scholar
[Ma1]Manning, A.. Topological entropy for geodesic flows. Ann. of Math. 110 (1979), 567–573.CrossRefGoogle Scholar
[Ma2]Manning, A.. Curvature bounds for the entropy of the geodesic flow on a surface. J. London Math. Soc. 24 (1981), 351–357.Google Scholar
[O]Oseledets, V. I.. A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trudy Moskov. Mat. Obshch. 19 (1968), 197–221 (in Russian, English translation: Trans. Moscow Math. Soc.).Google Scholar
[OS]Osserman, R. & Sarnak, P.. A new curvature invariant and entropy of geodesic flows. Invent. Math. 77 (1984), 455–462.CrossRefGoogle Scholar
[P]Pesin, Ya. B.. Lyapunov characteristic exponents and smooth ergodic theory. Usp. Mat. Nauk 32 (1977), 55–111 (in Russian, English translation: Russ. Math. Surveys).Google Scholar
[Pl]Plante, J.. Foliations with measure preserving holonomy. Ann. of Math. 102 (1975), 327–361.Google Scholar
[R]Ruelle, D.. Ergodic theory of differentiable dynamical systems. Publ. Math. IHES 50 (1979), 275–305.CrossRefGoogle Scholar
[S]Sarnak, P.. Entropy estimates for geodesic flows. Ergod. Th. & Dynam. Sys. 2 (1982), 513–524.Google Scholar
[Wa]Walters, P.. An Introduction to Ergodic Theory, Springer Verlag: Berlin-Heidelberg-New York, 1982.CrossRefGoogle Scholar
[Z]Zeghib, A.. Feuilletages géodésiques des variétés localement symetriques et applications. Thèse, Univ. de Dijon, 1985.Google Scholar