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Sur une notion d'autonomie de systèmes dynamiques, appliquée aux ensembles invariants des flots d' Anosov algébriques

Published online by Cambridge University Press:  19 September 2008

A. Zeghib
A. Zeghib
Affiliation:
CNRS: UMR 128, Ecole Normale Supérieure de Lyon, 46, Allée d' Italie, 69364 LYON Cedex 07, France
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Abstract

We introduce a notion of autonomous dynamical systems which generalizes algebraic dynamical systems. We show by giving examples and by describing some properties that this generalization is not a trivial one. We apply the methods then developed to algebraic Anosov systems. We prove that a C1-submanifold of finite volume, which is invariant by an algebraic Anosov system is ‘essentially’ algebraic.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 1 , February 1995 , pp. 175 - 207
Copyright
Copyright © Cambridge University Press 1995

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References

REFERENCES

[Ano]Anosov, D. V.. Geodesic flows on compact manifolds of negative curvature. Proc. Steklov Inst. Math., AMS Trans. (1969).Google Scholar
[Bes]Besse, A.. Manifolds all of Whose Geodesies are Closed. Springer: Berlin, 1978.CrossRefGoogle Scholar
[Ebe]Eberlein, P.. When is a geodesic flow of Anosov type?. J. Diff. Geom. 8 (1973), 437–463.Google Scholar
[Fat]Fathi, A.. Some compact invariant sets for hyperbolic linear automorphisms of torn. Ergod. Th. & Dynam. Sys. 8 (1988), 205–213.CrossRefGoogle Scholar
[Fed]Federer, H.. Geometric Measure Theory. Springer: Berlin, 1969.Google Scholar
[Fra]Franks, J.. Invariant sets of hyperbolic toral automorphisms. Amer. J. Math. 99 (1977), 1089–1095.CrossRefGoogle Scholar
[Ghy1]Ghys, E.. Flots d' Anosov dont les feuilletages stables sont différentiables. Ann. Sci. Ec Norm. Sup. 20 (1987), 251–270.CrossRefGoogle Scholar
[Ghy2]Ghys, E.. Dynamique des flots unipotents sur les espaces homogenes. Sém. Bourbaki. (11 1991), Exposé 747.Google Scholar
[Han]Hancock, S.. Construction of invariant sets for Anosov diffeomorphisms. J. London. Math. Soc. 18 (1978), 339–348.Google Scholar
[Hel]Helgason, S.. Differential Geometry, Lie Groups and Symmetric Spaces. Academic: New York, 1978.Google Scholar
[Hir]Hirsch, M.. On invariant subsets of hyperbolic sets. Essays in Topology and Related Topics. (1970), 126–146.CrossRefGoogle Scholar
[Hir, Pug]Hirsch, M. and Pugh, C.. Stable manifolds and hyperbolic sets. Proc. Symp. Pure Math. 14 (1970), 125–146.CrossRefGoogle Scholar
[Man1]Mañé, R.. Orbits of paths under toral automorphisms. Proc. Amer. Soc. 73 (1979), 121–125.CrossRefGoogle Scholar
[Man2]Mañé, R.. Invariant sets of Anosov diffeomorphisms. Invent. Math. 46 (1978), 147–152.CrossRefGoogle Scholar
[Man3]Mañé, R.. Quasi Anosov diffeomorphisms and hyperbolic manifolds. Trans. Amer. Math. Soc. 229 (1977), 351–370.CrossRefGoogle Scholar
[Mar]Margulis, G.. Discrete Subgroups of Semisimple Lie Groups. Springer: Berlin, 1991.CrossRefGoogle Scholar
[Mol]Molino, P.. Riemannian Foliations. Birkhauser: Boston, 1988.CrossRefGoogle Scholar
[Pla]Plante, J.. Anosov flows. Amer. J. Math. 94 (1972), 729–754.CrossRefGoogle Scholar
[Sel]Selgrade, I.. Isolated invariant sets for flows on vector bundles. Trans. Amer. Math. Soc. 203 (1975), 359.CrossRefGoogle Scholar
[Thu]Thurston, W.. Geometry and Topology of 3-manifolds. Princeton University (1978).Google Scholar
[Tom]Tomter, P.. Anosov flows on infra-homogeneous spaces. Proc. Symp. in Pure. Math. 14 (1970), 299–327.CrossRefGoogle Scholar
[Wal]Walters, P.. An Introduction to Ergodic Theory. Springer: Berlin, 1982.CrossRefGoogle Scholar
[Wol]Wolf, J.. Spaces of Constant Curvature. McGraw-Hill: New York, 1967.Google Scholar
[Zeg1]Zeghib, A.. Subsystems of Anosov systems.Google Scholar
[Zeg2]Zeghib, A.. Ensembles invariants des flots géodésiques des variétés localement symétriques. A paraître dans Ergod. Th. & Dynam. Sys.Google Scholar
[Zim]Zimmer, R.. Ergodic Theory and Semisimple Groups. Birkhauser: Boston, 1984.CrossRefGoogle Scholar