Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T04:03:39.207Z Has data issue: false hasContentIssue false
  • English
  • Français

Propriétés ergodiques, en mesure infinie, de certains systèmes dynamiques fibrés

Published online by Cambridge University Press:  19 September 2008

Y. Guivarc'h
Y. Guivarc'h
Affiliation:
IRMAR, Mathematiques Université de RennesI-35042 Rennes Cedex, France
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.

We study the ergodic properties of a class of dynamical systems with infinite invariant measure. This class contains skew-products of Anosov systems with ℝd. The results are applied to the K property of skew-products and also to the ergodicity of the geodesic flow on abelian coverings of compact manifolds with constant negative curvature.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 9 , Issue 3 , September 1989 , pp. 433 - 453
Copyright
Copyright © Cambridge University Press 1989

References

REFERENCES

[1]Adler, R.L. & Shields, P. C.. Skew products of Bernoulli shifts with rotations. Israël J. Math. 19 (1974), 228–236.CrossRefGoogle Scholar
[2]Bowen, R.. Equilibrium states and the ergodic theory of Anosov diSeomorphisms. Lecture Notes in Mathematics 470 (Springer-Verlag, Berlin, 1975).CrossRefGoogle Scholar
[3]Bowen, R.. Symbolic dynamics for hyperbolic flows. Amer. J. Math. 95 (1973), 429–460.CrossRefGoogle Scholar
[4]Bowen, R. & Ruelle, D.. The ergodic theory of axiom A flows. Inventiones Math. 29 (1975), 181–202.CrossRefGoogle Scholar
[5]Bunimovich, L.A. & Sinai, Ya G.. Statistical properties of Lorentz gas with periodic configurations. Commun. Math. Phys. 78 (1981), 479–497.CrossRefGoogle Scholar
[6]Choquet, G. & Deny, J.. Sur l'équation de convolution μ = σμ”. C.R.A.S. Paris 250A (1960), 799–801.Google Scholar
[7]Gottschalk, W.H. & Hedlund, G. A.. Topological dynamics. Amer. Math. Soc. 36 (1955).Google Scholar
[8]Guivarc'h, Y. & Hardy, J.. Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov. Annales de l'Inst. H. Poincaré 24 (1) (1988), 73–98.Google Scholar
[9]Guivarc'h, Y.. Applications d'un théorème limite local a transience et à; la récurrence de marches de Markov. Théorie du Potentiel. Lecture Notes in Mathematics ngicb1096 (Springer-Verlag, Berlin, 1983). 301–332.Google Scholar
[10]Guivarc'h, Y., Keane, M. & Roynette, B.. Marches aleatoires sur les groupes de Lie. Lecture Notes in Mathematics 624 (Springer-Verlag, Berlin, 1977).CrossRefGoogle Scholar
[11]Guivarc'h, Y.. Mouvement brownien sur les revêtements d'une variété compacte. C.R.A.S. Paris 292 (1981) 851–853.Google Scholar
[12]Kalikow, S.. T, T −1 transformation is not loosely Bernoulli. Ann. Math. 115 (1982), 393–409.CrossRefGoogle Scholar
[13]Katok, A.. Smooth non-Bernoulli K -automorphisms. Inventiones Math. 61 (1980), 291–300.CrossRefGoogle Scholar
[14]Le Page, E.. Theoremes limites pour les produits de matrices aleatoires. Lecture Notes in Mathematics 928, (Springer-Verlag, Berlin, 1982) 355–386.Google Scholar
[15]Lifshits, A.N.. Homology properties of y-systems. Math. Zametki 10 (5) (1971), 555–564.Google Scholar
[16]Lin, M.. Mixing for Markov operators. Z. Wahr. 19 (1971), 231–242.CrossRefGoogle Scholar
[17]Meilijson, I.. Mixing properties of a class of skew-products. I. J. Math. 19 (1974), 266–270.Google Scholar
[18]Rees, M.. Checking ergodicity of some geodesic flows with infinite Gibbs measure. Ergod. Th. … Dynam. Sys. 1 (1981), 107–133.CrossRefGoogle Scholar
[19]Rudolph, D.J.. Z" and R" cocycle extensions and complementary algebras. Preprint Depart, of Math., University Maryland.Google Scholar
[20]Rudolph, D.J.. Asymptotically brownian skew products give non loosely Bernoulli K-automorphisms. Preprint. University of Maryland.Google Scholar
[21]Smale, S.. Differentiate dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar
[22]Sinai, Ya G.. Gibbs measures in ergodic theory. Russian Math. Surveys 27 (4) (1972), 21–69.CrossRefGoogle Scholar
[23]Spitzer, F.. Principles of random walks. Van Nostrand, Princeton, 1964.CrossRefGoogle Scholar
[24]Sullivan, D.. The density at infinity of a discrete groupe of hyperbolic motions. I.H.E.S. Publ. Math. 50 (1979), 171–202.Google Scholar
[25]Thouvenot, J.-P.. Communication personnelle.Google Scholar
[26]Varopoulos, N.Th.. C.R.A.S. Paris 302 (1) 6 (1986), 203–205.Google Scholar