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Ergodic properties of the stable foliations

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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1514))

Abstract

We describe some properties of the harmonic measures associated with the stable and the strong stable foliations of a geodesic flow on a negatively-curved manifold.

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References

  1. A. Ancona: Negatively curved manifolds, elliptic operators and Martin boundary. Ann. of Maths. 125 (1987), 495–536.

    Article  MathSciNet  MATH  Google Scholar 

  2. M.T. Anderson and R. Schoen: Positive harmonic functions on complete manifolds of negative curvature. Ann. of Maths. 121 (1985) 429–461.

    Article  MathSciNet  MATH  Google Scholar 

  3. D.V. Anosov and Ya. Sinaï: Some smooth ergodic systems. Russian math. survey 22:5 (1967) 103–167.

    Article  MATH  Google Scholar 

  4. R. Bowen and B. Marcus: Unique ergodicity for horocycle foliations. Israël J. Maths. 26 (1977) 43–67.

    Article  MathSciNet  MATH  Google Scholar 

  5. L. Garnett: Foliations, the ergodic theorem and Brownian motion. J. Func. Anal. 51 (1983) 285–311.

    Article  MathSciNet  MATH  Google Scholar 

  6. E. Ghys, R. Langevin and P.G. Walczak: Entropie géométrique des feuilletages. Acta Math. 160 (1988) 105–142.

    Article  MathSciNet  Google Scholar 

  7. U. Hamenstädt: An explicit description of the harmonic measure. Math. Z. 205 (1990) 287–299.

    Article  MathSciNet  MATH  Google Scholar 

  8. U. Hamenstädt: Metric and topological entropies of geodesic flow. Preprint.

    Google Scholar 

  9. U. Hamenstädt: In preparation.

    Google Scholar 

  10. S. Hurder: Ergodic theory of foliations and a theorem of Sacksteder. Dynamical Systems Proceeding Spec. Year. College Park Md Springer L.N. maths 1342 (1988) 291–328.

    MathSciNet  MATH  Google Scholar 

  11. V.A. Kaimanovich: Brownian motion and harmonic functions on covering manifolds. An entropy approach. Soviet math. Doklady 33 (1986) 812–816.

    Google Scholar 

  12. V.A. Kaimanovich: Brownian motion on foliations: entropy, invariant measures, mixing. Funct. Anal. Appl. 22 (1988).

    Google Scholar 

  13. V.A. Kaimanovich: Invariant measures of the geodesic flow and measures at infinity on negetively curved manifolds. Ann. I.H.P. (Physique Théorique), 53 (1990) 361–393.

    MathSciNet  Google Scholar 

  14. A. Katok: Entropy and closed geodesics. Erg. Th. Dynam. Sys. 2 (1982) 339–365.

    MathSciNet  MATH  Google Scholar 

  15. A. Katok: Four applications of Conformal Equivalence to Geometry and Dynamics. Erg. Th. & Dynam. Sys. 8* (1988) 139–152.

    Article  MathSciNet  MATH  Google Scholar 

  16. G. Knieper: Horospherical measure and rigidity of manifolds of negative curvature. Preprint.

    Google Scholar 

  17. Y. Kifer, F. Ledrappier: Hausdorff Dimension of Harmonic Measures on Negatively Curved Manifolds. T.A.M.S. 318 (1990) 685–704.

    Article  MathSciNet  MATH  Google Scholar 

  18. F. Ledrappier: Propriété de Poisson et courbure négative. C.R.A.S. Paris 305 (1987) 191–194.

    MathSciNet  MATH  Google Scholar 

  19. F. Ledrappier: Ergodic properties of Brownian motion on covers of compact negatively-curved manifolds. Bol. Soc. Bras. Mat. 19 (1988) 115–140.

    Article  MathSciNet  MATH  Google Scholar 

  20. F. Ledrappier: Harmonic measures and Bowen-Margulis measures. Isr. J. Maths. 71 (1990) 275–287.

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Manning: Topological entropy for geodesic flows. Ann. Maths. 105 (1977) 81–105.

    Article  Google Scholar 

  22. G.A. Margulis: Applications of ergodic theory to the investigation of manifolds of negative curvature. Funct. Anal. Appl. 3 (1969) 335–336.

    Article  MathSciNet  MATH  Google Scholar 

  23. G.A. Margulis: Certain measures associated with U-flows on compact manifolds. Funct. Anal. Appl. 4 (1970) 55–67.

    Article  MathSciNet  MATH  Google Scholar 

  24. J. Plante: Foliations with measure preserving holomony. Ann. Maths. 102 (1975) 327–361.

    Article  MathSciNet  MATH  Google Scholar 

  25. J-J. Prat: Etude asymptotique et convergence angulaire du mouvement brownien sur une variété à courbure négative. CRAS Paris 280 (1985) 1539–1542.

    MathSciNet  MATH  Google Scholar 

  26. N. Varopoulos: Information theory and harmonic functions. Bulletin Sci. Maths. 110 (1986) 347–389.

    MathSciNet  MATH  Google Scholar 

  27. P.G. Walczak: Dynamics of the geodesic flow of a foliation. Ergod. Th. & Dynam. Sys. 8, (1988) 637–650.

    Article  MathSciNet  MATH  Google Scholar 

  28. C-B. Yue: Contribution to Sullivan’s conjecture. Preprint.

    Google Scholar 

  29. C-B. Yue: Brownian motion on Anosov foliation, integral formula and rigidity. Preprint.

    Google Scholar 

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Ulrich Krengel Karin Richter Volker Warstat

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© 1992 Springer-Verlag

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Ledrappier, F. (1992). Ergodic properties of the stable foliations. In: Krengel, U., Richter, K., Warstat, V. (eds) Ergodic Theory and Related Topics III. Lecture Notes in Mathematics, vol 1514. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0097534

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  • DOI: https://doi.org/10.1007/BFb0097534

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55444-8

  • Online ISBN: 978-3-540-47076-2

  • eBook Packages: Springer Book Archive

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