Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-05T06:37:21.918Z Has data issue: false hasContentIssue false
  • English
  • Français

Harmonic measures and the foliated geodesic flow for foliations with negatively curved leaves

Published online by Cambridge University Press:  05 August 2014

SÉBASTIEN ALVAREZ
SÉBASTIEN ALVAREZ*
Affiliation:
Institut de Mathématiques de Bourgogne, CNRS-UMR 5584, Université de Bourgogne, 21078 Dijon Cedex, France email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we define a notion of Gibbs measure for the geodesic flow tangent to a foliation with negatively curved leaves and associated to a particular potential $H$. We prove that there is a canonical bijective correspondence between these measures and Garnett’s harmonic measures.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 2 , April 2016 , pp. 355 - 374
Copyright
© Cambridge University Press, 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alvarez, S.. Gibbs u-states for the foliated geodesic flow and transverse invariant measures. Preprint,arXiv:1311.7121.Google Scholar
Alvarez, S.. Gibbs measures for foliated bundles with negatively curved leaves. Preprint, arXiv: 1311.3574.Google Scholar
Alvarez, S.. Mesures de Gibbs et mesures harmoniques pour les feuilletages aux feuilles courbées négativement. PhD Thesis, L’Université de Bourgogne, 2013, available online at http://tel.archives-ouvertes.fr/tel-00958080.Google Scholar
Anderson, M.T. and Schoen, R.. Positive harmonic functions on complete manifolds of negative curvature. Ann. of Math. 121(2) (1985), 429–461.CrossRefGoogle Scholar
Bakhtin, Y. and Martínez, M.. A characterization of harmonic measures on laminations by hyperbolic Riemann surfaces. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008), 1078–1089.CrossRefGoogle Scholar
Bonatti, C., Díaz, L. and Viana, M.. Dynamics Beyond Uniform Hyperbolicity. A global geometric and probabilistic perspective (Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, III). Springer, Berlin, 2005.Google Scholar
Connell, C. and Martínez, M.. Harmonic and invariant measures on foliated spaces. Preprint,http://mypage.iu.edu/∼connell/publications/HarmonicMeas.pdf.Google Scholar
Garnett, L.. Foliations, the ergodic theorem and Brownian motion. J. Funct. Anal. 51 (1983), 285–311.CrossRefGoogle Scholar
Ghys, E.. Topologie des feuilles génériques. Ann. of Math. 141(2) (1995), 387–422.CrossRefGoogle Scholar
Haefliger, A.. Foliations and compactly generated pseudogroups. Foliations: Geometry and Dynamics (Warsaw, 2000). World Scientific, River Edge, NJ, 2002, pp. 275–295.CrossRefGoogle Scholar
Ledrappier, F.. Propriété de poisson et courbure négative. C. R. Acad. Sci. Paris, Ser I. 305 (1987), 191–194.Google Scholar
Ledrappier, F.. Ergodic properties of Brownian motion on covers of compact negatively curved manifolds. Bol. Soc. Bras. Mat. 19 (1988), 115–140.CrossRefGoogle Scholar
Kaimanovich, V.. An entropy criterion for maximality of boundary of random walks on discrete groups. Dokl. Akad. Nauk SSSR 280 (1985), 193–197.Google Scholar
Mañé, R.. Ergodic Theory and Differentiable Dynamics. Springer, Berlin, 1987.CrossRefGoogle Scholar
Martínez, M.. Measures on hyperbolic surface laminations. Ergod. Th. & Dynam. Sys. 26 (2006), 847–867.CrossRefGoogle Scholar
Prat, J.J.. Étude asymptotique du mouvement Brownien sur une variété riemannienne á courbure négative. C. R. Acad. Sci. Paris, Sér. A–B 272 (1971), 1586–1589.Google Scholar
Rokhlin, V.A.. On the fundamental ideas of measure theory. Amer. Math. Soc. Transl. 10 (1962), 1–52.Google Scholar
Sullivan, D.. The dirichlet problem at infinity for a negatively curved manifold. J. Differential Geom. 18 (1983), 723–732.CrossRefGoogle Scholar