Invariant measures for the stable foliation on negatively curved periodic manifolds
[Mesures invariantes pour le feuilletage stable d’une variété périodique de courbure négative]
Ledrappier, François1
1 University of Notre Dame Department of Mathematics Notre Dame, IN 46556-4618 (USA)
Annales de l'Institut Fourier, Tome 58 (2008) no. 1, pp. 85-105.

Nous décrivons les mesures réversibles associées au feuilletage stable du flot géodésique sur une variété périodique de courbure négative. Nous étendons ainsi ce qui était connu pour les surfaces hyperboliques aux cas de courbure variable et de dimension supérieure.

We classify reversible measures for the stable foliation on manifolds which are infinite covers of compact negatively curved manifolds. We extend the known results from hyperbolic surfaces to varying curvature and to all dimensions.

DOI : 10.5802/aif.2345
Classification : 37D40, 37A40, 53C12
Keywords: Invariant measure, stable foliation, negative curvature
Mot clés : mesure invariante, feuilletage stable, courbure négative
Ledrappier, François 1

1 University of Notre Dame Department of Mathematics Notre Dame, IN 46556-4618 (USA) 
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Ledrappier, François. Invariant measures for the stable  foliation on negatively curved periodic manifolds. Annales de l'Institut Fourier, Tome 58 (2008) no. 1, pp. 85-105. doi : 10.5802/aif.2345. http://www.numdam.org/articles/10.5802/aif.2345/

[1] Babillot, M. On the classification of invariant measures for horospherical foliations on nilpotent covers of negatively curved manifolds, Random walks and geometry, (V.A. Kaimanovich, Ed.) de Gruyter, Berlin, 2004, pp. 319-335 | Zbl

[2] Bowen, R.; Marcus, B. Unique ergodicity for horocycle foliations, Israel J. Math., Volume 26 (1977), pp. 43-67 | DOI | MR | Zbl

[3] Cartan, É. Leçons sur la géométrie des espaces de Riemann, 1928 (Paris) | Zbl

[4] Coornaert, M.; Papadopoulos, A. Horofunctions and symbolic dynamics on Gromov hyperbolic groups, Glasgow Math.J., Volume 43 (2001), pp. 425-456 | DOI | MR | Zbl

[5] Dal’bo, F. Remarques sur le spectre des longueurs d’une surface et comptages, Bol. Soc. Brasil. Mat. (N.S.), Volume 30 (1999) no. 2, pp. 199-221 | DOI | Zbl

[6] Garnett, L. Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal., Volume 51 (1983), pp. 285-311 | DOI | MR | Zbl

[7] Ghys, E.; de la Harpe, P. Sur les groupes hyperboliques, d’après Mikhael Gromov, Progress in math., Volume 83, Birkhäuser, 1990 | Zbl

[8] Gromov, M. Hyperbolic Groups, Essays in Group Theory, Volume 8, Math. Sci. Res. Inst. Publ., 1987, pp. 75-263 | MR | Zbl

[9] Hamenstädt, U. Harmonic measures for compact negatively curved manifolds, Acta Mathematica, Volume 178 (1997), pp. 39-107 | DOI | MR | Zbl

[10] Hamenstädt, U. Ergodic properties of Gibbs measures on nilpotent covers, Ergod. Th. & Dynam. Sys., Volume 22 (2002), pp. 1169-1179 | DOI | MR | Zbl

[11] Helgason, S. Differential Geometry, Lie Groups and Symmetric spaces, Graduate Studies in Mathematics, Academic Press, New York, 2001 | MR | Zbl

[12] Kaimanovich, V. A. Brownian motion on foliations: Entropy, invariant measures, mixing, Funct. Anal. Appl., Volume 22 (1989), pp. 326-328 | DOI | MR | Zbl

[13] Karpelevich, F. I. The geometry of geodesics and the eigenfunctions of the Laplacian on symmetric spaces, Trans. Moskov. Math. Soc., Volume 14 (1965), pp. 48-185 | MR | Zbl

[14] Katok, A.; Hasselblatt, B. Introduction to the modern theory of dynamical systems, CUP, 1995 (Cambridge) | Zbl

[15] Ledrappier, F.; Sarig, O. Invariant measures for the horocycle flow on periodic hyperbolic surfaces (to appear, Isr. J. Math.) | Zbl

[16] Margulis, G. A. Discrete subgroups of semi-simple groups, Ergebnisse, Band 17, Springer-Verlag, 1991 | Zbl

[17] Otal, J.-P. Sur la géométrie symplectique de l’espace des géodésiques d’une variété à courbure négative, Rev. Mat. Iberoamericana, Volume 8 (1992), pp. 441-456 | Zbl

[18] Roblin, T. Ergodicité et équidistribution en courbure négative, Mémoires, 95, S.M.F., 2003 | Numdam | MR | Zbl

[19] Roblin, T. Un théorème de Fatou pour les densités conformes avec applications aux revêtements galoisiens en courbure négative, Isr. J. Math., Volume 147 (2005), pp. 333-357 | DOI | MR

[20] Sarig, O. Invariant measures for the horocycle flow on Abelian covers, Inv. Math., Volume 157 (2004), pp. 519-551 | DOI | MR | Zbl

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