Geodesic
Invariant measures for the horocycle flow on periodic hyperbolic surfaces
Ledrappier, François1 ; Sarig, Omri2
1 Department of Mathematics, University of Notre-Dame, Notre-Dame, IN 46556-4618
2 Mathematics Department, Pennsylvania State University, University Park, PA 16802
Electronic research announcements of the American Mathematical Society, Tome 11 (2005), pp. 89-94 Cet article a éte moissonné depuis la source American Mathematical Society

Voir la notice de l'article

We describe the ergodic invariant Radon measures for the horocycle flow on general (infinite) regular covers of finite volume hyperbolic surfaces. The method is to establish a bijection between these measures and the positive minimal eigenfunctions of the Laplacian of the covering surface.
DOI : 10.1090/S1079-6762-05-00151-4

Ledrappier, François 1 ; Sarig, Omri 2

1 Department of Mathematics, University of Notre-Dame, Notre-Dame, IN 46556-4618
2 Mathematics Department, Pennsylvania State University, University Park, PA 16802 
@article{10_1090_S1079_6762_05_00151_4,
     author = {Ledrappier, Fran\c{c}ois and Sarig, Omri},
     title = {Invariant measures for the horocycle flow on periodic hyperbolic surfaces},
     journal = {Electronic research announcements of the American Mathematical Society},
     pages = {89--94},
     year = {2005},
     volume = {11},
     doi = {10.1090/S1079-6762-05-00151-4},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S1079-6762-05-00151-4/}
}
TY  - JOUR
AU  - Ledrappier, François
AU  - Sarig, Omri
TI  - Invariant measures for the horocycle flow on periodic hyperbolic surfaces
JO  - Electronic research announcements of the American Mathematical Society
PY  - 2005
SP  - 89
EP  - 94
VL  - 11
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S1079-6762-05-00151-4/
DO  - 10.1090/S1079-6762-05-00151-4
ID  - 10_1090_S1079_6762_05_00151_4
ER  - 
%0 Journal Article
%A Ledrappier, François
%A Sarig, Omri
%T Invariant measures for the horocycle flow on periodic hyperbolic surfaces
%J Electronic research announcements of the American Mathematical Society
%D 2005
%P 89-94
%V 11
%U http://geodesic.mathdoc.fr/articles/10.1090/S1079-6762-05-00151-4/
%R 10.1090/S1079-6762-05-00151-4
%F 10_1090_S1079_6762_05_00151_4
Ledrappier, François; Sarig, Omri. Invariant measures for the horocycle flow on periodic hyperbolic surfaces. Electronic research announcements of the American Mathematical Society, Tome 11 (2005), pp. 89-94. doi: 10.1090/S1079-6762-05-00151-4

[1] Aaronson, Jon, Sarig, Omri, Solomyak, Rita Tail-invariant measures for some suspension semiflows Discrete Contin. Dyn. Syst. 2002 725 735

[2] Babillot, Martine On the classification of invariant measures for horosphere foliations on nilpotent covers of negatively curved manifolds 2004 319 335

[3] Babillot, Martine, Ledrappier, François Geodesic paths and horocycle flow on abelian covers 1998 1 32

[4] Bougerol, Philippe, Élie, Laure Existence of positive harmonic functions on groups and on covering manifolds Ann. Inst. H. Poincaré Probab. Statist. 1995 59 80

[5] Burger, Marc Horocycle flow on geometrically finite surfaces Duke Math. J. 1990 779 803

[6] Conze, J.-P., Guivarc’H, Y. Propriété de droite fixe et fonctions propres des opérateurs de convolution 1976

[7] Dani, S. G. Invariant measures of horospherical flows on noncompact homogeneous spaces Invent. Math. 1978 101 138

[8] Dani, S. G., Smillie, John Uniform distribution of horocycle orbits for Fuchsian groups Duke Math. J. 1984 185 194

[9] Furstenberg, Harry The unique ergodicity of the horocycle flow 1973 95 115

[10] Gromov, Mikhael Groups of polynomial growth and expanding maps Inst. Hautes Études Sci. Publ. Math. 1981 53 73

[11] Ji, L., Macpherson, R. Geometry of compactifications of locally symmetric spaces Ann. Inst. Fourier (Grenoble) 2002 457 559

[12] Kaimanovich, V. A. Ergodic properties of the horocycle flow and classification of Fuchsian groups J. Dynam. Control Systems 2000 21 56

[13] Kaĭmanovich, V. A. Brownian motion and harmonic functions on covering manifolds. An entropic approach Dokl. Akad. Nauk SSSR 1986 1045 1049

[14] Lin, Vladimir Ya., Pinchover, Yehuda Manifolds with group actions and elliptic operators Mem. Amer. Math. Soc. 1994

[15] Lyons, Terry, Sullivan, Dennis Function theory, random paths and covering spaces J. Differential Geom. 1984 299 323

[16] Ratner, Marina On Raghunathan’s measure conjecture Ann. of Math. (2) 1991 545 607

[17] Sarig, Omri Invariant Radon measures for horocycle flows on abelian covers Invent. Math. 2004 519 551

[18] Sullivan, Dennis The density at infinity of a discrete group of hyperbolic motions Inst. Hautes Études Sci. Publ. Math. 1979 171 202

Cité par Sources :