Invariant measures for the horocycle flow on periodic hyperbolic surfaces

Ledrappier, François; Sarig, Omri

doi:10.1090/S1079-6762-05-00151-4

Invariant measures for the horocycle flow on periodic hyperbolic surfaces

Authors: François Ledrappier and Omri Sarig
Journal: Electron. Res. Announc. Amer. Math. Soc. 11 (2005), 89-94
MSC (2000): Primary 37D40, 37A40; Secondary 31C12
DOI: https://doi.org/10.1090/S1079-6762-05-00151-4
Published electronically: November 15, 2005
MathSciNet review: 2183007
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: 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.

Jon Aaronson, Omri Sarig, and Rita Solomyak, Tail-invariant measures for some suspension semiflows, Discrete Contin. Dyn. Syst. 8 (2002), no. 3, 725–735. MR 1897877, DOI https://doi.org/10.3934/dcds.2002.8.725
Martine Babillot, On the classification of invariant measures for horosphere foliations on nilpotent covers of negatively curved manifolds, Random walks and geometry, Walter de Gruyter, Berlin, 2004, pp. 319–335. MR 2087786
Martine Babillot and François Ledrappier, Geodesic paths and horocycle flow on abelian covers, Lie groups and ergodic theory (Mumbai, 1996) Tata Inst. Fund. Res. Stud. Math., vol. 14, Tata Inst. Fund. Res., Bombay, 1998, pp. 1–32. MR 1699356
Philippe Bougerol and Laure Élie, Existence of positive harmonic functions on groups and on covering manifolds, Ann. Inst. H. Poincaré Probab. Statist. 31 (1995), no. 1, 59–80 (English, with English and French summaries). MR 1340031
Marc Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J. 61 (1990), no. 3, 779–803. MR 1084459, DOI https://doi.org/10.1215/S0012-7094-90-06129-0
J.-P. Conze and Y. Guivarc’h, Propriété de droite fixe et fonctions propres des opérateurs de convolution, Séminaire de Probabilités, I (Univ. Rennes, Rennes, 1976) Dépt. Math. Informat., Univ. Rennes, Rennes, 1976, pp. 22 (French). MR 0487265
S. G. Dani, Invariant measures of horospherical flows on noncompact homogeneous spaces, Invent. Math. 47 (1978), no. 2, 101–138. MR 578655, DOI https://doi.org/10.1007/BF01578067
S. G. Dani and John Smillie, Uniform distribution of horocycle orbits for Fuchsian groups, Duke Math. J. 51 (1984), no. 1, 185–194. MR 744294, DOI https://doi.org/10.1215/S0012-7094-84-05110-X
Harry Furstenberg, The unique ergodicity of the horocycle flow, Recent advances in topological dynamics (Proc. Conf., Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), Springer, Berlin, 1973, pp. 95–115. Lecture Notes in Math., Vol. 318. MR 0393339
Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73. MR 623534
L. Ji and R. MacPherson, Geometry of compactifications of locally symmetric spaces, Ann. Inst. Fourier (Grenoble) 52 (2002), no. 2, 457–559 (English, with English and French summaries). MR 1906482
V. A. Kaimanovich, Ergodic properties of the horocycle flow and classification of Fuchsian groups, J. Dynam. Control Systems 6 (2000), no. 1, 21–56. MR 1738739, DOI https://doi.org/10.1023/A%3A1009517621605
V. A. Kaĭmanovich, Brownian motion and harmonic functions on covering manifolds. An entropic approach, Dokl. Akad. Nauk SSSR 288 (1986), no. 5, 1045–1049 (Russian). MR 852647
Vladimir Ya. Lin and Yehuda Pinchover, Manifolds with group actions and elliptic operators, Mem. Amer. Math. Soc. 112 (1994), no. 540, vi+78. MR 1230774, DOI https://doi.org/10.1090/memo/0540
Terry Lyons and Dennis Sullivan, Function theory, random paths and covering spaces, J. Differential Geom. 19 (1984), no. 2, 299–323. MR 755228

[LeS1]LeS1 F. Ledrappier and O. Sarig: Unique ergodicity for non-uniquely ergodic horocycle flows. To appear in Disc. Cont. Dynam. Syst. Issue dedicated to A. Katok.

[LeS2]LeS F. Ledrappier and O. Sarig: Invariant measures for the horocycle flow on periodic hyperbolic surfaces. Preprint available at http://www.math.psu.edu/sarig/

Marina Ratner, On Raghunathan’s measure conjecture, Ann. of Math. (2) 134 (1991), no. 3, 545–607. MR 1135878, DOI https://doi.org/10.2307/2944357

[Ro]Rob T. Roblin: Un théorème de Fatou pour les densités conformes avec applications aux revêtements galoisiens en courbure négative. Preprint.

Omri Sarig, Invariant Radon measures for horocycle flows on abelian covers, Invent. Math. 157 (2004), no. 3, 519–551. MR 2092768, DOI https://doi.org/10.1007/s00222-004-0357-4
Dennis Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171–202. MR 556586