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
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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.
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[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/
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[Ro]Rob T. Roblin: Un théorème de Fatou pour les densités conformes avec applications aux revêtements galoisiens en courbure négative. Preprint.
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[ASS]ASS J. Aaronson, O. Sarig, R. Solomyak: Tail-invariant measures for some suspension semiflows. Discr. and Contin. Dyn. Sys. 8 (no. 3), 725–735 (2002).
[Ba]Ba M. Babillot: On the classification of invariant measures for horospherical foliations on nilpotent covers of negatively curved manifolds. In: Random walks and geometry (V.A. Kaimanovich, Ed.), de Gruyter, Berlin 2004, 319–335.
[BL]BL M. Babillot and F. Ledrappier: Geodesic paths and horocycle flows on Abelian covers. Lie groups and ergodic theory (Mumbai, 1996), 1–32, Tata Inst. Fund. Res. Stud. Math. 14, Tata Inst. Fund. Res., Bombay, 1998.
[BE]BE P. Bougerol and L. Élie: Existence of positive harmonic functions on groups and on covering manifolds. Ann. Inst. H. Poincaré Probab. Statist. 31 (1995), no. 1, 59–80.
[Bu]Bu M. Burger: Horocycle flow on geometrically finite surfaces. Duke Math. J. 61 (1990), no. 3, 779–803.
[CG]CG 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), Exp. No. 4, 22 pp. Dept. Math. Informat., Univ. Rennes, Rennes, 1976.
[D]D S. G. Dani: Invariant measures of horospherical flows on non-compact homogeneous spaces. Invent. Math. 47 (1978), no. 2, 101–138.
[DS]DS S. G. Dani and J. Smillie: Uniform distribution of horocycle orbits for Fuchsian groups. Duke Math. J. 51 (1984), 185–194.
[F]F H. Furstenberg: The unique ergodicity of the horocycle flow. Springer Lecture Notes 318 (1972), 95–115.
[Gr]Gr M. Gromov: Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. No. 53 (1981), 53–73.
[JM]JM L. Ji and R. MacPherson: Geometry of compactifications of locally symmetric spaces. Ann. Inst. Fourier (Grenoble) 52 (2002), no. 2, 457–559.
[Kai1]Ka V. Kaimanovich: Ergodic properties of the horocycle flow and classification of Fuchsian groups. J. Dynam. Control Systems 6 (2000), no. 1, 21–56.
[Kai2]Ka1 V. Kaimanovich: Brownian motion and harmonic functions on covering manifolds. An entropic approach. Dokl. Akad. Nauk SSSR 288 (1986), no. 5. Engl. Transl. in Soviet Math. Doklady 33 (1986), 812–816.
[LP]LP V. Lin and Y. Pinchover: Manifolds with group actions and elliptic operators. Memoirs of the AMS 112 (1994), 78pp.
[LS]LS T. Lyons and D. Sullivan: Function theory, random paths and covering spaces. J. Differential Geom. 19 (1984), no. 2, 299–323.
[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/
[Ra]R M. Ratner: On Raghunathan’s measure conjecture. Ann. of Math. (2) 134 (1991), no. 3, 545–607.
[Ro]Rob T. Roblin: Un théorème de Fatou pour les densités conformes avec applications aux revêtements galoisiens en courbure négative. Preprint.
[Sa]Sa O. Sarig: Invariant measures for the horocycle flow on Abelian covers. Inv. Math. 157 (2004), 519–551.
[Su]SuIHES D. Sullivan: The density at infinity of a discrete group of hyperbolic motions. Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171–202.
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Additional Information
François Ledrappier
Affiliation:
Department of Mathematics, University of Notre-Dame, Notre-Dame, IN 46556-4618
Email:
ledrappier.1@nd.edu
Omri Sarig
Affiliation:
Mathematics Department, Pennsylvania State University, University Park, PA 16802
Email:
sarig@math.psu.edu
Received by editor(s):
July 27, 2005
Published electronically:
November 15, 2005
Additional Notes:
F.L. is supported by NSF grant DMS-0400687
O.S. is supported by NSF grant DMS-0500630
Dedicated:
Pour Martine
Communicated by:
Boris Hasselblatt
Article copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.