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Statistical behaviour of the leaves of Riccati foliations

Published online by Cambridge University Press:  21 May 2009

Laboratoire de Topologie, UMR 5584 du CNRS B.P. 47 870, 21078 Dijon Cedex, France (email:
CIMAT A.P. 402, Guanajuato, 36000, México (email:,
CIMAT A.P. 402, Guanajuato, 36000, México (email:,


We introduce the geodesic flow on the leaves of a holomorphic foliation with leaves of dimension one and hyperbolic, corresponding to the unique complete metric of curvature −1 compatible with its conformal structure. We do these for the foliations associated to Riccati equations, which are the projectivization of the solutions of linear ordinary differential equations over a finite Riemann surface of hyperbolic type S, and may be described by a representation ρ:π1(S)→GL(n,ℂ). We give conditions under which the foliated geodesic flow has a generic repeller–attractor statistical dynamics. That is, there are measures μ and μ+ such that for almost any initial condition with respect to the Lebesgue measure class the statistical average of the foliated geodesic flow converges for negative time to μ and for positive time to μ+ (i.e. μ+ is the unique Sinaï, Ruelle and Bowen (SRB)-measure and its basin has total Lebesgue measure). These measures are ergodic with respect to the foliated geodesic flow. These measures are also invariant under a foliated horocycle flow and they project to a harmonic measure for the Riccati foliation, which plays the role of an attractor for the statistical behaviour of the leaves of the foliation.

Research Article
Copyright © Cambridge University Press 2009

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[1]Bakhtin, Y. and Martinez, M.. A characterization of harmonic measures on laminations by hyperbolic Riemann surfaces. Preprint, Arxiv:0611235. Ann. Inst. H. Poincaré (2008) to appear.CrossRefGoogle Scholar
[2]Bonatti, Ch. and Gómez-Mont, X.. Sur le comportement statistique des feuilles de certains feuilletages holomorphes. Monogr. Enseign. Math. 38 (2001), 1541.Google Scholar
[3]Bonatti, Ch., Gómez-Mont, X. and Viana, M.. Généricité d’exposants de Lyapunov non-nuls pour des produits déterministes de matrices. Ann. Inst. H. Poincaré 20 (2003), 579624.CrossRefGoogle Scholar
[4]Bonatti, Ch. and Viana, M.. Lyapunov Exponents with multiplicity 1 for deterministic products of matrices. Ergod. Th. & Dynam. Sys. 24 (2004), 12951330.CrossRefGoogle Scholar
[5]Bowen, R. and Ruelle, D.. The ergodic theory of axiom A flows. Invent. Math. 29 (1975), 181202.CrossRefGoogle Scholar
[6]Coddington, E. and Levinson, N.. Theory of Ordinary Differential Equations. McGraw-Hill, New York, 1955.Google Scholar
[7]Deroin, B. and Kleptsyn, V.. Random conformal dynamical systems. Geom. Funct. Anal. 17 (2007), 10431105.CrossRefGoogle Scholar
[8]Fornaess, J. E. and Sibony, N.. Harmonic currents of finite energy and laminations. Geom. Funct. Anal. 15 (2005), 9621003.CrossRefGoogle Scholar
[9]Fornaess, J. E. and Sibony, N.. Unique ergodicity of harmonic currents on singular foliations of P 2. Preprint arXiv:math/0606744, 2006.Google Scholar
[10]Fornaess, J. E. and Sibony, N.. Riemann surface laminations with singularities. J. Geom. Anal. 18(2) (2008), 400442.CrossRefGoogle Scholar
[11]Garnett, L.. Foliations, the ergodic theorem and Brownian motions. J. Funct. Anal. 51(3) (1983), 285311.CrossRefGoogle Scholar
[12]Ghys, E. and de la Harpe, P.. Sur Les Groupes Hyperboliques d’apres Mikhael Gromov (Progress in Mathematics, 83). Birkhäuser, Zurich, 1990.CrossRefGoogle Scholar
[13]Hedlund, G.. Fuchsian groups and transitive horocycles. Duke Math. J. 2 (1936), 530542.CrossRefGoogle Scholar
[14]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54). Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[15]Katok, S.. Fuchsian Groups. The University of Chicago Press, Chicago, IL, 1992.Google Scholar
[16]Krengel, U.. Ergodic Theorems. With a Supplement by Antoine Brunel (de Gruyter studies in Mathematics, 6). Walter de Gruyter & Co., Berlin, 1985, vii+357pp.CrossRefGoogle Scholar
[17]Lawson, B.. The qualitative theory of foliations. CBMS Reg. Conf. Math. 27 (1978).Google Scholar
[18]Martinez, M.. Measures on hyperbolic surface laminations. Ergod. Th. & Dynam. Sys. 26 (2006), 847867.CrossRefGoogle Scholar
[19]Martinez, M. and Verjovsky, A.. Hedlund’s theorem for compact minimal laminations. Preprint arXiv:0711.2307. 2007.Google Scholar
[20]Maskit, B.. Kleinian groups. Grundlehren der Mathematishen Wissenschaften (Fundamental Principles of Mathematical Sciences, 287). Springer, Berlin, 1988.Google Scholar
[21]Ruelle, D.. A measure associated with axiom A attractors. Amer. J. Math. 98 (1976), 619654.CrossRefGoogle Scholar
[22]Sinaï, Y.. Markov partitions and C-diffeomorphisms. Funct. Anal. Appl. 2 (1968), 6489.CrossRefGoogle Scholar
[23]Viana, M.. Almost all cocycles over any hyperbolic system have non-vanishing Lyapunov exponents. Ann. of Math. (2) 167 (2008), 643680.CrossRefGoogle Scholar