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Statistical behaviour of the leaves of Riccati foliations

Published online by Cambridge University Press:  21 May 2009

CH. BONATTI ,
X. GÓMEZ-MONT  and
R. VILA-FREYER
CH. BONATTI
Affiliation:
Laboratoire de Topologie, UMR 5584 du CNRS B.P. 47 870, 21078 Dijon Cedex, France (email: [email protected])
X. GÓMEZ-MONT
Affiliation:
CIMAT A.P. 402, Guanajuato, 36000, México (email: [email protected], [email protected])
R. VILA-FREYER
Affiliation:
CIMAT A.P. 402, Guanajuato, 36000, México (email: [email protected], [email protected])
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Abstract

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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 1 , February 2010 , pp. 67 - 96
Copyright
Copyright © Cambridge University Press 2009

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References

[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), 15–41.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), 579–624.CrossRefGoogle Scholar
[4]Bonatti, Ch. and Viana, M.. Lyapunov Exponents with multiplicity 1 for deterministic products of matrices. Ergod. Th. & Dynam. Sys. 24 (2004), 1295–1330.CrossRefGoogle Scholar
[5]Bowen, R. and Ruelle, D.. The ergodic theory of axiom A flows. Invent. Math. 29 (1975), 181–202.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), 1043–1105.CrossRefGoogle Scholar
[8]Fornaess, J. E. and Sibony, N.. Harmonic currents of finite energy and laminations. Geom. Funct. Anal. 15 (2005), 962–1003.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), 400–442.CrossRefGoogle Scholar
[11]Garnett, L.. Foliations, the ergodic theorem and Brownian motions. J. Funct. Anal. 51(3) (1983), 285–311.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), 530–542.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), 847–867.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), 619–654.CrossRefGoogle Scholar
[22]Sinaï, Y.. Markov partitions and C-diffeomorphisms. Funct. Anal. Appl. 2 (1968), 64–89.CrossRefGoogle Scholar
[23]Viana, M.. Almost all cocycles over any hyperbolic system have non-vanishing Lyapunov exponents. Ann. of Math. (2) 167 (2008), 643–680.CrossRefGoogle Scholar