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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: bonatti@u-bourgogne.fr)
X. GÓMEZ-MONT
Affiliation:
CIMAT A.P. 402, Guanajuato, 36000, México (email: gmont@cimat.mx, vila@cimat.mx)
R. VILA-FREYER
Affiliation:
CIMAT A.P. 402, Guanajuato, 36000, México (email: gmont@cimat.mx, vila@cimat.mx)
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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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