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Statistical behaviour of the leaves of Riccati foliations

Published online by Cambridge University Press:  21 May 2009

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)

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
Copyright
Copyright © Cambridge University Press 2009

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