Statistical Behaviour of the Leaves of Riccati Foliations

TL;DR

This paper studies the statistical dynamics of geodesic flows on leaves of Riccati foliations, establishing conditions for the existence of unique ergodic measures that describe typical leaf behavior.

Contribution

It introduces a framework for analyzing the statistical properties of geodesic flows on Riccati foliation leaves and identifies conditions for unique ergodic measures with attractor dynamics.

Findings

Existence of measures m- and m+ describing leaf dynamics

Convergence of statistical averages to these measures for almost all initial conditions

Invariant measures are ergodic and project to harmonic measures

Abstract

We introduce the geodesic flow on the leaves of a holomorphic foliation with leaves of dimension 1 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 projectivisation of the solutions of linear ordinary differential equations over a finite Riemann surface of hyperbolic type S, and may be described by a representation r:pi_1(S) -> GL(n,C). We give conditions under which the foliated geodesic flow has a generic repellor-attractor statistical dynamics. That is, there are measures m- and m+ 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 m- and for positive time to m+ (i.e. m+ is the unique 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 behavior of the leaves of the foliation.