Remarks on the dynamics of the horocycle flow for homogeneous foliations by hyperbolic surfaces

TL;DR

This paper explores the dynamics of horocycle flows on foliated manifolds by hyperbolic surfaces, extending classical results and providing elementary proofs within a homogeneous framework, with applications to Riemannian and Lie foliations.

Contribution

It extends Hedlund's classical minimality theorem to product groups and offers an elementary proof avoiding Ratner's Orbit-Closure Theorem, focusing on non-homogeneous situations and foliations.

Findings

Extended minimality results to PSL(2,R) and Lie groups

Provided elementary proofs for non-homogeneous cases

Illustrated with examples like hyperbolic torus bundles

Abstract

This article is a first step towards the understanding of the dynamics of the horocycle flow on foliated manifolds by hyperbolic surfaces. This is motivated by a question formulated by M. Martinez and A. Verjovsky on the minimality of this flow assuming that the natural affine foliation is minimal too. We have tried to offer a simple presentation, which allows us to update and shed light on the classical theorem proved by G. A. Hedlund in 1936 on the minimality of the horocycle flow on compact hyperbolic surfaces. Firstly, we extend this result to the product of PSL(2,R) and a Lie group G, which places us within the homogeneous framework investigated by M. Ratner. Since our purpose is to deal with non-homogeneous situations, we do not use Ratner's famous Orbit-Closure Theorem, but we give an elementary proof. We show that this special situation arises for homogeneous Riemannian and Lie foliations, reintroducing the foliation point of view. Examples and counter-examples take an important place in our work, in particular, the very instructive case of the hyperbolic torus bundles over the circle. Our aim in writing this text is to offer to the reader an accessible introduction to a subject that was intensively studied in the algebraic setting, although there still are unsolved geometric problems.