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

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