Fernando Alcalde (joint work with Françoise Dal'Bo, Matilde Martínez, and Alberto Verjovsky)

Dynamics of the horocycle flow for homogeneous and non-homogeneous foliations by hyperbolic surfaces.

Congreso: Workshop Geometry and Dynamics of Foliations (Foliations 2014)

ICMAT, Madrid. September 1-5, 2014

Abstract: The aim of this talk is to present some progress towards the understanding of the dynamics of the horocycle flow on compact foliated manifolds by hyperbolic surfaces. This is motivated by a question formulated by Matilde Martnez and Alberto Verjovsky on the minimality of this flow when the action of the ane group generated by the combined action of the geodesic and horocycle fows is minimal too.

Firstly, we shall extend the classical theorem proved by Gustav A. Hedlund in 1936 on the minimality of the horocycle flow on compact hyperbolic surfaces to homogeneous manifolds for the product of PSL(2,R) and any connected Lie group G.  We shall give an elementary proof that does not use the famous Ratner's Orbit-Closure Theorem. We shall also show that this is always the case for homogeneous Riemannian and Lie foliations. This is a joint work with Françoise Dal'Bo.

Examples and counter-examples will take an important place in our talk. They will serve to illustrate our result, as well as a theorem by Martínez and Verjovsky that characterises the minimality of the affine action. We shall use another classical example to briefly describe some work in progress with Dal'Bo, Martínez and Verjovsky in the non-homogeneous case.