2014/10/27 by Fernando Alcalde Cuesta, Cuesta, Fernando Alcalde, Cuesta +1 · 2 citations
Mathematics · #37C85 #37D40 #57R30 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1410.7181
openalex publication_date 2014/10/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This article is a first step towards the understanding of the dynamics of the\nhorocycle flow on foliated manifolds by hyperbolic surfaces. This is motivated\nby a question formulated by M. Martinez and A. Verjovsky on the minimality of\nthis flow assuming that the natural affine foliation is minimal too. We have\ntried to offer a simple presentation, which allows us to update and shed light\non the classical theorem proved by G. A. Hedlund in 1936 on the minimality of\nthe horocycle flow on compact hyperbolic surfaces. Firstly, we extend this\nresult to the product of PSL(2,R) and a Lie group G, which places us within the\nhomogeneous framework investigated by M. Ratner. Since our purpose is to deal\nwith non-homogeneous situations, we do not use Ratner's famous Orbit-Closure\nTheorem, but we give an elementary proof. We show that this special situation\narises for homogeneous Riemannian and Lie foliations, reintroducing the\nfoliation point of view. Examples and counter-examples take an important place\nin our work, in particular, the very instructive case of the hyperbolic torus\nbundles over the circle. Our aim in writing this text is to offer to the reader\nan accessible introduction to a subject that was intensively studied in the\nalgebraic setting, although there still are unsolved geometric problems.\n