With cedram.org Confluentes Mathematici Search for an article Search within the site English français Table of contents for this issue | Previous article | Next article Masseye Gaye; Cheikh LoSur l’inexistence d’ensembles minimaux pour le flot horocyclique(On the non-existence of minimal sets for the horocycle flow)Confluentes Mathematici, 9 no. 1 (2017), p. 95-104, doi: 10.5802/cml.37 Article PDF Class. Math.: 37D40, 20H10, 14H55, 30F35Keywords: surface hyperbolique, ensemble minimal, orbite horocyclique, point limite. Résumé - AbstractThe topological dynamics of the horocycle flow $h_\mathbb{R}$ on the unitary tangent bundle of a geometrically finite surface $S$ is well known. In particular in this case a $h_\mathbb{R}$-minimal set exists always. When the surface is geometrically infinite, the behaviour of this flow depends on the action of the fundamental group $\pi _1(S)$ of the surface on the boundary of the hyperbolic plane. The aim of this paper is to give a non-existence condition of $h_\mathbb{R}$-minimal sets and use it to construct a family of geometrically infinite surfaces on which the horocyclic flow has no minimal sets. Bibliography[1] A. F. Beardon, The geometry of discrete groups, Springer-Verlag, New York, 1983. [2] F. Dal’Bo, Trajectoires géodésiques et horocycliques, Savoirs Actuels, EDPS-CNRS, 2007. [3] F. Dal’Bo et A.N. Starkov, On a classification of limit points of infinitely generated Schottky groups, J. Dyn. control Sys., 6(4) :561–578, 2000. [4] E. Ghys, Dynamique des flots unipotents sur les espaces homogènes, Sém. Bourbaki, 34 :93–136, 1991-1992. [5] G. A. Hedlund, Fuchsian groups and transitive horocycles, Duke Math. J., 2(3) :530–542, 1936. [6] M. Kulikov, The horocycle flow without minimal sets, C.R. A.S., 1338 :477–480, 2004. [7] S. Matsumoto, Horocycle flow without minimal sets, J. Math. Sci. Univ. Tokyo, 23(3) :661–673, 2016. [8] A. N. Starkov, Fuchsian groups from the dynamical viewpoint, J. Dyn. control Sys., 1(3) :427–445, 1995. © 2012Confluentes Mathematici Published by the Institut Camille Jordan CNRS - UMR 5208 and the Unité de Mathématiques Pures et Appliquées CNRS - UMR 5669 of Université de Lyon eISSN : 1793-7434