Journal of Modern Dynamics (JMD)

Horocycle flows for laminations by hyperbolic Riemann surfaces and Hedlund's theorem

Pages: 113 - 134, Volume 10, 2016      doi:10.3934/jmd.2016.10.113

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Matilde Martínez - Instituto de Matemática y Estadística Rafael Laguardia, Facultad de Ingeniería, Universidad de la República, J. Herrera y Reissig 565, C.P. 11300 Montevideo, Uruguay (email)
Shigenori Matsumoto - Department of Mathematics, College of Science and Technology, Nihon University, 1-8-14 Kanda, Surugadai, Chiyoda-ku, Tokyo, 101-8308, Japan (email)
Alberto Verjovsky - Universidad Nacional Autónoma de México, Apartado Postal 273, Admon. de correos #3, C.P. 62251 Cuernavaca, Morelos, Mexico (email)

Abstract: We study the dynamics of the geodesic and horocycle flows of the unit tangent bundle $(\hat M, T^1\mathfrak{F})$ of a compact minimal lamination $(M,\mathfrak{F})$ by negatively curved surfaces. We give conditions under which the action of the affine group generated by the joint action of these flows is minimal and examples where this action is not minimal. In the first case, we prove that if $\mathfrak{F}$ has a leaf which is not simply connected, the horocyle flow is topologically transitive.

Keywords:  Hyperbolic surfaces, horocycle and geodesic flows, hyperbolic laminations, minimality.
Mathematics Subject Classification:  Primary: 37C85, 37D40, 57R30.

Received: April 15 2015;      Revised: February 23 2016;      Available Online: May 4 2016.