Journal of Modern Dynamics (JMD)

Escape of mass in homogeneous dynamics in positive characteristic

Pages: 369 - 407, Volume 11, 2017      doi:10.3934/jmd.2017015

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Alexander Kemarsky - Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark (email)
Frédéric Paulin - Laboratoire de mathématiques d’Orsay, UMR 8628 Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay Cedex, France (email)
Uri Shapira - Mathematics Department, Technion, Israel Institute of Technology, Haifa, 32000, Israel (email)

Abstract: We show that in positive characteristic the homogeneous probability measure supported on a periodic orbit of the diagonal group in the space of $2$-lattices, when varied along rays of Hecke trees, may behave in sharp contrast to the zero characteristic analogue: For a large set of rays, the measures fail to converge to the uniform probability measure on the space of $2$-lattices. More precisely, we prove that when the ray is rational there is uniform escape of mass, that there are uncountably many rays giving rise to escape of mass, and that there are rays along which the measures accumulate on measures which are not absolutely continuous with respect to the uniform measure on the space of $2$-lattices.

Keywords:  Homogeneous measures, positive characteristic, lattices, local fields, escape of mass, Hecke tree, Bruhat-Tits tree, equidistribution.
Mathematics Subject Classification:  Primary: 20G25, 37A17, 20E08, 22F30; Secondary: 20H20, 20G30, 20C08, 37D40.

Received: December 2015;      Revised: December 2016;      Available Online: May 2017.