# American Institute of Mathematical Sciences

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

## Escape of mass in homogeneous dynamics in positive characteristic

 1 Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark 2 Laboratoire de mathématiques déOrsay, UMR 8628 Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay Cedex, France 3 Mathematics Department, Technion, Israel Institute of Technology, Haifa, 32000 Israel

Received  December 20, 2015 Revised  December 05, 2016 Published  May 2017

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.

Citation: Alexander Kemarsky, Frédéric Paulin, Uri Shapira. Escape of mass in homogeneous dynamics in positive characteristic. Journal of Modern Dynamics, 2017, 11: 369-407. doi: 10.3934/jmd.2017015
##### References:

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##### References:
The modular ray ${\rm {PGL}}_2(k_\infty[Y])\backslash\!\backslash{\mathbb{T}}_\infty$
Back and forth paths in cuspidal rays
A partition of the Hecke sphere $S_\nu(n)$
Rational Bruhat-Tits rays
Sector-spheres in Hecke trees
Iterated construction of nested sectors
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