Escape of mass in homogeneous dynamics in positive characteristic

Alexander Kemarsky; Frédéric Paulin; Uri Shapira

doi:10.3934/jmd.2017015

Article Contents

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

This volume Previous Article Normal forms for non-uniform contractions Next Article A disconnected deformation space of rational maps

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

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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.

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Mathematics Subject Classification: Primary: 20G25, 37A17, 20E08, 22F30; Secondary: 20H20, 20G30, 20C08, 37D40.

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Figure 1. The modular ray ${\rm {PGL}}_2(k_\infty[Y])\backslash\!\backslash{\mathbb{T}}_\infty$

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Figure 2. Back and forth paths in cuspidal rays

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Figure 3. A partition of the Hecke sphere $S_\nu(n)$

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Figure 4. Rational Bruhat-Tits rays

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Figure 5. Sector-spheres in Hecke trees

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Figure 6. Iterated construction of nested sectors

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