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Effectivity of uniqueness of the maximal entropy measure on $p$ -adic homogeneous spaces

Published online by Cambridge University Press:  11 February 2015

RENE RÜHR
Affiliation:
Eidgenössische Technische Hochschule Zürich, Rämistrasse 101, 8092 Zürich, Switzerland email reneruehr@gmail.com
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Abstract

We consider the dynamical system given by an $\text{Ad}$ -diagonalizable element $a$ of the $\mathbb{Q}_{p}$ -points $G$ of a unimodular linear algebraic group acting by translation on a finite volume quotient $X$ . Assuming that this action is exponentially mixing (e.g. if $G$ is simple) we give an effective version (in terms of $K$ -finite vectors of the regular representation) of the following statement: If ${\it\mu}$ is an $a$ -invariant probability measure with measure-theoretical entropy close to the topological entropy of $a$ , then ${\it\mu}$ is close to the unique $G$ -invariant probability measure of $X$ .

Type
Research Article
Copyright
© Cambridge University Press, 2015 

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