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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 Open the ORCID record for RENE RÜHR [Opens in a new window]
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
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 6 , September 2016 , pp. 1972 - 1988
Copyright
© Cambridge University Press, 2015 

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