Effectivity of Uniqueness of the Maximal Entropy Measure on $p$-adic homogeneous spaces

TL;DR

This paper proves that in certain $p$-adic homogeneous spaces, measures with entropy close to the maximum are nearly the unique invariant measure, using effective mixing properties.

Contribution

It provides an effective version of measure uniqueness in $p$-adic homogeneous spaces under exponential mixing assumptions.

Findings

Measures with near-maximal entropy are close to the unique invariant measure.

The result applies to actions with exponential mixing, including simple groups.

The approach uses $K$-finite vectors of the regular representation.

Abstract

We consider the dynamical system given by a diagonalizable element $a$ of a closed linear unimodular algebraic subgroup $G$ of the special linear group over the $p$-adic numbers 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 $\mu$ is an $a$-invariant probability measure with measure-theoretical entropy close to the topological entropy of $a$ then $\mu$ is close to the unique $G$-invariant probability measure of $X$.