Published online by Cambridge University Press: 11 February 2015
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
$X$ .