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
$X$.