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
$X$
.
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