  Consider an homogeneous space under a locally compact group G and a lattice
in G. Then the lattice naturally acts on the homogeneous space. Looking at a
dense orbit, one may wonder how to describe its repartition. One then adopt a
dynamical point of view and compare the asymptotic distribution of points in
the orbits with the natural measure on the space. In the setting of Lie groups
and their homogeneous spaces, several results showed an equidistribution of
points in the orbits using Ratner's rigidity of polynomial dynamics in
homogeneous spaces.
  We address here this problem in the setting of p-adic and S-arithmetic
groups.
