  We extend results of Y. Benoist and J.-F. Quint concerning random walks on
homogeneous spaces of simple Lie groups to the case where the measure defining
the random walk generates a semigroup which is not necessarily Zariski dense,
but satisfies some expansion properties for the adjoint action. Using these
dynamical results, we study diophantine properties of typical points on some
self-similar fractals in R^d. As examples, we show that for any self-similar
fractal K in R^d satisfying the open set condition (for instance any translate
or dilate of Cantor's middle thirds set or of a Koch snowflake), almost every
point with respect to the natural measure on K is not badly approximable.
Furthermore, almost every point on the fractal is of generic type, which
implies (in the one dimensional case) that its continued fraction expansion
contains all finite words with the frequencies predicted by Gauss measure. We
prove analogous results for matrix multiplication, and for the case of fractals
defined by Mobius transformation.
