Abstract:
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 $\KK \subset \R^d$ 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 $\KK$ is not badly approximable. Furthermore, almost every point on the fractal is of generic type, which means (in the one-dimensional case) that its continued fraction expansion contains all finite words with the frequencies predicted by the Gauss measure. Joint work with David Simmons.