Random walks on homogeneous spaces and Diophantine approximation on fractals

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 $${\mathbb {R}}^d$$ . As examples, we show that for any self-similar fractal $$\mathcal {K}\subseteq {\mathbb {R}}^d$$ 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 $$\mathcal {K}$$ 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. We prove analogous results for matrix approximation, and for the case of fractals defined by Mobius transformations.