Random walks on homogeneous spaces and diophantine approximation on fractals

TL;DR

This paper extends dynamical results on random walks in homogeneous spaces to cases with less restrictive conditions and applies these to study Diophantine approximation properties of typical points on self-similar fractals, showing most are not badly approximable.

Contribution

It generalizes previous results to broader classes of measures and applies these to analyze Diophantine properties on various fractals, including those defined by Mobius transformations.

Findings

Almost every point on certain self-similar fractals is not badly approximable.

Typical points on these fractals are of generic type with predicted continued fraction frequencies.

Results extend to matrix multiplication and fractals defined by Mobius transformations.

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 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.