Random walks on tori and normal numbers in self similar sets

Yiftach Dayan, Arijit Ganguly, Barak Weiss

Published 2020-02-02Version 1

We show that random walks on a $d$-dimensional torus by affine expanding maps whose linear parts commute, under a certain condition imposed on their translation parts, have a unique stationary measure. We then use this result to show that given an IFS of contracting similarity maps of $\mathbb{R}^{d}$ with a uniform contraction ratio $\frac{1}{D}$, where $D$ is some integer $>1$, under some suitable condition on the linear parts of the maps in the IFS, almost every point in the attractor (w.r.t. any Bernoulli measure) has an equidistributed orbit under the map $x\mapsto Dx\,\text{(mod }\mathbb{Z}^{d})$ w.r.t. Haar measure on $\mathbb{T}^{d}$. In the 1-dim case, this conclusion amounts to normality to base $D$. As an example, we obtain that w.r.t. a natural measure, almost every point in an irrational dilation of the middle thirds Cantor set is normal to base 3.