Random walks on tori and normal numbers in self similar sets

TL;DR

This paper investigates random walks on tori with affine expanding maps, proving unique stationary measures and showing that typical points in certain self-similar sets are equidistributed or normal, under irrationality conditions.

Contribution

It establishes conditions under which Haar measure is unique and typical points are equidistributed or normal in self-similar sets with affine expanding maps.

Findings

Haar measure is the unique stationary measure for the studied random walks.

Almost every point in certain self-similar sets is equidistributed under the linear map.

In one dimension, points are normal to the base of the expansion.

Abstract

We study random walks on a $d$-dimensional torus by affine expanding maps whose linear parts commute. Assuming an irrationality condition on their translation parts, we prove that the Haar measure is the unique stationary measure. We deduce that if $K \subset \mathbb{R}^d$ is an attractor of a finite iterated function system of $n\geq 2$ maps of the form $x \mapsto D^{-r_i} x + t_i \ (i=1, \ldots, n)$, where $D$ is an expanding $d\times d$ integer matrix, and is the same for all the maps, and $r_{i} \in\mathbb{N}$, under an irrationality condition on the translation parts $t_i$, almost every point in $K$ (w.r.t. any Bernoulli measure) has an equidistributed orbit under the map $x\mapsto Dx$ (multiplication mod $\mathbb{Z}^{d}$). In the one-dimensional case, this conclusion amounts to normality to base $D$. Thus for example, almost every point in an irrational dilation of the middle-thirds Cantor set is normal to base 3.