RANDOM WALKS ON TORI AND NORMAL NUMBERS IN SELF-SIMILAR SETS

Yiftach Dayan, Arijit Ganguly, Barak Weiss

Research output: Contribution to journalArticlepeer-review

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 ⊂ ℝd is an attractor of a finite iterated function system of n ≥ 2 maps of the form x ↦→ D−1 x + ti (i = 1, …, n), where D is an expanding d×d integer matrix, and is the same for all the maps, under an irrationality condition on the translation parts ti, almost every point in K (w.r.t. any Bernoulli measure) has an equidistributed orbit under the map x ↦→ Dx (multiplication mod ℤd). 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.

Original languageEnglish
Pages (from-to)467-493
Number of pages27
JournalAmerican Journal of Mathematics
Volume146
Issue number2
DOIs
StatePublished - Apr 2024

All Science Journal Classification (ASJC) codes

  • General Mathematics

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