## 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 ⊂ ℝ^{d} is an attractor of a finite iterated function system of n ≥ 2 maps of the form x ↦→ D^{−1} x + t_{i} (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 t_{i}, almost every point in K (w.r.t. any Bernoulli measure) has an equidistributed orbit under the map x ↦→ Dx (multiplication mod 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.

Original language | English |
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Pages (from-to) | 467-493 |

Number of pages | 27 |

Journal | American Journal of Mathematics |

Volume | 146 |

Issue number | 2 |

DOIs | |

State | Published - 1 Apr 2024 |

Externally published | Yes |

## ASJC Scopus subject areas

- General Mathematics