On normal numbers and self-similar measures

TL;DR

This paper proves that under certain conditions involving self-similar measures and Pisot numbers, the image of such measures under smooth transformations is supported on numbers that are normal in a specific base.

Contribution

It establishes a new link between self-similar measures, Pisot numbers, and normality in a given base, extending understanding of measure transformations and normal numbers.

Findings

Measures supported on self-similar sets become normal in base β after smooth transformations.

The result applies to a broad class of self-similar measures and diffeomorphisms.

Conditions involve algebraic independence of contraction ratios and Pisot numbers.

Abstract

Let $\lbrace f_i(x)=s_i \cdot x+t_i \rbrace$ be a self-similar IFS on $\mathbb{R}$ and let $\beta >1$ be a Pisot number. We prove that if $\frac{\log |s_i|}{\log \beta}\notin \mathbb{Q}$ for some $i$ then for every $C^1$ diffeomorphism $g$ and every non-atomic self similar measure $\mu$, the measure $g\mu$ is supported on numbers that are normal in base $\beta$.