On normal numbers and self-similar measures

Abstract

Let $\{f_i(x)=s_i\cdot x+t_i\}$ be a self-similar IFS on $\mathbb{R}$ and let ${\hat{I}}^2>1$ be a Pisot number. We prove that if $\frac{\log |s_i|}{\log {\hat{I}}^2}\notin \mathbb{Q}$ for some $i$ then for every $C^1$ diffeomorphism $g$ and every non-atomic self similar measure $\hat{I}¼$, the measure $g\hat{I}¼$ is supported on numbers that are normal in base ${\hat{I}}^2$.

This article supersedes arXiv:2107.02699