Doubling property of self-similar measures with overlaps

Abstract

Recently, Yang, Yuan and Zhang [Doubling properties of self-similar measures and Bernoulli measures on self-affine Sierpinski sponges, Indiana Univ. Math. J., 73 (2024), 475-492] characterized when a self-similar measure satisfying the open set condition is doubling. In this paper, we study when a self-similar measure with overlaps is doubling. Let $m\ge 2$ and let $\beta >1$ be the Pisot number satisfying ${\beta }^m={\sum }_{j=0}^{m-1}{\beta }^j$. Let $\mathbf{p}=(p_1,p_2)$ be a probability weight and let ${\mu }_{\mathbf{p}}$ be the self-similar measure associated to the IFS $\{S_1(x)=x/\beta ,S_2(x)=x/\beta +(1-1/\beta ),\}.$ Yung [...,Indiana Univ. Math. J., ] proved that when $m=2$, ${\mu }_{\mathbf{p}}$ is doubling if and only if $\mathbf{p}=(1/2,1/2)$. We show that for $m\ge 3$, ${\mu }_{\mathbf{p}}$ is always non-doubling.