Doubling property of self-similar measures with overlaps

TL;DR

This paper investigates when self-similar measures with overlaps are doubling, revealing that for certain parameters they are always non-doubling, extending previous results on measures satisfying the open set condition.

Contribution

It extends the characterization of doubling properties from measures satisfying the open set condition to those with overlaps, especially for Pisot number-based self-similar measures.

Findings

For m=2, measure is doubling if and only if p=(1/2,1/2).

For m≥3, the measure is always non-doubling.

Provides a complete classification for these self-similar measures.

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 $m\geq 2$ 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 $m\geq 3$, $\mu_{\mathbf{p}}$ is always non-doubling.