Doubling property of self-similar measures with overlaps

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.