标题： 自相似测度的倍增性质 显示英文标题

标题： Doubling property of self-similar measures with overlaps

摘要： 最近，杨、袁和张[自相似测度和自仿射谢尔宾斯基海绵上的伯努利测度的二倍性质，印第安纳大学数学杂志，73 (2024), 475-492]表征了满足开集条件的自相似测度何时是二倍测度。在本文中，我们研究了具有重叠的自相似测度何时是二倍测度。设$m\geq 2$并让$\beta>1$是满足$\beta^m=\sum_{j=0}^{m-1}\beta^j$的皮索数。设$\mathbf{p}=(p_1,p_2)$是一个概率权重，让$\mu_{\mathbf{p}}$是与IFS$\{ S_1(x)={x}/{\beta}, S_2(x)={x}/{\beta}+(1-{1}/{\beta}),\}.$关联的自相似测度，杨[...，印第安纳大学数学杂志] J.，] 证明了当 $m=2$， $\mu_{\mathbf{p}}$ 是倍增的当且仅当 $\mathbf{p}=(1/2,1/2)$。 我们证明了对于 $m\geq 3$， $\mu_{\mathbf{p}}$ 总是非倍增的。 显示英文摘要

摘要： 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.