Revista de la
Unión Matemática Argentina
Spectrality of planar Moran–Sierpinski-type measures
Qian Li and Min-Min Zhang

Volume 67, no. 1 (2024), pp. 65–80    

Published online: March 12, 2024

https://doi.org/10.33044/revuma.2932

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Abstract

Let $\{M_n\}_{n=1}^{\infty}$ be a sequence of expanding positive integral matrices with $M_n= \begin{pmatrix} p_n & 0\\0 & q_n \end{pmatrix}$ for each $n\ge 1$, and let $D=\left\{\begin{pmatrix} 0\ 0 \end{pmatrix}, \begin{pmatrix} 1\ 0 \end{pmatrix}, \begin{pmatrix} 0\ 1 \end{pmatrix} \right\}$ be a finite digit set in $\mathbb{Z}^2$. The associated Borel probability measure obtained by an infinite convolution of atomic measures \[ \mu_{\{M_n\},D}=\delta_{M_1^{-1}D}*\delta_{(M_2M_1)^{-1}D}*\cdots*\delta_{(M_n\cdots M_2M_1)^{-1}D}*\cdots \] is called a Moran–Sierpinski-type measure. We prove that, under certain conditions, $\mu_{\{M_n\}, D}$ is a spectral measure if and only if $3\mid p_n$ and $3\mid q_n$ for each $n\geq2$.

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