Spectrality of a class of moran measures on $\mathbb{R}^2$

Abstract

We investigate spectral properties of planar Moran measures $μ_{\{M_n\},\{D_n\}}$ generated by sequences of expanding matrices $\{M_n\}\subset GL(2,\mathbb{Z})$ and digit sets $\{D_n\}\subset\mathbb{Z}^2$, where each digit set has the form $$ D_n = \left\{ \begin{pmatrix} 0 \\ 0 \end{pmatrix}, \begin{pmatrix} α_{n_1} \\ α_{n_2} \end{pmatrix}, \begin{pmatrix} β_{n_1} \\ β_{n_2} \end{pmatrix}, \begin{pmatrix} -α_{n_1}-β_{n_1} \\ -α_{n_2}-β_{n_2} \end{pmatrix} \right\} $$ satisfying $α_{n_1}β_{n_2}-α_{n_2}β_{n_1} \ne 0 \pmod{2}$. Under the hypotheses $|\det(M_n)| > 4$ for all $n\geq 1$, $\sup_{n\geq 1}\|M_n^{-1}\| < 1$, and $\{D_n\}$ is finite, we establish the following characterization: $$ μ_{\{M_n\},\{D_n\}} \text{ is a spectral measure} \Longleftrightarrow M_n \in GL(2,2\mathbb{Z}) \text{ for all } n\geq 2. $$ Furthermore, for the critical case $|\det(M_n)| = 4$, we derive a complete spectral criterion for a significant class of Moran measures through combinatorial analysis of digit sets. These results extend current understanding of spectral self-affine measures to Moran-type constructions.