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

TL;DR

This paper characterizes when certain planar Moran measures are spectral, linking this property to the matrices involved, and provides a complete criterion for a critical case, extending the theory of spectral measures.

Contribution

It establishes a precise spectral characterization for Moran measures generated by specific matrix and digit set conditions, including a complete criterion for the critical determinant case.

Findings

Spectrality is equivalent to matrices being in GL(2,2Z) for all n≥2.

Complete spectral criterion derived for the case | ext{det}(M_n)|=4.

Extends spectral measure theory to Moran-type constructions.

Abstract

We investigate spectral properties of planar Moran measures $\mu_{\{M_n\},\{D_n\}}$ 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 $$ D_n = \left\{ \begin{pmatrix} 0 \\ 0 \end{pmatrix}, \begin{pmatrix} \alpha_{n_1} \\ \alpha_{n_2} \end{pmatrix}, \begin{pmatrix} \beta_{n_1} \\ \beta_{n_2} \end{pmatrix}, \begin{pmatrix} -\alpha_{n_1}-\beta_{n_1} \\ -\alpha_{n_2}-\beta_{n_2} \end{pmatrix} \right\} $$ satisfying $\alpha_{n_1}\beta_{n_2}-\alpha_{n_2}\beta_{n_1} \ne 0 \pmod{2}$. 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: $$ \mu_{\{M_n\},\{D_n\}} \text{ is a spectral measure} \Longleftrightarrow M_n \in GL(2,2\mathbb{Z}) \text{ for all } n\geq 2. $$ 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.