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

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 $$D_n=\left\{\left(\begin{array}{c}0\\ 0\end{array}\right),\left(\begin{array}{c}{\alpha }_{n_1}\\ {\alpha }_{n_2}\end{array}\right),\left(\begin{array}{c}{\beta }_{n_1}\\ {\beta }_{n_2}\end{array}\right),\left(\begin{array}{c}-{\alpha }_{n_1}-{\beta }_{n_1}\\ -{\alpha }_{n_2}-{\beta }_{n_2}\end{array}\right)\right\}$$ satisfying ${\alpha }_{n_1}{\beta }_{n_2}-{\alpha }_{n_2}{\beta }_{n_1}\ne 0(\mathrm{mod}2)$. Under the hypotheses $|\det (M_n)|>4$ for all $n\ge 1$, ${\sup }_{n\ge 1}\Vert M_n^{-1}\Vert <1$, and $\{D_n\}$ is finite, we establish the following characterization: $${\mu }_{\{M_n\},\{D_n\}}\text{ is a spectral measure}⟺M_n\in GL(2,2\mathbb{Z})\text{ for all }n\ge 2.$$ 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.

21 pages