Fourier bases of a class of planar self-affine measures

Ming-Liang Chen, Jing-Cheng Liu and Zhi-Yong Wang

Vol. 327 (2023), No. 1, 55–81

DOI: 10.2140/pjm.2023.327.55

Abstract

Let ${\mu }_{M,D}$ be the planar self-affine measure generated by an expansive integer matrix $M\in M_2(\mathbb{Z})$ and a noncollinear integer digit set

$$D=\left\{\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \end{array}\right),\left(\begin{array}{c}\hfill {\alpha }_1\hfill \\ \hfill {\alpha }_2\hfill \end{array}\right),\left(\begin{array}{c}\hfill {\beta }_1\hfill \\ \hfill {\beta }_2\hfill \end{array}\right),\left(\begin{array}{c}\hfill -{\alpha }_1-{\beta }_1\hfill \\ \hfill -{\alpha }_2-{\beta }_2\hfill \end{array}\right)\right\}.$$

We show that ${\mu }_{M,D}$ is a spectral measure if and only if there exists a matrix $Q\in M_2(ℝ)$ such that $(\tilde{M},\tilde{D})$ is admissible, where $\tilde{M}=QM{Q}^{-1}$ and $\tilde{D}=QD$. In particular, when ${\alpha }_1{\beta }_2-{\alpha }_2{\beta }_1\notin 2\mathbb{Z}$, ${\mu }_{M,D}$ is a spectral measure if and only if $M\in M_2(2\mathbb{Z})$. This completely settles the spectrality of the self-affine measure ${\mu }_{M,D}$.

Keywords

self-affine measure, spectral measure, spectrum, admissible

Mathematical Subject Classification

Primary: 28A25, 28A80

Secondary: 42C05, 46C05

Milestones

Received: 17 September 2023

Revised: 8 December 2023

Accepted: 22 January 2024

Published: 18 February 2024