The spectrality of self-affine measure under the similarity transformation of $G{L}_n(p)$

Abstract

Let ${\mu }_{M,D}$ be the self-affine measure generated by an expanding integer matrix $M\in M_n(\mathbb{Z})$ and a finite digit set $D\subset {\mathbb{Z}}^n$. It is well known that the two measures ${\mu }_{M,D}$ and ${\mu }_{\tilde{M},\tilde{D}}$ have the same spectrality if $\tilde{M}=B^{-1}MB$ and $\tilde{D}=B^{-1}D$, where $B\in M_n(\mathbb{R})$ is a nonsingular matrix. This fact is usually used to simplify the digit set $D$ or the expanding matrix $M$. However, it often transforms integer digit set $D$ or expanding matrix $M$ into real, which brings many difficulties to study the spectrality of ${\mu }_{\tilde{M},\tilde{D}}$. In this paper, we introduce a similarity transformation of general linear group $G{L}_n(p)$ for some self-affine measures, and discuss their spectrality. This kind of similarity transformation can keep the integer properties of $D$ and $M$ simultaneously, which leads to many advantages in discussing the spectrality of self-affine measures. As an application, we extend some well-known spectral self-affine measures to more general forms.

References in corpus (1)

• Complex Hadamard matrices and the Spectral Set Conjecture

Cited by in corpus (1)

• Spectrality of generalized Sierpinski-type self-affine measures