自仿射测度在$GL_n(p)$相似变换下的谱性
The spectrality of self-affine measure under the similarity transformation of $GL_n(p)$
摘要：
令$μ_{M,D}$是由扩展整数矩阵$M\in M_n(\mathbb{Z})$和有限位集$D\subset\mathbb{Z}^n$生成的自仿射测度。众所周知，如果$\tilde{M}=B^{-1}MB$和$\tilde{D}=B^{-1}D$，则两个测度$μ_{M,D}$和$μ_{\tilde{M},\tilde{D}}$具有相同的谱性，其中$B\in M_n(\mathbb{R})$是非奇异矩阵。这一事实通常用于简化数字集$D$或扩展矩阵$M$。然而，由于需要将整数数字集$D$或扩展矩阵$M$变换为实数，这给$μ_{\tilde{M},\tilde{D}}$的谱性质研究带来了困难。本文引入了一般线性群$GL_n(p)$对某些自仿射测度的相似变换，并讨论了它们的谱性。这种相似变换可以同时保持$D$和$M$的整数性质，从而在讨论自仿射测度的谱性时具有许多优点.作为应用，我们将一些著名的谱自仿射测度推广到更一般的形式。
英文摘要：
Let $μ_{M,D}$ 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 $μ_{M,D}$ and $μ_{\tilde{M},\tilde{D}}$ 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 $μ_{\tilde{M},\tilde{D}}$. In this paper, we introduce a similarity transformation of general linear group $GL_n(p)$ 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.
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