论文标题
在$ gl_n（p）$的相似性转换下自我措施的光谱
The spectrality of self-affine measure under the similarity transformation of $GL_n(p)$
论文作者
论文摘要
令$μ_{m，d} $为M_n（\ Mathbb {Z}）$的扩展整数矩阵$ m \生成的自我处理度量，并且有限的数字集$ d \ subset \ subset \ mathbb {z}^n $。众所周知，这两种度量$μ_{m，d} $和$μ_ {\ tilde {m}，\ tilde {d}} $具有相同的光谱，如果$ \ tilde {m} = b^{ - 1} mb $ and $ b^wre-wre m_n（\ mathbb {r}）$是一个非语音矩阵。这个事实通常用于简化数字集$ d $或扩展的矩阵$ m $。但是，它通常会将整数数字设置$ d $或将矩阵$ m $扩展到真实中，这带来了许多困难来研究$μ_ {\ tilde {m}，\ tilde {d}} $的光谱。在本文中，我们引入了一般线性群$ gl_n（p）$的相似性转换，以进行某些自我措施，并讨论它们的光谱。这种相似性转换可以同时使整数属性保持$ d $和$ m $，这在讨论自我措施措施的光谱方面带来了许多优势。作为一种应用，我们将一些众所周知的光谱自我措施扩展到更通用的形式。
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.
