平面具有四元数字集自相似测度的谱结构研究

数学物理学报 ›› 2025, Vol. 45 ›› Issue (4): 1023-1040.

平面具有四元数字集自相似测度的谱结构研究

吕军()

新疆农业大学数理学院乌鲁木齐 830052

收稿日期:2024-08-26 修回日期:2025-01-26 出版日期:2025-08-26 发布日期:2025-08-01
作者简介:E-mail: lvjun136248@sina.com
基金资助:
国家自然科学基金(12371087);自治区高校基本科研业务费科研项目(XJEDU2025P045)

The Study on Spectral Structure of Planar Self-Similar Measures with Four Element Digit Sets

Lü Jun()

College of Mathematics and Physics, XinJiang Agricultural University, Urumqi 830052

Received:2024-08-26 Revised:2025-01-26 Online:2025-08-26 Published:2025-08-01
Supported by:
NSFC(12371087);Basic Research Funds for Autonomous Region Universities Research Projects(XJEDU2025P045)

摘要/Abstract

摘要：

设$Q=\begin{pmatrix} b & 0\\ 0 & b \end{pmatrix}$是一整扩张矩阵且设$\mathcal{D}=\left\{\begin{pmatrix} 0 \\ 0 \end{pmatrix},\,\,\begin{pmatrix} 1 \\ 0 \end{pmatrix},\,\,\begin{pmatrix} 0 \\ 1 \end{pmatrix},\,\,\begin{pmatrix} -1 \\ -1 \end{pmatrix} \right\}$ 是一四元数字集,作者考虑由上述整扩张矩阵$Q$和四元数字集$\mathcal{D}$所生成的自相似测度$\mu_{Q,\mathcal{D}}$ 的谱结构.测度$\mu_{Q,\mathcal{D}}$是谱测度当且仅当$b=2q$,$q\in \mathbb{N} $,并且此测度的谱是如下集

$\Lambda(Q,\mathcal{C}_q)=\left\{\sum_{k=0}^{n}Q^{k}\mathcal{C}_{q}:\,\,n\geq 1\right\}:=\mathcal{C}_{q}+Q\mathcal{C}_{q}+Q^{2}\mathcal{C}_{q}+\cdots,\,\,\text{所有有限和},$

其中$\mathcal{C}_{q}=q\left\{\begin{pmatrix} 0 \\ 0 \end{pmatrix},\,\,\begin{pmatrix} 1 \\ 0 \end{pmatrix},\,\,\begin{pmatrix} 0 \\ 1 \end{pmatrix},\,\,\begin{pmatrix} 1 \\ 1 \end{pmatrix} \right\}$. 该文通过极大树映射研究了测度$\mu_{Q,\mathcal{D}}$的极大正交集的结构并且在此基础上探讨了其谱特征矩阵的相关问题.

关键词: 极大树映射, 谱特征矩阵, 谱测度, 谱结构, 自相似测度

Abstract:

Let $Q=\begin{pmatrix} b & 0\\ 0 & b \end{pmatrix}$ be an integer expanding matrix and let $\mathcal{D}=\left\{\begin{pmatrix} 0 \\ 0 \end{pmatrix},\begin{pmatrix} 1 \\ 0 \end{pmatrix},\begin{pmatrix} 0 \\ 1 \end{pmatrix},\begin{pmatrix} -1 \\ -1 \end{pmatrix} \right\}$ be a four element digit set. We considered the spectral structure of self-similar measure $\mu_{Q,\mathcal{D}}$ which generated by an integer expanding matrix $Q$ and a four element digits $\mathcal{D}$. It is well known that $\mu_{Q,\mathcal{D}}$ is a spectral measure if and only if $b=2q$ for some $q\in\mathbb{N}$. The spectrum for this spectral measure is the following set

$\Lambda(Q,\mathcal{C}_q)=\left\{\sum_{k=0}^{n}Q^{k}\mathcal{C}_{q}:\,\,n\geq 1\right\}:=\mathcal{C}_{q}+Q\mathcal{C}_{q}+Q^{2}\mathcal{C}_{q}+\cdots,\,\,\text{all}\,\,\text{finite}\,\,\text{sums},$

where $\mathcal{C}_{q}=q\left\{\begin{pmatrix} 0 \\ 0 \end{pmatrix},\,\,\begin{pmatrix} 1 \\ 0 \end{pmatrix},\,\,\begin{pmatrix} 0 \\ 1 \end{pmatrix},\,\,\begin{pmatrix} 1 \\ 1 \end{pmatrix} \right\}$. In this paper, we investigate the structure of the maximum orthogonal set of $\mu_{Q,\mathcal{D}} $ through the maximum tree mapping and based on this, the relevant issues of its spectral eigenmatrix were discussed.

Key words: maximum tree mapping, spectral eigenmatrix, spectral measure, spectral structure, self-similar measure

中图分类号:

O174.2

吕军. 平面具有四元数字集自相似测度的谱结构研究[J]. 数学物理学报, 2025, 45(4): 1023-1040.

Lü Jun. The Study on Spectral Structure of Planar Self-Similar Measures with Four Element Digit Sets[J]. Acta mathematica scientia,Series A, 2025, 45(4): 1023-1040.

参考文献 21

[1]	An L X, He X G, Tao L. Spectrality of the planar Sierpinski family. J Math Anal Appl, 2015, 4.2(2): 725-732
[2]	Chen M L, Yan Z H. On the spectrality of self-affine measures with four digits on $\mathbb{R}^{2}$. Internat J Math, 2021, 32(1): 215004
[3]	Chen S, Tang M W. Spectrality and non-spectrality of planar self-similar measures with four-element digit sets. Fractals, 2020, 28(7):2050130
[4]	Dai X R. When does a Bernoulli convolution admit a spectrum? Adv Math, 2012, 2.1(3/4): 1681-1693
[5]	Dai X R. Spectra of Cantor measures. Math Ann, 2016, 3.6(3): 1621-1647
[6]	Dai X R, He X G, Lai C K. Spectral property of Cantor measures with consecutive digits. Adv Math, 2013, 2.2: 187-208
[7]	Dai X R, He X G, Lau K S. On spectral N-Bernoulli meausure. Adv Math, 2014, 2.9: 511-531
[8]	Dai X R, Fu X Y, Yan Z H. Spectrality of self-affine Sierpinski-type measures on $\mathbb{R}^2$. Appl Comput Harmon Anal, 2020, 52: 63-81
[9]	Deng Q R, Lau K S. Sierpinski-type spectral self-similar measures. J Funct Anal, 2015, 2.9(5): 1310-1326
[10]	Dutkay D, Jorgensen P. Fourier series on fractals: a parallel with wavelet theory. Contemp Math, 2008, 4.4: 75-101
[11]	Dutkay D, Haussermann J, Lai C K. Hadamard triples generate self-affine spectral measures. Trans Amer Math Soc, 2019, 3.1(2): 1439-1481
[12]	Dutkay D, Han D G, Sun Q Y. On the spectra of a Cantor measure. Adv Math, 2009, 2.1(1): 251-276
[13]	Dutkay D, Lai C K. Uniformity of measures with Fourier frames. Adv Math, 2014, 2.2: 684-707
[14]	He X G, Lai C K, Lau K S. Exponential spectra in $L^2(\mu)$. Appl Comput Harmon Anal, 2013, 34(3): 327-338
[15]	He X G, Tang M W, Wu Z Y. Spectral structure and spectral eigenvalue problems of a class of self-similar spectral measures. J Funct Anal, 2019, 2.7(10): 3688-3722
[16]	Jorgensen P, Pedersen S. Dense analytic subspaces in fractal $L^2$ spaces. J Anal Math, 1998, 75: 185-228
[17]	Jorgensen P, Kornelson K, Shuman K. Affine systems: asymptotics at infinity for fractal measures. Acta Appl Math, 2007, 98: 181-222
[18]	Łaba, Wang Y. On spectral Cantor measures. J Funct Anal, 2002, 1.3: 409-420
[19]	Li J J, Wu Z Y. On spectral structure and spectral eigenvalue problems for a class of self similar spectral measure with product form. Nonlinearity, 2022, 35: 3095-3117
[20]	Li J L. Spectra of a class of self-affine measures. J Funct Anal, 2011, 2.0: 1086-1095
[21]	Li H X, Li Q. Spectral structure of planar self-similar measures with four-element digit set. J Math Anal Appl, 2022, 5.3(1): 126202