${{\mathbb{R}}^{n}}$中一类具有N元数字集的自仿测度的谱性

数学物理学报 ›› 2020, Vol. 40 ›› Issue (3): 667-675.

${{\mathbb{R}}^{n}}$中一类具有N元数字集的自仿测度的谱性

李红光¹(),张鹏飞^2,*()

¹ 怀化学院数学与计算科学学院湖南怀化 418008
² 湖南师范大学数学与统计学院长沙 410081

收稿日期:2019-01-31 出版日期:2020-06-26 发布日期:2020-07-15
通讯作者: 张鹏飞 E-mail:lhg20052008@126.com;pfzhang@link.cuhk.edu.hk
作者简介:李红光, E-mail:lhg20052008@126.com
基金资助:
国家自然科学基金(11831007)

Spectral Property of Some Self-Affine Measures with N-Element Digits on ${{\mathbb{R}}^{n}}$

Hongguang Li¹(),Pengfei Zhang^2,*()

¹ School of Mathematics and Computational Science, Huaihua College, Hunan Huaihua 418008
² College of Mathematics and Statistics, Hunan Normal University, Changsha 410081

Received:2019-01-31 Online:2020-06-26 Published:2020-07-15
Contact: Pengfei Zhang E-mail:lhg20052008@126.com;pfzhang@link.cuhk.edu.hk
Supported by:
the NSFC(11831007)

摘要/Abstract

摘要：

假设$R \in M_n({\Bbb Z})$为扩张矩阵和$N$元数字集${\cal D}=\{0, a_1, a_2, \cdots, a_{N-1}\}u \equiv \{0, 1, \cdots, $ $N-1\}u \ ({\rm mod}N) $,这里$u\in {\Bbb Z}^n\setminus\{0\}$.该文主要研究由${\cal D}$和$R$生成的自仿测度$\mu_{R, {\cal D}}$的谱性,得到了$\mu_{R, {\cal D}}$为谱测度的一个充分条件.对于一些特殊情况,得到了$\mu_{R, {\cal D}}$为谱测度的一个充分必要条件,并给出其谱的具体表达式.

关键词: N元数字集, 自仿测度, 谱测度

Abstract:

Let $R \in M_n({\Bbb Z})$ be an expanding matrix and ${\cal D}=\{0, a_1, a_2, \cdots, a_{N-1}\}u \equiv \{0, 1, \cdots, N-1\}u \ ({\rm mod}N) $ be a $N$-element digit set, where $u\in {\Bbb Z}^n\setminus\{0\}$. In this paper, we study the spectral property of the self-affine measures $\mu_{R, {\cal D}}$ which is generated by ${\cal D}$ and $R$, and obtain a sufficient condition such that $\mu_{R, {\cal D}}$ is a spectral measure. Moreover, for a special case, we give a necessary and sufficient condition such that $\mu_{R, {\cal D}}$ is a spectral measure, and the exact spectrum of $\mu_{R, {\cal D}}$ is given.

Key words: N-element digits, Self-affine measures, Spectral measures

中图分类号:

O174.2

李红光,张鹏飞. ${{\mathbb{R}}^{n}}$中一类具有N元数字集的自仿测度的谱性[J]. 数学物理学报, 2020, 40(3): 667-675.

Hongguang Li,Pengfei Zhang. Spectral Property of Some Self-Affine Measures with N-Element Digits on ${{\mathbb{R}}^{n}}$[J]. Acta mathematica scientia,Series A, 2020, 40(3): 667-675.

参考文献 12

1	Jorgensen P E T , Pederson S . Dense analytic subspaces in fractal L²-spaces. J Anal Math, 1998, 75, 185- 228 doi: 10.1007/BF02788699
2	Dai X R . When does a Bernoulli convolution admit a spectrum?. Adv Math, 2012, 231, 1681- 1693 doi: 10.1016/j.aim.2012.06.026
3	Li J L . Non-spectral problem for a class of planar self-affine measures. J Funct Anal, 2008, 255, 3125- 3148
4	Li J L . On the μ_{M, D}-orthogonal exponentials. Nonlinear Anal, 2010, 73, 940- 951 doi: 10.1016/j.na.2010.04.017
5	Li J L . Spectra of a class of self-affine measures. J Funct Anal, 2011, 260, 1086- 1095 doi: 10.1016/j.jfa.2010.12.001
6	Li J L . Spectral self-affine measures on the spatial Sierpinski gasket. Monatsh Math, 2015, 176, 293- 322 doi: 10.1007/s00605-014-0725-0
7	Li J L . Non-spectrality of self-affine measures on the spatial Sierpinski gasket. J Math Anal Appl, 2015, 432, 1005- 1017 doi: 10.1016/j.jmaa.2015.07.032
8	Dutkay D E , Jorgensen P E T . Fourier frequencies in affine iterated function systems. J Funct Anal, 2007, 247, 110- 137 doi: 10.1016/j.jfa.2007.03.002
9	Dutkay D E , Haussermann J , Lai C K . Hadamard triples generate self-affine spectral measures. Trans Amer Math Soc, 2019, 371, 1439- 1481
10	Liu J C , Luo J . Spectral property of self-affine measure on ${{\mathbb{R}}^{n}}$. J Funct Anal, 2017, 272, 599- 612 doi: 10.1016/j.jfa.2016.10.011
11	Ramsey F P . On a problem of formal logic. Proc London Math Soc, 1929, 30 (2): 264- 286
12	Strichartz R . Remarks on "Dense analytic subspaces in fractal L²-spaces". J Anal Math, 1998, 75, 229- 231 doi: 10.1007/BF02788700

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