On spectral eigenmatrix problem for the planar self-affine measures with three digits

被引:0
作者
Liu, Jing-Cheng [1 ]
Liu, Ming [1 ]
Tang, Min-Wei [1 ]
Wu, Sha [2 ]
机构
[1] Hunan Normal Univ, Sch Math & Stat, Key Lab Comp & Stochast Math, Minist Educ, Changsha 410081, Hunan, Peoples R China
[2] Hunan Univ, Sch Math, Changsha 410082, Hunan, Peoples R China
基金
中国国家自然科学基金;
关键词
Self-affine measure; Spectral measure; Spectrum structure; Spectral eigenmatrix; EIGENVALUE PROBLEMS; PROPERTY;
D O I
10.1007/s43034-024-00386-1
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
Let mu M,D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{M,D}$$\end{document} be a self-affine measure generated by an iterated function systems {phi d(x)=M-1(x+d)(x is an element of R2)}d is an element of D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\phi _d(x)=M<^>{-1}(x+d)\ (x\in \mathbb {R}<^>2)\}_{d\in D}$$\end{document}, where M is an element of M2(Z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\in M_2(\mathbb {Z})$$\end{document} is an expanding integer matrix and D={(0,0)t,(1,0)t,(0,1)t}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D = \{(0,0)<^>t,(1,0)<^>t,(0,1)<^>t\}$$\end{document}. In this paper, we study the spectral eigenmatrix problem of mu M,D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{M,D}$$\end{document}, i.e., we characterize the matrix R which R Lambda\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R\Lambda $$\end{document} is also a spectrum of mu M,D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{M,D}$$\end{document} for some spectrum Lambda\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document}. Some necessary and sufficient conditions for R to be a spectral eigenmatrix are given, which extends some results of An et al. (Indiana Univ Math J, 7(1): 913-952, 2022). Moreover, we also find some irrational spectral eigenmatrices of mu M,D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{M,D}$$\end{document}, which is different from the known results that spectral eigenmatrices are rational.
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页数:22
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