Topological structure of fractal tilings attached to number system

数系分形镶嵌的拓扑结构

基本信息

批准号：
12640017
负责人：
AKIYAMA Shigeki
金额：
$ 1.66万
依托单位：
Niigata University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640017/
关键词：
Pisot number Tiling Number system Zeta function Symbolic Dynamical system Fractal Tiling Pisot number Number System L function zeta function

项目摘要

Natural extension of number theoretical approximating algorithm and its Markov partition give clues to connect number theory and symbolic dynamical system. During this research period it is shown that finiteness and its weaker form of Pisot number system play important roles in constructing such natural extension of beta expansion. In the first paper of the list, we classified all cubic Pisot units having this finiteness property. In the 6-th paper, it is shown that weakly finiteness assures a non-overlapping self-affine tiling attached to beta expansions. It is conjectured that all Pisot units have weakly finiteness property. On the other hand, the theory of tilings attached to canonical number systems was also developed. In the 2-nd paper with J.Thuswaldner, we have studied closely topological structure of tilings attached to quadratic canonical number systems. If the number of triple points of a tile is 4 or 6 then the tile should be homeomorphic to a disk. We classified at last these disklike cases by coefficients of irreducible polynomials. The 5-th paper with A.Pethoe, we gave a new sufficient condition of canonical number system. When the constant term is sufficiently large, this result gives a sharp bound. Other papers with Y.Tanigawa and S.Egami are devoted to the analytic continuation of Euler-Zagier's multiple zeta function and their non-negative values. K.Matsumoto gave a generalization of them. We expect these researches would give positive ideas to define a natural zeta function attached to tilings.

数字理论近似算法及其马尔可夫分区的自然扩展为连接数字理论和符号动力学系统的线索提供了线索。在这一研究期间，这表明有限性及其较弱的PISOT数字系统在构建Beta扩展的自然扩展方面起着重要作用。在列表的第一篇论文中，我们将所有具有这种有限性属性的立方PISOT单元分类。在第6篇论文中，据表明，弱有限确保了与Beta扩展相关的非重叠的自我诉讼瓷砖。猜想所有PISOT单元具有弱有限的属性。另一方面，还发展了与规范数量系统相关的瓷砖理论。在与J.Thuswaldner的2 nd论文中，我们研究了附着在二次规范数字系统上的瓷砖的拓扑结构。如果瓷砖的三重点的数量为4或6，则瓷砖应同态磁盘。我们终于通过不可还原多项式的系数对这些磁盘状的情况进行了分类。第5篇文章与A. pethoe，我们给出了规范编号系统的新条件。当恒定项足够大时，该结果会产生急剧的结合。其他具有Y.Tanigawa和S.Egami的论文专门致力于Euler-Zagier多重Zeta函数及其非负值的分析延续。 K.Matsumoto对它们进行了概括。我们预计这些研究将提供积极的想法，以定义斜利所附加的自然Zeta功能。