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.

数论逼近算法的自然推广及其马尔可夫划分为连接数论和符号动力系统提供了线索。在本研究期间，结果表明皮索数系的有限性及其弱形式在构建β展开式的这种自然扩展中发挥着重要作用。在列表的第一篇论文中，我们对具有这种有限性的所有三次皮索单位进行了分类。在第六篇论文中，表明弱有限性确保了附加到 beta 扩展的非重叠自仿射平铺。据推测，所有Pisot单位都具有弱有限性。另一方面，附加于规范数字系统的平铺理论也得到了发展。在与 J.Thuswaldner 合作的第二篇论文中，我们仔细研究了二次规范数系统上的平铺的拓扑结构。如果瓦片的三点数是 4 或 6，那么该瓦片应该与圆盘同胚。我们最后通过不可约多项式的系数对这些盘状情况进行了分类。在与A.Pethoe的第五篇论文中，我们给出了规范数系的一个新的充分条件。当常数项足够大时，该结果会出现尖锐的界限。 Y.Tanikawa 和 S.Egami 的其他论文致力于 Euler-Zagier 多重 zeta 函数及其非负值的解析延拓。 K.Matsumoto 对它们进行了概括。我们期望这些研究能够给出积极的想法来定义附加到平铺的自然 zeta 函数。