2001 Fiscal Year Final Research Report Summary
Topological structure of fractal tilings attached to number system
| Project/Area Number |
12640017
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Niigata University |
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Principal Investigator |
AKIYAMA Shigeki Faculty of Science, Niigata University, Associate Professor, 理学部, 助教授 (60212445)
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| Co-Investigator(Kenkyū-buntansha) |
MATSUMOTO Kohji Graduate School of Mathematics, Nagoya University, Professor, 大学院・多元数理科学研究科, 教授 (60192754)
TANIGAWA Yoshio Graduate School of Mathematics, Nagoya University, Associate Professor, 大学院・多元数理科学研究科, 助教授 (50109261)
ITO Shunji Faculty of Liberal Arts, Tsuda College, Professor, 学芸学部, 教授 (30055321)
YOSHIHARA Hisao Faculty of Science, Niigata University, Professor, 理学部, 教授 (60114807)
AKASHI Shigeo Faculty of Science, Niigata University, Professor, 理学部, 教授 (30202518)
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| Project Period (FY) |
2000 – 2001
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| Keywords | Pisot number / Tiling / Number system / Zeta function / Symbolic Dynamical system / Fractal |
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| Research Abstract |
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.
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