2004 Fiscal Year Final Research Report Summary
Symbolic Dynamical System and Number Theoretical Tilings
| Project/Area Number |
14540015
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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 Niigata University, Faculty of Science, Associate Professor, 理学部, 助教授 (60212445)
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| Co-Investigator(Kenkyū-buntansha) |
ITO Shunji Kanazawa University, Faculty of Technology, Professor, 工学部, 教授 (30055321)
TANIGAWA Yoshio Nagoya University, Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (50109261)
MATSUMOTO Kohji Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (60192754)
YOSHIHARA Hisao Niigata University, Faculty of Science, Professor, 理学部, 教授 (60114807)
TAKEUCHI Teruo Niigata University, Faculty of Science, Professor, 理学部, 教授 (10018848)
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| Project Period (FY) |
2002 – 2004
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| Keywords | Symbolic Dynamical System / Tiling / Number System / Pisot Number / Fractal / Zeta Function |
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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 weak finiteness of Pisot number system is equivalent to the non-overlapping tiling property, which is essential in above construction. In the fifth paper of the list, we proved all cubic Pisot units have this weak finiteness. It is conjectured by N.Sidorov that all Pisot numbers have weakly finiteness property.
On the other hand, the theory of tilings associated to canonical number systems was also developed. In the first paper with H.Rao, a variant of finiteness is studied in canonical number system setting and we gave some family with this property. One important development has been made that we reached to the concept of shift radix system, which generalize known number systems and provide us a kind of moduli space of good number systems. On this subject, several joint projects with A.Pethoe, J.Thuswaldnder, H.Brunotte, K.Scheicher and W.Steiner are on going. Several topological studies of tiles are published including the one with N.Gjini and the one with J.Luo and J.Thuswaldner. The classification of canonical number system is linked to the power integral basis problem on which a joint study had begun with T.Takeuchi
Substitution dynamics introduced by Sh.Ito and P.Arnoux has a strong connection to this study. Its algebraic structure permits homological study. With the help of H.Yoshihara, we are trying to extend of this method. With Y.Tanigawa, we studied the fractional part of powers of a given Salem number which intimately related to Pisot numbers. It is tempting to introduce zeta functions associated to aperiodic structures on which we are collaborating with K.Matsumoto. A comprehensive survey was written in ‘Sugaku' published by the Japanese Mathematical Society.
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