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2001 Fiscal Year Final Research Report Summary

Topological structure of fractal tilings attached to number system

Research Project

Project/Area Number 12640017
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionNiigata University

Principal Investigator

AKIYAMA Shigeki  Faculty of Science, Niigata University, Associate Professor, 理学部, 助教授 (60212445)

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)
Project Period (FY) 2000 – 2001
KeywordsPisot number / Tiling / Number system / Zeta function / Symbolic Dynamical system / Fractal
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.

  • Research Products

    (12 results)

All Other

All Publications (12 results)

  • [Publications] Shigeki Akiyama: "Cubic Pisot Units with finite beta expansions"Algebraic Number Theory and Diophantine Analysis, ed. by F.Halter-Koch and R.F. Tichy, de Gruyter. 11-26 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Shigeki Akiyama: "Topological properties of two-dimensional number systems"Journal de Theorie des Nombres de Bordeaux. Vol.12. 69-79 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Shigeki Akiyama: "Analytic continuation of multiple zeta-functions and their values at non-positive integers"Acta Arithmetica. Vol.98, no.2. 107-116 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Shigeki Akiyama: "Multiple zeta values at non-positive integers"The Ramanujan Journal. Vol.5, no.4. (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Shigeki Akiyama: "On Canonical Number Systems"Theoretical Computer Science. Vol.270. 921-933 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Shigeki Akiyama: "On the boundary of self affine tilings generated by Pisot numbers"Journal of the Mathematical Society of Japan. Vol.54. (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Shigeki Akiyama: "Cubic Pisot Units with finite beta expansions"Algebraic Number Theory and Diophantine Analysis, ed. By F.Halter-Koch and R.F. Tichy, de Gruyter. 11-26 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shigeki Akiyama and Joerg M. Thuswaldner: "Topological properties of two-dimensional number systems"Journal de Theorie des Nombres de Bordeaux. Vol. 12. 69-79 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shigeki Akiyama, Shigeki Egami and Yoshio Tanigawa: "Analytic continuation of multiple zeta-functions and their values at non-positive, integers"Acta Arithmetica. Vol. 98, no. 2. 107-116 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shigeki Akiyama and Yoshio Tanigawa: "Multiple zeta values at non-positive integers"The Ramanujan Journal. Vol. 5, no. 4 (to appear). (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shigeki Akiyama and Attila Pethoe: "On Canonical Number Systems"Theoretical Computer Science. Vol. 270. 921-933 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shigeki Akiyama: "On the boundary of self-affine tilings generated by Pisot numbers"J. Math. Soc. Japan. Vol. 54 (to appear). (2002)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2003-09-17  

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