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Symbolic Dynamical System and Number Theoretical Tilings

Research Project

Project/Area Number 14540015
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  Niigata University, Faculty of Science, Associate Professor, 理学部, 助教授 (60212445)

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)
明石 重男  新潟大学, 理学部, 教授 (30202518)
Project Period (FY) 2002 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥3,900,000 (Direct Cost: ¥3,900,000)
Fiscal Year 2004: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2003: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 2002: ¥1,300,000 (Direct Cost: ¥1,300,000)
KeywordsSymbolic Dynamical System / Tiling / Number System / Pisot Number / Fractal / Zeta Function
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.

Report

(4 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (30 results)

All 2004 Other

All Journal Article (18 results) Publications (12 results)

  • [Journal Article] New criteria for canonical number systems2004

    • Author(s)
      Shigeki Akiyama
    • Journal Title

      Acta Arithmetica 111, 1

      Pages: 5-25

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On boundary connectedness of connected tiles2004

    • Author(s)
      Jun Luo
    • Journal Title

      Mathematical Proceedings Cambridge Philosophical Society 137, 2

      Pages: 397-410

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Salem numbers and uniform distribution modulo 12004

    • Author(s)
      Shigeki Akiyama
    • Journal Title

      Publicationes Mathematicae Debrecen 64, 3-4

      Pages: 329-341

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the connectedness of self-affine tilings2004

    • Author(s)
      Shigeki Akiyama
    • Journal Title

      Archiv der Mathematik 82, 2

      Pages: 153-163

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A certain finiteness property of Pisot number systems2004

    • Author(s)
      Shigeki Akiyama
    • Journal Title

      Journal of Number Theory 107, 1

      Pages: 135-160

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] 記号力学系と数系タイル張り2004

    • Author(s)
      秋山茂樹
    • Journal Title

      雑誌「数学」 (日本数学会) 56, 4

      Pages: 351-365

    • NAID

      10013753684

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] New criteria for canonical number systems2004

    • Author(s)
      Shigeki Akiyama, Rao Hui
    • Journal Title

      Acta Arith. vol.111, no.1

      Pages: 5-25

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On boundary connectedness of connected tiles2004

    • Author(s)
      Jun Luo, Shigeki Akiyama, Joerg Thuswaldner
    • Journal Title

      Math.Proc.Cambridge Phil.Soc. vol.137, no.2

      Pages: 397-410

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Salem numbers and uniform distribution modulo 12004

    • Author(s)
      Shigeki Akiyama, Yoshio Tanigawa
    • Journal Title

      Publ.Math.Debrecen vol.64/3-4

      Pages: 329-341

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the connectedness of self-affine tilings2004

    • Author(s)
      Shigeki Akiyama, Nertila Gjini
    • Journal Title

      Archiv der Math. vol.82

      Pages: 153-163

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A certain finiteness property of Pisot number systems2004

    • Author(s)
      Shigeki Akiyama, Hui Rao, Wolfgang Steiner
    • Journal Title

      Journal of Number Theory vol.107-1

      Pages: 135-160

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Symbolic dynamical system and tilings associated to number systems2004

    • Author(s)
      Shigeki Akiyama
    • Journal Title

      Sugaku vol.56 No.4

      Pages: 351-365

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] New criteria for canonical number systems2004

    • Author(s)
      Shigeki Akiyama
    • Journal Title

      Acta Arithmetica 111,1

      Pages: 5-25

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On boundary connectedness of connected tiles2004

    • Author(s)
      Jun Luo
    • Journal Title

      Mathematical Proceedings Cambridge Philosophical Society 137,2

      Pages: 397-410

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Salem numbers and uniform distribution modulo 12004

    • Author(s)
      Shigeki Akiyama
    • Journal Title

      Publicationes Mathematicae Debrecen 64,3-4

      Pages: 329-341

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On the connectedness of self-affine tilings2004

    • Author(s)
      Shigeki Akiyama
    • Journal Title

      Archiv der Mathematik 82,2

      Pages: 153-163

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A certain finiteness property of Pisot number systems2004

    • Author(s)
      Shigeki Akiyama
    • Journal Title

      Journal of Number Theory 107,1

      Pages: 135-160

    • Related Report
      2004 Annual Research Report
  • [Journal Article] 記号力学系と数系タイル張り2004

    • Author(s)
      秋山茂樹
    • Journal Title

      雑誌「数学」(日本数学会) 56,4

      Pages: 351-365

    • NAID

      10013753684

    • Related Report
      2004 Annual Research Report
  • [Publications] Shigeki Akiyama: "On the boundary of self affine tilings generated by Pisot numbers"Journal Mathematical Society of Japan. 54. 283-308 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] Shigeki Akiyama: "Cubic CNS Polynomials, notes on a conjecture of W.J.Gilbert"Journal of Mathematical Analysis and its Applications. 281. 402-415 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Shigeki Akiyama: "New criteria for canonical number systems"Acta Arithmetica. 111,1. 5-25 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Jun Luo: "On boundary connectedness of connected tiles"Mathematical Proceedings Cambridge Philosophical Society. (発表予定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Shigeki Akiyama: "Salem numbers and uniform distribution modulo 1"Publicationes Mathematicae Debrecen. (発表予定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Shigeki Akiyama: "A survey on topological properties of tiles related to number systems"Geometriae Dedicata. (発表予定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Shigeki Akiyama: "Multiple zeta values at non-positive integers"The Ramanujan Journal. Vol.5 no.4. 327-351 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] Shigeki Akiyama: "On Canonical Number Systems"Theoretical Computer Science. Vol.270. 921-933 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Shigeki Akiyama: "On analytic continuation of multiple L-functions and related zeta-functions"Analytic number theory, ed. C.Jia and K.Matsumoto, Kluwer. 1-16 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Shigeki Akiyama: "On the boundary of self affine tilings generated by Pisot numbers"Journal of the Mathematical Society of Japan. Vol.54,no.2. 283-308 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Shigeki Akiyama: "Cubic CNS Polynomials, Notes on a conjecture of W.J.Gilbert"Journal of Mathematical Analysis and its Applications. (発売予定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Jun Luo: "On boundary connectedness of connected tiles"Mathematical Proceedings of the Cambridge Philosophical Society. (発売予定). (2003)

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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