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Pisot conjecture on substitutive dynamical systems

Research Project

Project/Area Number 24540012
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionUniversity of Tsukuba

Principal Investigator

Akiyama Shigeki  筑波大学, 数理物質系, 教授 (60212445)

Co-Investigator(Kenkyū-buntansha) KOMATSU Kazushi  高知大学, 理学部, 准教授 (00253336)
Project Period (FY) 2012-04-01 – 2016-03-31
Project Status Completed (Fiscal Year 2015)
Budget Amount *help
¥5,200,000 (Direct Cost: ¥4,000,000、Indirect Cost: ¥1,200,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2013: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2012: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywordsピゾ数 / 置換規則 / 自己誘導構造 / タイリング / 自然拡大 / 概周期構造 / 準結晶 / Pisot 数 / ベータ展開 / sofic system / 不変測度 / Pisot数 / シフト基数系
Outline of Final Research Achievements

Substitutive dynamical system models self-inducing structure appears in general dynamical systems. It is generated by a substitution rule on finite letters. Such system realized as self-similar tiling can not be strongly mixing and whether it is weakly mixing or not is described by whether the dilation is a Pisot number.
We studied the well-known conjecture in this area which asserts that every irreducible Pisot substituion is pure discrete. As a result with J.Y.Lee we elucidated relationship between several coincidence conditions. Further with J.Y.Lee and F.Gahler, we did a systematic valification of the conjecture up to trace 2. All of them are pure discrete, we did not find a counter example.

Report

(5 results)
  • 2015 Annual Research Report   Final Research Report ( PDF )
  • 2014 Research-status Report
  • 2013 Research-status Report
  • 2012 Research-status Report
  • Research Products

    (66 results)

All 2016 2015 2014 2013 2012 Other

All Int'l Joint Research (4 results) Journal Article (18 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 18 results,  Acknowledgement Compliant: 5 results) Presentation (36 results) (of which Int'l Joint Research: 7 results,  Invited: 25 results) Book (2 results) Remarks (4 results) Funded Workshop (2 results)

  • [Int'l Joint Research] Leoben University(Austria)

    • Related Report
      2015 Annual Research Report
  • [Int'l Joint Research] 香港中文大学(HongKong)

    • Related Report
      2015 Annual Research Report
  • [Int'l Joint Research] Cathoric Kwandong University/Dongguk University(韓国)

    • Related Report
      2015 Annual Research Report
  • [Int'l Joint Research] 華中科技大学/華中師範大学(China)

    • Related Report
      2015 Annual Research Report
  • [Journal Article] Rotational beta expansion: Ergodicity and Soficness2016

    • Author(s)
      Shigeki Akiyama and Jonathan Caalim
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 印刷中

    • NAID

      130005310478

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Strong coincidence and overlap coincidence2016

    • Author(s)
      Shigeki Akiyama
    • Journal Title

      Discrete and Continuous Dynamical System - A

      Volume: 印刷中

    • NAID

      120007129305

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Characterization of the numbers which satisfy the height reducing property2015

    • Author(s)
      Shigeki Akiyama, Joerg Thuswaldner and Toufik Zaimi
    • Journal Title

      Indagationes Mathematicae

      Volume: 26 Issue: 1 Pages: 24-27

    • DOI

      10.1016/j.indag.2014.03.003

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Journal Article] Comments on the height reducing property II2015

    • Author(s)
      Shigeki Akiyama, Joerg Thuswaldner and Toufik Zaimi
    • Journal Title

      Indagationes Mathematicae

      Volume: 26 Issue: 1 Pages: 28-39

    • DOI

      10.1016/j.indag.2014.07.002

    • NAID

      120007136476

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Journal Article] A family of non-sofic beta expansions2014

    • Author(s)
      Shigeki Akiyama
    • Journal Title

      Ergodic Theory and Dynamical Systems

      Volume: - Issue: 2 Pages: 1-12

    • DOI

      10.1017/etds.2014.60

    • NAID

      120007135447

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Determining pure discrete spectrum on some self-affine tilings2014

    • Author(s)
      Shigeki Akiyama, Franz Gaehler and Jeong-Yup Lee
    • Journal Title

      Discrete Mathematics and Theoretical Computer Science

      Volume: 16 Pages: 305-316

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Correlation clustering of graphs and integers2014

    • Author(s)
      Shigeki Akiyama, Laszlo Aszalos, Lajos Hajdu, Attila Pethoe
    • Journal Title

      Infocommunication Journal

      Volume: 6 Pages: 3-12

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] On the distribution of polynomials with bounded roots II. Polynomials with integer coefficients2014

    • Author(s)
      Shigeki Akiyama and Attila Petho
    • Journal Title

      Uniform Distribution Theory

      Volume: 9 Pages: 5-19

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Mean divisibility of multinomial coefficients2014

    • Author(s)
      Shigeki Akiyama
    • Journal Title

      Journal of Number Theory

      Volume: 136 Pages: 438-459

    • DOI

      10.1016/j.jnt.2013.10.005

    • NAID

      120007130548

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Overlap coincidence to strong coincidence in substitution tiling dynamics2014

    • Author(s)
      Shigeki Akiyama and Jeong-Yup Lee
    • Journal Title

      European Journal of Combinatorics

      Volume: 39 Pages: 233-243

    • DOI

      10.1016/j.ejc.2014.01.009

    • NAID

      120007136736

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Isometries of the special orthogonal group2013

    • Author(s)
      Toshikazu Abe, Shigeki Akiyama and Osamu Hatori
    • Journal Title

      Linear Algerba and Its Applications

      Volume: 439 Issue: 1 Pages: 174-188

    • DOI

      10.1016/j.laa.2013.03.008

    • NAID

      120007136918

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the least common multiple of Lucas subsequences2013

    • Author(s)
      Shigeki Akiyama and Florian Luca
    • Journal Title

      Acta Arithmetica

      Volume: 161 Issue: 4 Pages: 327-349

    • DOI

      10.4064/aa161-4-2

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Discrete Spectra and Pisot numbers2013

    • Author(s)
      Shigeki Akiyama and Vilmos Komornik
    • Journal Title

      Journal of Number Theory

      Volume: 133 Issue: 2 Pages: 375-390

    • DOI

      10.1016/j.jnt.2012.07.015

    • NAID

      120007137130

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Discretized rotation has infinitely many periodic orbits2013

    • Author(s)
      Shigeki Akiyama and Attila Petho
    • Journal Title

      NonLinearity

      Volume: 26 Issue: 3 Pages: 871-880

    • DOI

      10.1088/0951-7715/26/3/871

    • NAID

      120007137059

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Browder's convergence for one-parameter nonexpansive semigroups2012

    • Author(s)
      Shigeki Akiyama, Tomonari Suzuki
    • Journal Title

      Canadian Mathematical Bulletin

      Volume: 55 Issue: 1 Pages: 15-25

    • DOI

      10.4153/cmb-2011-071-9

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Height reducing problem on algebraic integers2012

    • Author(s)
      Shigeki Akiyama, Paulius Drungilas and Jonas Jankauskas
    • Journal Title

      Functiones et Approximatio

      Volume: 47-1 Pages: 105-119

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Primitive matrices over polynomial semirings2012

    • Author(s)
      Shigeki Akiyama and Horst Brunotte
    • Journal Title

      Linear Algebra and its Applications

      Volume: 436-9 Issue: 9 Pages: 3568-3596

    • DOI

      10.1016/j.laa.2011.12.029

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] A note on aperiodic Ammann tiles2012

    • Author(s)
      Shigeki Akiyama
    • Journal Title

      Discrete & Computational Geometry

      Volume: 48 Issue: 3 Pages: 702-710

    • DOI

      10.1007/s00454-012-9418-4

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Presentation] Rotational beta expansion and self-similar tilings2016

    • Author(s)
      Shigeki Akiyama
    • Organizer
      The 1-st Hong Kong/Kyoto Workshop on Fractal Geometry and Related Areas
    • Place of Presentation
      香港科技大学, China
    • Year and Date
      2016-03-21
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Rotational beta expansion and self-similar tilings2016

    • Author(s)
      Shigeki Akiyama
    • Organizer
      Substitutions and continued fractions
    • Place of Presentation
      Paris, France
    • Year and Date
      2016-03-07
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Some class of non-sofic beta numbers2016

    • Author(s)
      Shigeki Akiyama
    • Organizer
      数論とエルゴード理論
    • Place of Presentation
      金沢大学サテライトプラザ(石川県金沢市)
    • Year and Date
      2016-02-07
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Topology of tiles and boundary parametrization2015

    • Author(s)
      Shigeki Akiyama
    • Organizer
      Sino-Japan Workshop on Dynamic Systems and Fractals
    • Place of Presentation
      Nimbo, China
    • Year and Date
      2015-12-28
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] An Erdos problem on polynomials2015

    • Author(s)
      Shigeki Akiyama
    • Organizer
      エルゴード理論とその周辺
    • Place of Presentation
      慶応大学(神奈川県横浜市)
    • Year and Date
      2015-11-25
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research
  • [Presentation] A problem of Tarski and invariant measure of rotational beta transform2015

    • Author(s)
      Shigeki Akiyama
    • Organizer
      Mini Workshop on Symbolic Dynamics and Related Topics
    • Place of Presentation
      広島大学(広島県東広島市)
    • Year and Date
      2015-10-23
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Quasi-crystal generated by polynomials: Erdos's problem2015

    • Author(s)
      Shigeki Akiyama
    • Organizer
      Aperiodic tile sets and related topics
    • Place of Presentation
      京都大学数理解析研究所(京都府京都市)
    • Year and Date
      2015-10-21
    • Related Report
      2015 Annual Research Report
  • [Presentation] Descretized rotation : Does digital analogy approximate the real world ?2015

    • Author(s)
      Shigeki Akiyama
    • Organizer
      CUHK Colloquium
    • Place of Presentation
      香港中文大, China
    • Year and Date
      2015-09-11
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Beta expansion involving rotational action2015

    • Author(s)
      Shigeki Akiyama
    • Organizer
      Measurable and Topological Dynamical System
    • Place of Presentation
      Daejeong, Korea
    • Year and Date
      2015-07-01
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Invariant measures of rotational beta expansions2015

    • Author(s)
      Shigeki Akiyama
    • Organizer
      Fractal and Numeration
    • Place of Presentation
      Admont, Austria
    • Year and Date
      2015-06-10
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Invariant measures of rotational beta expansions2015

    • Author(s)
      秋山茂樹
    • Organizer
      華中科技大学セミナー
    • Place of Presentation
      華中科技大学, 中国
    • Year and Date
      2015-03-27
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Beta expansion with rotation2015

    • Author(s)
      秋山茂樹
    • Organizer
      華中師範大学セミナー
    • Place of Presentation
      華中師範大学, 中国
    • Year and Date
      2015-03-23
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Beta expansion with rotation2015

    • Author(s)
      秋山茂樹
    • Organizer
      Workshop on "beta-transformation and related topics"
    • Place of Presentation
      九州大学
    • Year and Date
      2015-03-10
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Almost periodicity and coincidences in tiling dynamics I, II2015

    • Author(s)
      秋山茂樹
    • Organizer
      研究集会「数論とエルゴード理論」
    • Place of Presentation
      金沢大学
    • Year and Date
      2015-02-07
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] 概周期構造とタイル張り力学系2014

    • Author(s)
      秋山茂樹
    • Organizer
      研究集会「複素解析的ベクトル場・葉層構造とその周辺」
    • Place of Presentation
      京都教育大
    • Year and Date
      2014-12-14
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Beyond the Pisot Conjecture: an outlook2014

    • Author(s)
      秋山茂樹
    • Organizer
      Lorenz Center Workshop
    • Place of Presentation
      Lorenz Center Workshop, オランダ
    • Year and Date
      2014-09-05
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Mathematics on quasi-crystal structure2014

    • Author(s)
      秋山茂樹
    • Organizer
      Sapporo summer conference on dynamics of patterns in materials science
    • Place of Presentation
      北海道大学
    • Year and Date
      2014-08-01
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] 準結晶の数理とタイル張り2014

    • Author(s)
      秋山茂樹
    • Organizer
      東北大WPI-AIMRセミナー
    • Place of Presentation
      東北大WPI-AIMR
    • Year and Date
      2014-06-26
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Discretized Rotation: Intersection of number theory and dynamical system2014

    • Author(s)
      秋山茂樹
    • Organizer
      International Conference of Discrete Mathematics and Applied Sciences
    • Place of Presentation
      University of the Thai Chamber of Commerce, Thai
    • Year and Date
      2014-05-22
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Numeration systems and associated tilings dynamical systems I,II,III2013

    • Author(s)
      Shigeki Akiyama
    • Organizer
      Summer School `Number Theory and Dynamics'
    • Place of Presentation
      Grenoble University, France
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] 置換規則の力学系2013

    • Author(s)
      Shigeki Akiyama
    • Organizer
      京都大学集中講義
    • Place of Presentation
      京都大学理学部数学教室(京都)
    • Related Report
      2013 Research-status Report
  • [Presentation] Height reducing problem

    • Author(s)
      Shigeki Akiyama
    • Organizer
      Number Theory Seminer
    • Place of Presentation
      Leiden University, Netherland
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Periodic orbits of discretized rotation

    • Author(s)
      Shigeki Akiyama
    • Organizer
      Workshop on Dynamical Systems, Automata and Algorithms
    • Place of Presentation
      Picardie Jules Vernes University, Amiens, France
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Height Reducing Problem

    • Author(s)
      Shigeki Akiyama
    • Organizer
      2nd Pacific Rim Mathematical Association Congress
    • Place of Presentation
      Shanghai Jiao Tong University, China
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] 離散回転の周期軌道の無限性

    • Author(s)
      Shigeki Akiyama
    • Organizer
      第7回日中数論セミナー準備会
    • Place of Presentation
      近畿大学(飯塚)
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Strong coincidence and overlap coincidence

    • Author(s)
      Shigeki Akiyama
    • Organizer
      タイル張り力学系とその周辺
    • Place of Presentation
      京都大学数理解析研究所(京都)
    • Related Report
      2013 Research-status Report
  • [Presentation] On the mean divisibility of multinomial sequences

    • Author(s)
      Shigeki Akiyama
    • Organizer
      第7回日中数論セミナー
    • Place of Presentation
      九州大学(福岡)
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] 離散回転の力学系

    • Author(s)
      Shigeki Akiyama
    • Organizer
      京都大学談話会
    • Place of Presentation
      京都大学理学部数学教室(京都)
    • Related Report
      2013 Research-status Report
  • [Presentation] Beta expansion and algebraic coincidence

    • Author(s)
      Shigeki Akiyama
    • Organizer
      Measurable and Topological Dynamical Systems
    • Place of Presentation
      慶応大学理工学部(東京)
    • Related Report
      2013 Research-status Report
  • [Presentation] Algebraic coincidence for beta expansion and its dual tiling

    • Author(s)
      Shigeki Akiyama
    • Organizer
      Sino-Japanese Workshop on Fractals and Dynamical Systems
    • Place of Presentation
      中国科学院(中国)
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] On Shift Radix Systems

    • Author(s)
      Shigeki Akiyama
    • Organizer
      Periodic Orbits in Dynamical Systems
    • Place of Presentation
      Erwin Schrodinger International Institute for Mathematical Physics (オーストリア・ウィーン)
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Revisiting Ammann tiles

    • Author(s)
      Shigeki Akiyama
    • Organizer
      Numeration and Substitution 2012
    • Place of Presentation
      京都大学数理解析研究所(京都府)
    • Related Report
      2012 Research-status Report
  • [Presentation] Recognizing the boundary of fractal tile

    • Author(s)
      Shigeki Akiyama
    • Organizer
      Topology of tiling spaces and related topics
    • Place of Presentation
      京都大学数理解析研究所(京都府)
    • Related Report
      2012 Research-status Report
  • [Presentation] On the period of discretized rotations

    • Author(s)
      Shigeki Akiyama
    • Organizer
      Ergodic Theory and Metric Number Theory
    • Place of Presentation
      日本女子大学(東京都)
    • Related Report
      2012 Research-status Report
  • [Presentation] Periodic orbits of discretized rotations

    • Author(s)
      Shigeki Akiyama
    • Organizer
      Advances on fractals and related topics
    • Place of Presentation
      香港中文大学(中国・香港)
    • Related Report
      2012 Research-status Report
  • [Presentation] Height reducing problem of algebraic numbers

    • Author(s)
      Shigeki Akiyama
    • Organizer
      Diophatine Analysis and Related topics
    • Place of Presentation
      日本大学理工学部(東京都)
    • Related Report
      2012 Research-status Report
  • [Book] Numeration and Substitution 2014, 数理研講究録別冊 B462014

    • Author(s)
      Shigeki Akiyama, Valerie Bertha, Takao Komatsu, Hui Rao 編
    • Total Pages
      247
    • Publisher
      京都大学数理解析研究所
    • Related Report
      2014 Research-status Report
  • [Book] Summer School on the Theory of Uniform Distribution. Papers from the school held in Kyoto, August 23-25, 20102012

    • Author(s)
      秋山茂樹
    • Total Pages
      118
    • Publisher
      京都大学数理解析研究所
    • Related Report
      2012 Research-status Report
  • [Remarks]

    • URL

      http://math.tsukuba.ac.jp/~akiyama/index_j.html

    • Related Report
      2015 Annual Research Report
  • [Remarks] http://math.tsukuba.ac.jp/~akiyama/index_j.html

    • Related Report
      2014 Research-status Report
  • [Remarks] 個人 Web Page

    • URL

      http://math.tsukuba.ac.jp/~akiyama/index_j.html

    • Related Report
      2013 Research-status Report
  • [Remarks] 個人Web Page

    • URL

      http://math.tsukuba.ac.jp/~akiyama/index_j.html

    • Related Report
      2012 Research-status Report
  • [Funded Workshop] Aperiodic tile sets and related topics2015

    • Place of Presentation
      京都大学数理解析研究所(京都府京都市)
    • Year and Date
      2015-10-19
    • Related Report
      2015 Annual Research Report
  • [Funded Workshop] Natural extention of arithmetic algorithms and S-adic system2015

    • Place of Presentation
      玉原国際セミナーハウス(群馬県沼田市)
    • Year and Date
      2015-07-20
    • Related Report
      2015 Annual Research Report

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Published: 2013-05-31   Modified: 2022-03-01  

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