The structure of the boundary of the Rauzy fractal

2012 Fiscal Year Final Research Report

• Project/Area Number: 22540117
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Yokohama National University
• Principal Investigator: FURUKADO Maki 横浜国立大学, 国際社会科学研究科, 助手 (20303068)
• Co-Investigator(Kenkyū-buntansha): ITO Shunji 金沢大学, 自然科学研究科, 研究員 (30055321)
• Project Period (FY): 2010 – 2012
• Keywords: サブスティテューション / タイリングサブスティテューション / フラクタル / 準周期タイリング

Research Abstract

The tiling substitutions obtained by the Rauzy induction of4-interval exchange transformations with the period of length 8 sometimes generate theoverlapping patch. We avoid this phenomenon by using the “blocking method” and wesucceeded in generating 8 tilings.

Research Products

• [Journal Article] Algebraic numbers, free group automorphisms and substitutions on the plane (2011)
  • Author(s): Pierre Arnoux,古門麻貴, Edmund Harriss,伊藤俊次
  • Journal Title: Transactions of the American Mathematical Society Volume: vo1.363,no. 9 Pages: 4651-4699
  • DOI: DOI:10.1090/SOOO2-9947-2011-05188-3
  • Peer Reviewed
• [Journal Article] Substitutions from Rauzy induction on 4-interval exchange transformations and quasi-periodic tilings (2011)
  • Author(s): Pierre Arnoux,古門麻貴,伊藤俊次
  • Journal Title: 京都大学数理解析研究所講究録「準周期秩序の数理」 Volume: 1725 Pages: 152-174
• [Presentation] On the condition for the generation of the stepPed surfaces via modified Jacobi-Perron expansions I, II (2013)
  • Author(s): 古門麻貴
  • Organizer: Workshop 「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)
  • Year and Date: 2013-02-09
• [Presentation] Non-Pisot substitutions from Rauzy induction of 4 interval exchange transformations and quasi-periodic tilings, Conference (2013)
  • Author(s): 古門麻貴
  • Organizer: Subtile 2013
  • Place of Presentation: CIRM(フランス)
  • Year and Date: 2013-01-15
• [Presentation] Anew multidimensiona1slow continued fraction algorithm andstepped surface,共同研究 (2012)
  • Author(s): 古門麻貴,伊藤俊次,斉藤朝輝,田村純一,安冨真一
  • Organizer: タイリング空間のトポロジーとその周辺
  • Place of Presentation: 京都大学数理解析研究所(京都府)
  • Year and Date: 2012-10-04
• [Presentation] The conditions for generating stepPed surfaces in terms of the modified Jacobi-Perron algorithm (2012)
  • Author(s): 古門麻貴,伊藤俊次,安冨真一
  • Organizer: 共同研究「タ イリング空間のトポロジーとその周辺」
  • Place of Presentation: 京都大学数理解析研究所(京都府)
  • Year and Date: 2012-10-04
• [Presentation] Non-Pisot substitutions from RID and quasi-periodic tilings (2012)
  • Author(s): 古門麻貴,伊藤俊次
  • Organizer: AMS Sectional Meeting AMS Special Session on Tilings, Substitutions, and Bratteli-Vershik Transformations
  • Place of Presentation: George Washington University(アメリカ)
  • Year and Date: 2012-03-17
• [Presentation] Non-Pisot substitutions from RID and quasi-periodic tilings (2012)
  • Author(s): 古門麻貴
  • Organizer: Workshop「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)
  • Year and Date: 2012-02-17
• [Presentation] Non-Pisot substitutions from RID and quasi-periodic tilings (2011)
  • Author(s): 古門麻貴,伊藤俊次
  • Organizer: 共同研究「自己相似タイリングの剛性定理とその周辺」
  • Place of Presentation: 京都大学数理解析研究所(京都府)
  • Year and Date: 2011-12-21
• [Presentation] Non-Pisot substitutions from RID and quasi-periodic tilings (2011)
  • Author(s): 古門麻貴,伊藤俊次
  • Organizer: 研究集会「エル ゴード理論とその周辺」
  • Place of Presentation: 大阪大学大学院理学研究科(大阪府)
  • Year and Date: 2011-12-09
• [Presentation] On the admissible condition of orbits generated by Farey type algorithm with a value function (2011)
  • Author(s): 古門麻貴,伊藤俊次
  • Organizer: 2011年度東京力学系セミナー「夏季集中セミナー」
  • Place of Presentation: 弘前大学工学部(青森県)
  • Year and Date: 2011-08-07
• [Presentation] On characteristic polynomials of 4-1etter substitutions from Rauzy induction (2010)
  • Author(s): 古門麻貴
  • Organizer: 2010年度東京力学系セミナー「夏季集中セミナー」
  • Place of Presentation: 東京工業高等専門学校(東京都)
  • Year and Date: 2010-09-04
• [Presentation] The substitutions related to the Rauzy induction on 4-interval exchange transformations and tilings II, Workshop (2010)
  • Author(s): 古門麻貴, 伊藤俊次
  • Organizer: Substitutive Tiling and Fractal Geometry
  • Place of Presentation: Sun Yat sen University(中国)
  • Year and Date: 2010-07-08
• [Presentation] The substitutions related to the Rauzy induction of 4-interval exchange transformations and tilings II, Workshop (2010)
  • Author(s): 古門麻貴, 伊藤俊次
  • Organizer: Mathematics of Quasi-Periodic Order
  • Place of Presentation: 京都大学数理解析研究所(京都府)
  • Year and Date: 2010-06-23