Ergodic theory of number theoretic transformations toward the construction of infinite ergodic theory

• Project/Area Number: 15K17559
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Japan Women's University
• Principal Investigator: Natsui Rie 日本女子大学, 理学部, 准教授 (60398633)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: エルゴード理論 / 測度論的数論 / 複素連分数変換 / ユークリッドアルゴリズム / 数論的アルゴリズム / 複素連分数 / Hurwitz連分数

Outline of Final Research Achievements

Toward the goal finding a new invariant which is captured the difference in the notion between determinism and randomness on measurable dynamical systems with infinite invariant measure, this research is focused on the randomness of number generated from number theoretic transformations. In particular, this research obtained the result about the properties of complexity for nearest integer complex continued fraction transformations over imaginary quadratic field and number theoretic algorithms over non-Archimedean field.

Academic Significance and Societal Importance of the Research Achievements

未だに一般的体系が構築されていない無限大不変測度を持つ可測力学系に対するエルゴード理論において、determinismとrandomnessの概念の違いを捉える新たな不変量を見つけるために具体的な数論的変換から生まれる数の持つランダム性に着目し、その複雑性に関する研究成果を得た。数論的変換から生まれる数の複雑性に関する本研究成果は理論計算機科学と密接に結びついており、現代において必要不可欠である計算機の高速性や精度の向上に繋がることを期待する。

Research Products

• [Journal Article] On the construction of the natural extension of the Hurwitz complex continued fraction map. (2019)
  • Author(s): Hiromi Ei, Shunji Ito, Hitoshi Nakada, Rie Natsui
  • Journal Title: Monatshefte fur Mathematik Volume: 188 Issue: 1 Pages: 37-86
  • DOI: 10.1007/s00605-018-1229-0
  • Peer Reviewed
• [Journal Article] On the Construction of Continued Fraction Normal Series in Positive Characteristic (2017)
  • Author(s): Dong Han KIM, Hitoshi NAKADA and Rie NATSUI
  • Journal Title: Tokyo Journal of Mathematics Volume: 39
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Presentation] On the group extension of complex continued fraction maps (2019)
  • Author(s): Rie Natsui
  • Organizer: St Virgil FWF/JSPS Meeting
  • Int'l Joint Research
• [Presentation] On the construction of the natural extensions of the nearest integer complex continued fraction maps (2018)
  • Author(s): Rie Natsui
  • Organizer: The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications
  • Int'l Joint Research / Invited
• [Presentation] On the ergodic properties of the nearest integer continued fraction map over an imaginary quadratic field (2018)
  • Author(s): Rie Natsui
  • Organizer: St Virgil FWF/JSPS Meeting
  • Int'l Joint Research
• [Presentation] On absolutely continuous invariant measures for complex continued fraction maps (2016)
  • Author(s): Hiromi Ei and Rie Natsui
  • Organizer: Substitutions and continued fractions
  • Place of Presentation: LIAFA（パリ）
  • Year and Date: 2016-03-08
  • Int'l Joint Research