Ergodic theory of the measurable dynamical systems with infinite invariant measure and its applications to metric number theory

• Project/Area Number: 19549005
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Global analysis
• Research Institution: Japan Women's University
• Principal Investigator: NATSUI Rie Japan Women's University, 理学部, 助教 (60398633)
• Project Period (FY): 2007 – 2009
• Project Status: Completed (Fiscal Year 2009)
• Budget Amount: ¥3,820,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥720,000); Fiscal Year 2009: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2008: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2007: ¥700,000 (Direct Cost: ¥700,000)
• Keywords: 力学系理論 / エルゴード理論 / ergodic theory / metric number theory / continued fractions / Diophantine approximation / 可測力学系 / Diophantine近似 / 連分数

Research Abstract

In the measurable dynamical system with the infinite invariant measure, this research aims to characteristic of randomness, and tried the new interpretation from the point of view of the ergodic theory. In particular, this research could get the results for the complexity of numbers for number theoretic transformations, called α-continued fraction transformations and number theoretic algorithm, as the concrete research model.

Research Products

• [Journal Article] Generalized Brjuno functions associated toα-continued fractions (2010)
  • Author(s): Laura Luzzi, Stefano Marmi, Hitoshi Nakada, Rie Natsui
  • Journal Title: Journal of Approximation Theory 162号 Pages: 24-41
  • Peer Reviewed
• [Journal Article] Generalized Brjuno functions associated to α-continued fractions (2010)
  • Author(s): L.Luzzi, S.Marmi, H.Nakada, R.Natsui
  • Journal Title: Journal of Approximation Theory 162 Pages: 24-41
  • Peer Reviewed
• [Journal Article] Asymptotic behavior of the number of solutions for non-Archimedean Diophantine approximations with restricted denominators (2008)
  • Author(s): Valerie Berthe, Hitoshi Nakada, Rie Natsui
  • Journal Title: Finite Fields and Their Applications 14号 Pages: 849-866
  • Peer Reviewed
• [Journal Article] The non-monotonicity of the entropy ofα-continued fraction transformations (2008)
  • Author(s): Hitoshi Nakada, Rie Natsui
  • Journal Title: Nonlinearity 21号 Pages: 1207-1225
  • Peer Reviewed
• [Journal Article] The non-monotonicity of the entropy of α-continued fraction transformations (2008)
  • Author(s): Hitoshi Nakada, Rie Natsui
  • Journal Title: Nonlinearity 21 Pages: 1207-1225
  • Peer Reviewed
• [Journal Article] Asymptotic behavior of the number of solutions for non-Archimedean Diophantine approximations with restricted denominators (2008)
  • Author(s): Valerie Berthe, Hitoshi Nakada, Rie Natsui
  • Journal Title: Finite Fields and Their Applications 14 Pages: 849-866
  • Peer Reviewed
• [Presentation] On the Euclidean algorithm and Farey map for formal power series (2009)
  • Author(s): Rie Natsui
  • Organizer: Recent Topics in Ergodic Theory
  • Place of Presentation: 日本大学(東京)
  • Year and Date: 2009-11-06
• [Presentation] On the Euclidean algorthm and farey map for formal power series (2009)
  • Author(s): Rie Natsui
  • Organizer: Recent Topics in Ergodic Theory
  • Place of Presentation: 日本大学(東京)
  • Year and Date: 2009-11-06
• [Presentation] Euclidean algorithm in positive characteristic and associated Farey map (2009)
  • Author(s): 夏井利恵
  • Organizer: Dynamics of complex systems 2009
  • Place of Presentation: 北海道大学(北海道)
  • Year and Date: 2009-08-31
• [Presentation] Non-monotonicity of entropy of alpha-continued fraction transformations (2009)
  • Author(s): Rie Natsui
  • Organizer: Measurable and Topological Dynamical Systems in Asia
  • Place of Presentation: Ajou University (Korea, Suwon)
  • Year and Date: 2009-06-18
• [Presentation] Non-monotonicity of entropy of alpha-continued fraction transformations (2009)
  • Author(s): Rie Natsui
  • Organizer: Measurable and Topological Dynamical Systems in Asia
  • Place of Presentation: Ajou University(Korea、Suwon)
  • Year and Date: 2009-06-18
• [Presentation] Asymptotic behavior of the number of solutions for non-Archimedean Diophantine approximations with restricted denominators (2008)
  • Author(s): 夏井利恵
  • Organizer: エルゴード理論とその周辺
  • Place of Presentation: 日本大学(東京)
  • Year and Date: 2008-11-29
• [Presentation] Asymptotic behavior of the number of solutions for non-Archimedean Diophantine approximations with restricted denominators (2008)
  • Author(s): 夏井利恵
  • Organizer: エルゴート理論とその周辺
  • Place of Presentation: 日本大学文理学部
  • Year and Date: 2008-11-29
• [Presentation] α-Rosen continued fractionsについて (2008)
  • Author(s): 夏井利恵
  • Organizer: 数論とエルゴード理論
  • Place of Presentation: 金沢大学(石川)
  • Year and Date: 2008-02-10
• [Presentation] α-Rosen continued fractionsについて (2008)
  • Author(s): 夏井利恵
  • Organizer: 数論とエルゴード理論
  • Place of Presentation: 金沢大学
  • Year and Date: 2008-02-10
• [Presentation] On the entropy of alpha-continued fraction transformations (2007)
  • Author(s): Rie Natsui
  • Organizer: Ergodic Theory and its applications
  • Place of Presentation: 慶應義塾大学(神奈川)
  • Year and Date: 2007-12-21
• [Presentation] On the entropy of alpha-continued fraction transformations (2007)
  • Author(s): 夏井利恵
  • Organizer: エルゴード理論とその応用
  • Place of Presentation: 慶磨義塾大学
  • Year and Date: 2007-12-21
• [Presentation] α-連分数変換から作られる確率過程の1-parameter familyのentropyの変化について (2007)
  • Author(s): 夏井利恵, 仲田均
  • Organizer: 2007日本数学会秋季総合分科会統計数学分科会
  • Place of Presentation: 東北大学(宮城)
  • Year and Date: 2007-09-21
• [Presentation] α-連分数変換から作られる確率過程の1-parameter familyのentropyの変化について (2007)
  • Author(s): 夏井利恵
  • Organizer: 2007日本数学会秋季総合分科会
  • Place of Presentation: 東北大学
  • Year and Date: 2007-09-12