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A study on probabilistic methods ingeometric mecemine theory

Research Project

Project/Area Number 10640121
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionYamanashi University

Principal Investigator

NAKAMURA Munehika  Yamanashi University Faculty of Education and Human Sciences, Associate professor, 教育人間科学部, 助教授 (10227944)

Co-Investigator(Kenkyū-buntansha) MUTO Hideo  Yamanashi University, Associate professor, 教育人間科学部, 助教授 (20143646)
NAKAI Yoshiobu  Yamanashi University, professor, 教育人間科学部, 教授 (40022652)
SUZUKI Toshio  Yamanashi University, professor, 教育人間科学部, 教授 (20020472)
KUBO Izumu  Graduate school of Science, Hiroshima University, Professor, 大学院・理学研究科, 教授 (70022621)
Project Period (FY) 1998 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2000: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 1999: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 1998: ¥1,100,000 (Direct Cost: ¥1,100,000)
KeywordsHausdorff dimension / packing dimension / transfer operators / Cantor sets / ランダムな自己相似集合 / 歪んだカントール集合 / 熱力学形式 / ギブス測度 / 統計的自己相似集合 / スペクトル / マルチフラクタル / チータ・ワイル和 / 零点
Research Abstract

In this research, with the aim to establish the foundation of the theory of dimensions and measures of fractals such as Cantor sets, we tried to apply the probabilistic methods to their analysis. Concretely obtained results are the following.
Firstly, on statistically self-similar Cantor sets, we developed mathematically rigorous theory of the Hausdorff dimension, packing dimension and the measures associated with these dimensions. Here we utilized the statistical mechanical notions of the thermodynamic formalism such as transfer operators or Gibbs measures, and characterized the dimensions through them. Namely, on statistically perturbed Cantor sets, we showed that Hausdorff dimensions and packing dimensions are both equal to the zeroes of the pressures defined by the logarithmic potential of the derivatives of generating maps on symbolic dynamics.
Secondly, in deterministic cases we analyzed the perturbed Cantor sets. We constructed examples of Cantor sets which Hausdorff dimensions do not coincide with packing ones. We also showed that the dimensions of measures are equal to the entropy of the corresponding symbolic dynamics divided by the Lyapunov exponents with respect to the considered measures. Further-more we treated self-similar sets such that the derivatives of generating maps are non-Holder continuous, and showed that a dimension formula of the pressure holds.
Associated with the above, we studied the spectra of the transfer operators, or as their extension, the quasi-transfer operators, which play a crucial role in the analysis of self-similar sets. Here we treated the operators whose potentials do not satisfy the condition of the Holder continuity. Defining the pressure of the operators appropriately, we proved that the spectral sets of the opearators are equal to the disks in the complex plane of which radii are equal to their pressures.

Report

(4 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • 1998 Annual Research Report
  • Research Products

    (29 results)

All Other

All Publications (29 results)

  • [Publications] T.Suguhi,E.Watanabe: "Numerical computation of the Jordan cannonical form"IMACS series in compritational and applied mathematics. 4. 65-70 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 中井喜信: "3次のテータ・ワイル和(I)"山梨大学教育人間科学部研究報告. 49. 1-4 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Nakamura: "A multifractal formalism on asfic structured Cantor sets"山梨大学教育人間科学部紀要. 1. No.1 1-7 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Muto,T.Suzuki,T.Suzuki: "A new method to compute genes of polynomials"数理解析研究所講究録. 1091. 36-44 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 中井喜信: "3次のテータ・ワイル和"数理解析研究所講究録. 1091. 298-307 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 鈴木智博,鈴木俊夫,武藤秀夫: "数値積分誤差を用いた新しい多項式の零点の解法"日本応用数理学会論文誌. 9. 65-76 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 鈴木俊夫: "多項式の零点の新しい数値計算法の收束証明"山梨大学教育人間科学部紀要. 1. No.2 1-5 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 中村宗敬: "縮少写像原理とその応用について"山梨大学教育人間科学部紀要. 2. 1-5 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Muto: "An associated surface of Dlaunay Surface in H^3"山梨大学教育人間科学部紀要. 2. 5-11 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Ikeda,M.Nakamura: "On dimension of measures on perturbed Cantor sets"Topology and its application. (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Suzuki and E.Watanabe: "Numerical computation ot the Jordan cannonical form"IMACS series in computational and applied mathematics. Vol.4. 65-70 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Y.Nakai: "A candidate for cubic Thera-Weyl sums (I)"Memoirs of the Faculty of the Education and Human Sciences, Yamanashi University. Vol.49, No.2. 1-4 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Nakamura: "A mmultifractal formalism on sofic structured Cantor sets"Bulletin of the Faculty of the Education and Human Sciences, Yamanashi University. Vol.1, No.1. 1-7 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Hideo Muto, Tomohiro Suzuki, Tosdhio Suzuki: "A new method to compute zeros of polynomials"RIMS Kokyuroku. Vol.1091. 36-44 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Y.Nakai: "On cubic Theta-Weyl sums"RIMS Kokyuroku. Vol.1091. 298-307 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Tomohiro Suzuki, Toshio Suzuki and H.Muto: "A new method to compute the zeros of polynomilals."Nihon Ouyou Suurigakkai Ronbunshi. Vol.9. 65-76 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Suzuki: "A proof of the convergence of a new method for computing zeros of polynomials"Bulletin of the Faculty of the Education and Human Sciences, Yamanashi University. Vol.1, No.2. 1-5 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Nakamura: "On a contraction mapping theorem and its applications to transfer operators"Bulletin of the Faculty of the Education and Human Sciences, Yamanashi University. Vol.2, No.1. 1-5 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Hideo Muto: "An associated surface of Dlaunay surface in H^3"Bulletin of the Faculty of the Education and Human Sciences, Yamanashi University. Vol.2, No.1. 5-11 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Ikeda and M.Nakamura: "On dimension of measures on perturbed Cantor sets""Topology and its application".. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Nakamura: "On a contraction mapping theorem and its applications"山梨大学教育人間科学部紀要. 2・1. 1-4 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Muto: "An associated surface of Delaunay surface in IH^3"山梨大学教育人間科学部紀要. 2・1. 5-11 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Suzuki: "A proof of the convergence of a new method for computing zeros of polynomials"山梨大学教育人間科学部紀要. 1・2. 1-5 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Ikeda and M.Nakamusra: "On dimension of the measures on perturbed Cantor sets"Topology and its Application. (to appsear). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Munetaka NAKAMURA: "A multifractal formalism on sofic structured Cantor sets"山梨大学教育人間科学部紀要. 1・1. 1-7 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Hideo Muto,Tomohiro Suzuki, Toshio Suzuki: "A new method to compute zeros of polynomials"数理解析研究所講究録. 1091. 36-44 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 中井喜信: "3次のテータ・ワイル和"数理解析研究所講究録. 1091. 298-307 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Suzuki, E.Watanabe: "Numerical computation of the Jordan caronical form" IMACS series in computational and applied mathamatics, vol.4, Iterated methods in Scientific Computation. 4. 65-70 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 中井喜信: "3次のテータ・ワイル和(I)" Memoris of the faculty of Education & Human Sciences. 49. 1-4 (1998)

    • Related Report
      1998 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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