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Mathematical Fundation of Fractals

Research Project

Project/Area Number 14340034
Research Category

Grant-in-Aid for Scientific Research (B)

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

Principal Investigator

KIGAMI Jun  Kyoto Univ., Graduate School of Informatics, Professor, 情報学研究科, 教授 (90202035)

Co-Investigator(Kenkyū-buntansha) KUMAGAI Takashi  Kyoto Univ., Research Institute for Mathematical Sciences, Associate Professor, 数理解析研究所, 助教授 (90234509)
SHISHIKURA Mistuhiro  Kyoto Univ., Graduate School of Science, Professor, 理学研究科, 教授 (70192606)
OSADA Hirofumi  Kyushu Univ., Faculty of Mathematics, Professor, 数理学研究院, 教授 (20177207)
HATTORI Tetsuya  Tohoku Univ., Graduate School of Science, Professor, 理学研究科, 教授 (10180902)
ITO Shunji  Kanazawa Univ., Graduate School of Natural Science, Professor, 自然科学研究科, 教授 (30055321)
佐藤 坦  九州大学, 数理学研究院, 教授 (30037254)
Project Period (FY) 2002 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥14,300,000 (Direct Cost: ¥14,300,000)
Fiscal Year 2004: ¥4,200,000 (Direct Cost: ¥4,200,000)
Fiscal Year 2003: ¥4,400,000 (Direct Cost: ¥4,400,000)
Fiscal Year 2002: ¥5,700,000 (Direct Cost: ¥5,700,000)
KeywordsFractal / Dynamical System / Self-similar set / Tiling / Laplacian / heat kernel / 拡散過程 / ジュリア集合
Research Abstract

The purpose of this project is to study fractal from various mathematical viewpoints, for example, analysis, probability, ergode theory, dynamical systems and applied mathematics. We had two conferences in accordance with the purpose of this project. The first one held in the first year of the project. We discussed what was the main issues and how we should approach them. The second one held in in the last year of the project was to get together all the results we obtained in this project. The followings are the selection of results from this project. Kigami has shown that under the volume doubling condition, the upper Li-Yau type estimate of heat kernels is equivalent to the local Nash inequality and the escape time estimate. Kumagai along with Barlow and Bass has shown that the Li-Yau type heat kernel estimate is stable under a perturbation. Ito has studied beta-transform and the associated tiling of the Euclidean space. Kameya has made clear the relation between Julia sets and the self-similar sets. Hino has shown that the energy measure associated with the self-similar Dirichlet form on the Sierpinski gasket is mutually singular with any self-similar measure. Finally Kigami and Kameyama have obtained a relation between the topological property of a self-similar set and the asymptotic behavior of a diffusion process on it.

Report

(4 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (22 results)

All 2004 Other

All Journal Article (11 results) Publications (11 results)

  • [Journal Article] Local Nash inequality and inhomogeneity of heat kernels2004

    • Author(s)
      J.Kigami
    • Journal Title

      Proc.London Math.Soc.(3) 89

      Pages: 525-544

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Synchronised Similar Triangles for Three-Body Orbit with Zero Angular Momentum2004

    • Author(s)
      T.Fujiwara, H.Fukuda, A.Kameyama, H.Ozaki, M.Yamada
    • Journal Title

      J.Phys.A : Math.Gen 37

      Pages: 10571-10584

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Construction of diffusion processes on fractals, d-sets, and general metric spaces

    • Author(s)
      T.Kumagai, K.T.Strum
    • Journal Title

      Journal of Mathematics of Kyoto University (掲載予定)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Atomic surfaces, tilings and coincidence 1, Irreducible case

    • Author(s)
      S.Ito, H.Rao
    • Journal Title

      Israel J.Math. (掲載予定)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On singularity of energy measures on self-similar sets

    • Author(s)
      M.Hino
    • Journal Title

      Probability Theory and Related Fields (掲載予定)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Construction of diffusion processes on fractals, d-sets, and general metric spaces

    • Author(s)
      T.Kumagai, K.T.Strum
    • Journal Title

      Journal of Mathematics of Kyoto University (Publishing schedule)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Atomic surfaces, tilings and coincidence 1, Irreducible case

    • Author(s)
      S.Ito, H.Rao
    • Journal Title

      Israel J.Math. (Publishing schedule)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On singularity of energy measures on self-similar sets

    • Author(s)
      M.Hino
    • Journal Title

      Probability Theory and Related Fields (Publishing schedule)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Construction of diffusion process on fractals, d-sets, and general metric spaces

    • Author(s)
      T.Kumagai, K.T.Strum
    • Journal Title

      Journal of Mathematics of Kyoto University (掲載予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Atomic surfaces, tilings and coincidence 1, Irreducible case

    • Author(s)
      S.Ito, H.Rao
    • Journal Title

      Israel J.Math. (掲載予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On singularity of energy measures on self-similar sets

    • Author(s)
      M.Hino
    • Journal Title

      Probability Theory and Related Fields (掲載予定)

    • Related Report
      2004 Annual Research Report
  • [Publications] Hambly, B.M., Kumagai, T.: "Diffusion processes on fractal fields : heat kernel estimates and large deviations"Probab. Theory Related Fields. 127. 305-352 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Ito, Shunji, Fujii, Junko, Higashino, Hiroko, Yasutomi, Shin-ichi: J. Number Theory. 99. 255-283 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Hino, Masanori, Ramrez, Jos A.: "Small-time Gaussian behavior of symmetric diffusion semigroups"Ann.Probab.. 31. 1254-1295 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Kigami, Jun: "Harmonic analysis for resistance forms"J.Funct.Anal.. 204-NO.2. 399-444 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Kameyama, Atsushi: "On Julia sets of postcritically finite branched coverings. I. Coding of Julia sets"J.Math.Soc.Japan. 55. 439-454 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 服部 哲弥: "ランダム・ウォークと繰り込み群"共立出版(出版予定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Matsumoto, S., Shishikura, M.: "Minimal sets of certain annular homeomorphisms"Hiroshima Math.J.. 32-2. 207-215 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Bass, R.F., Kumagai, T.: "Laws of the iterated logarithm for the range of random walks in two and three dimensions"Ann.Probab.. 30-3. 1369-1396 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Hambly, B.M., Kigami, I., Kumagai, T.: "Multifractal formalisms for thelocal spectral and walk dimensions"Math.Proc.Cambridge Philos.Soc.. 132-3. 555-571 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Denker, M., Sato, H.: "Reflections on harmonic analysis of the Sierpinski gasket"Math.Nachr.. 241. 32-55 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Osada, H.: "Harnack inequalities for exotic Brownian motions"Kyushu J.Math.. 56-2. 363-380 (2002)

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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