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Studies on complex dynamics of transcendental entire functions

Research Project

Project/Area Number 09640199
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field 解析学
Research InstitutionKOCHI UNIVERSITY

Principal Investigator

MOROSAWA Shunsuke  Faculty of Science, Associate Professor, 理学部, 助教授 (50220108)

Co-Investigator(Kenkyū-buntansha) OGOMA Tetsusi  Faculty of Science, KOCHI UNIVERSITY Professor, 理学部, 教授 (20127921)
KATO Kazuhisa  Faculty of Science, KOCHI UNIVERSITY Professor, 理学部, 教授 (20036578)
NIIZEKI Shozo  Faculty of Science, KOCHI UNIVERSITY Professor, 理学部, 教授 (60036572)
THOGE Kazuya  Kazuya Kanazawa University, Graduate School of Natural Science and Technology, Associate Professor, 大学院自然科学研究科, 助教授 (30260558)
OHTSUBO Kasuya  Faculty of Science, KOCHI UNIVERSITY Professor, 理学部, 助教授 (20136360)
Project Period (FY) 1997 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥2,600,000 (Direct Cost: ¥2,600,000)
Fiscal Year 1999: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 1998: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 1997: ¥1,500,000 (Direct Cost: ¥1,500,000)
Keywordscomplex dynamics / Fatou set / Julia set / transcendental entire function / value distribution theory / order / wandering domain / Baker domain / 分岐理論 / 表現空間 / タイヒミュラー空間
Research Abstract

The summary of research results is as follows.
1. First, we consider the family of transcendental entire function {fィイD2λィエD2(z) =λz exp(z)}, which is a one complex parameter family. In the case that λ is a real parameter, we study its dynamical property by using the theory of unimodal maps. On this way we proved that there exists many λ for which the dynamics of fィイD2λィエD2 is chaotic.
2. In the case that λ is a complex parameter, we considered the parameter space. In particular, we decided a domain of the parameter space where the Fatou set of the dynamics corresponding to a point consists of bounded components. The method of proof is to study a property of certain curves in the Julia set. By the argument similar to this, we constructed a transcendental entire function whose Julia set is a Sierpinski carpet.
3 . One of the difference between the dynamics of transcendental entire functions and the dynamics of rational functions is that Fatou sets of the dynamics of transcendental entire functions may have Baker domains and/or wandering domains. We constructed a transcendental entire function whose Fatou set contains cyclic Baker domains whose period is greater than one.
4. A transcendental entire function is always considered as a limit function of locally uniform convergence of a sequence of polynomials. We considered how the convergence of sequences of Fatou sets and Julia set. In particular, we considered it in the case that the Fatou set of a transcendental entire function has Baker domains or wandering domains. In general, locally uniform convergence of a sequence of polynomial does not imply convergence of a sequence of Fatou sets or Julia sets. Under some conditions, we showed a sequence of components of Fatou sets converges to a Baker domain. We also showed that if a sequence of components of Fatou sets satisfies certain conditions then the Fatou sets of the limit function contains a wandering domain.

Report

(4 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • 1997 Annual Research Report
  • Research Products

    (35 results)

All Other

All Publications (35 results)

  • [Publications] S.Morosawa: "Note on the iteration of f_μ(z)=z exp(z+μ)"Sci.Bul.Josai.Univ.,Special Issue. 4. 11-16 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S.Morosawa: "An example of cyclic Baker domains"Men.Fac.Sci.Kochi Univ.(Math.). 20. 123-126 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S.Morosawa: "Local connectedness of Julia sets for transcendental entire functions"Proc.Inter.Conf.on Nonlinear Analysis and Convex Analysis. 266-273 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S.Morosawa: "The Coretheodory convergence of fatou components of Polynomials to Baker domains or wandering domains"Proc.the Second Congress ISAAC. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K.Kato: "Hausdorff dimensiona of generalized cookie-cutter Cantor sets"Far East J.Appl.Math.. 2. 191-202 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K.Tohge: "Logarithmic derivatives of meromorphic or algedroid solutions of some homogeneous linear differential equations"Analysis. 19. 273-297 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S.Morosawa: "Holomorphic Dynamics"Cambridge University Press. 338 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Morosawa: "Note on the iteration of fィイD2μィエD2(Ζ) = Ζ exp(Ζ + μ)"Sci. Bul. Josai. Univ., Special Issue. Vol. 4. 11-14 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Morosawa: "Bounded Fatou components of transcendental entire functions"Res. Rep. the Nevanlinna Theory and its Applications II. 65-69 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Morosawa: "Local connectedness of Julia sets for transcendental entire functions"Proc. Inter. Conf. on Nonlinear Analysis and Convex Analysis. 266-273 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Morosawa: "The Caratheodory convergence of Fatou components of Polynomials to Baker domains or wandering domains"Proc. the Second Congress ISAAC. in printing.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Kato: "Hausdorff dimensions of generalized cookie cutter Cantor sets"Far East J. Appl. Math.. Vol. 2. 191-202 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K . Tohge: "Logarithmic derivatives of meromorphic or algebroid solutions of some homogeneous linear differential equations"Analysis. Vol. 19. 273-297 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Tohge: "Nevaldinna theory and linear differential equations"Res. Rep. the Nevanlinna Theory and its Applications II. 1999. 87-94

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Tohge: "On meromorphic solutions of linear differential equations with at least one transcendental coefficient"Proc. the Second Congress ISAAC. in printing.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Morosawa, Y. Nishimura M. Taniguchi and T. Ueda: "Holomorphic Dynamics"Cambridge University Press. 388 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S.Morosawa: "Bounded Fatou Components of transcendental entire functions"Res.Rep.the Nevanlinna Theory and its Applications II. 65-69 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] S.Morosawa: "Local connectedness of Julia sets for transcendental entire functions"Proc.Inter.Conf.on Nonlinear Analysis and Convex Analysis. 266-273 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] S.Morosawa: "The Caratheodory covergence of Fatou components of Polynomials to Baker domains or wandering domains"Proc.the Second Congress ISAAC. (印刷中).

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Tohge: "Logarithemic derivatives of meromorphic or algebroid solutions of some himogeneous linear differential equations"Analysis. 19. 273-297 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Tohge: "Nevanlinna theory and linear differential equations"Res.Rep.the Nevanlinna Theory and its Applications II. 87-94 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Tohge: "On meromorphic solutions of linear differential equations with at least one transcendental coefficient"Proc.the Second Congress ISAAC. (印刷中).

    • Related Report
      1999 Annual Research Report
  • [Publications] S.Morosawa: "Holomorphic Dynamics"Cambridge University Press. 338 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] 諸澤 俊介: "An example of cyclic Baker domains" Mem.Fac.Sci.Kochi Univ.(Math.). 20. 123-126 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 諸澤 俊介: "Local connectedness of Julia sets for transcendental entire functions" Proc.Int.Conf.Nonlinear Analysis and Convex Analysis. 印刷中. 印刷中 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 新関 章三: "On a convergence theorem in L_p-space and some sci.of its application" Mem.Fac.Kochi Univ.(Math.). 19. 45-55 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 加藤 和久: "Hausdorff dimensions of generaalized cookie-cutter Cantor sets" Far East J.Appl.Math.2. 191-202 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 小駒 哲司: "On a Problem of Nagata related to Zariski's problem" Osaka J.Math.35. 487-491 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 大坪 義夫: "Optimality in multi-person stopping problem with general rewards" Proc.Int.Conf.on Nonlinear Analysis and Convex Analysis. 印刷中. 印刷中 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 諸澤 俊介: "Note on the iteration of fu(z)=zexp(z+μ)" Sci.Bull.Josai Univ.Special Issoe. (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] 新関 章三: "On a convergence theorem in Lp-space and some of its application" Mem.Fac.Sci.Kochi Univ.(Math.). 19巻. 45-55 (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] 小駒 哲司: "On a problem of Nagata related to Zariski's problem" Osaka J.Math.35巻. (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] 藤解 和也: "On the complex oscillation of some linear differential equation" J.Math.Aral.Appl.206巻. 503-517 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 藤解 和也: "Hypertrancendency of meromorphic solotions of a fonctional equation" Rep.Researches Nippon Inst.Tech.27巻. 479-484 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 長沼 英久: "On the mathematical structure of D_<2n>-laffices" Research Rep.Kochi Univ. 46巻. 1-8 (1997)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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