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Studies on complex dynamics of transcendental entire functions and the value distribution theory

Research Project

Project/Area Number 12640182
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKOCHI UNIVERSITY

Principal Investigator

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

Co-Investigator(Kenkyū-buntansha) OHTSUBO Yoshio  Faculty of Science, Kochi University, Professor, 理学部, 教授 (20136360)
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  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)
TANIGUCHI Masahiko  Kyoto University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (50108974)
Project Period (FY) 2000 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2001: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2000: ¥1,700,000 (Direct Cost: ¥1,700,000)
Keywordscomplex dynamics / Fatou set / Julia set / transcendental entire function / value distribution theory / semihyperbolicity / wandering domain / singular value / 超越整関数 / 超越整関数の位数 / ベーカー領域
Research Abstract

The summary of research results is as follows.
1. We consider semihyperbolic transcendental entire functions with professor Walter Bergweiler. Building on work by Mane, Carleson, Jones, and Yoccoz introduced the concept of semihyperbolicity for polynomials. It is a generalization of the concept today so-called hyperbolicity, which was introduced by Fatou. It has a close relationship with behavior of singular orbits. In the case of polynomials or rational functions, only critical values are singular values, whose sets are always finite. On the other hand, in the case of transcendental entire functions, asymptotic values are also singular values and sets of singular values may be infinite. First, we show a characterization of semihyperbolic transcendental entire functions are different from that of polynomials by constructing an example. Furthermore, transcendental entire functions may have wandering domains. We show that semihyperbolic transcendental entire functions never have wandering … More domains where the limit functions of iterate are finite. Moreover, we give a sufficient condition that the Julia sets of semihyperbolic transcendental entire functions are locally connected.
2. We consider the family of transcendental entire functions {f(z ; a,b,c) = abz + e^<bz> + c}. Functions of this family are of infinite singular type. Hence, they may have Baker domains or wandering domains. We give a range of real parameters where the Fatou sets are coincide with a Baker domains. Furthermore, we show that the area of the Julia set are equal to zero in that case. We also give an integral representation of the functions of the family.
3. In dynamics of polynomials, there never exist Baker domains nor wandering domains. To the contrary, in that of transcendental entire functions, there may exists those. On the other hand, any transcendental entire function can be approximated by some sequences of polynomials in the sense of locally uniformly convergence. We consider the Caratheodory convergence of Fatou sets and the Hausdorff convergence of Julia sets of such sequences of polynomials to transcendental entire functions which have Baker domains or wandering domains.
4. We consider subhyperbolic rational functions. In particular, we consider the boundaries of Fatou components of such rational functions which are simply connected. We construct an example of a subhyperbolic rational function whose Julia set has a topological property called Sierpinski carpet. Less

Report

(3 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • Research Products

    (27 results)

All Other

All Publications (27 results)

  • [Publications] S.Morosawa, M.Taniguchi: "Non-trivial deformation of an entire function abz+e^<bz>+c"Taiwanese Journal of Mathematics. 5. 641-646 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Niizeki: "On the convergence of sequence of sets and the continuity of measure"Mem. Fac. Sci. Kochi Univ.(Math.). 23. 9-17 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Ohtsubo, K.Toyonaga: "Optimal policy for minimizing risk models in Markov decision processes"J. Mathematical Analysis and Applications. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Taniguchi: "Explicit representation of structurally finite entire functions"Proc. Japan Acad. Ser. A. 77. 68-70 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J.Heittokangas, R.Korhonen, I.Laine, J.Rieppo, K.Tohge: "Complex difference equations of Malmquist type"Computational Methods and Function Theory. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K.Ishizaki, I.Laine, S.Shimomura, K.Tohge: "Riccati differential equations with elliptic coefficients, II"Tohoku Math. J.. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S. Morosawa: "Julia sets of subhyperbolic rational functions"Complex Variables. Vol.41. 151-162 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S. Morosawa: "The Caratheodory convergence of Fatou components of Polynomials to Baker domains or wandering domains"Proceedings of the Second Congress ISAAC, Kluwer Acad, Pub.. Vol.1. 347-356 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S. Morosawa and M. Taniguchi: "Non-trivial deformation of an entire function abz + e^<bz> + c"Taiwanese Journal of Mathematics. Vol.5. 641-646 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K. Kato: "On attractors of hyperbolic iterated function systems"Mem. Fac. Sci. Kochi Univ. (Math.). Vol.22. 29-43 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] E. Fujikawa, H. Shiga and M. Taniguchi: "Discreteness of the mapping class group for Riemann surfaces of infinite analytic type"Proceedings of the Second Congress ISAAC, Kluwer Acad. Pub.. Vol.2. 1015-1022 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M. Taniguchi: "Explicit representation of structurally finite entire functions"Proc. Japan Acad. Ser. A. Vol.77. 68-70 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K. Ishizaki, I. Laine, S. Shimomura and K. Tohge: "Riccati differential equations with elliptic coefficients"Result. Math.. Vol.38. 58-71 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo and K. Tohge: "Complex difference equations of Malmquist type"Computational Methods and Function Theory. (in printing).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K. Ishizaki, I. Laine, S. Shimomura and K. Tohge: "Riccati differential equations with elliptic coeffcients, II"Tohoku Math. J.. (in printing).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Morosawa, M.Taniguchi: "Non-trivial deformation of an entire function abz+e^<bz>+c"Taiwanese Journal of Mathematics. 5. 641-646 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Niizeki: "On the convergence of sequence of sets and the continuity of measure"Mem. Fac. Sci. Kochi Univ.(Math.). 23. 9-17 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Ohtsubo, K.Toyonaga: "Optimal policy for minimizing risk models in Markov decision processes"J. Mathematical Analysis and Applications. (印刷中).

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Taniguchi: "Explicit representation of structurally finite entire functions"Proc. Japan Acad. Ser. A. 77. 68-70 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] J.Heittokangas, R.Korhonen, I.Laine J.Rieppo, K.Tohge: "Complex difference equations of Malmquist type"Computational Methods and Function Theory. (印刷中).

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Ishizaki, I.Laine S.Shimomura, K.Tohge: "Riccati differential equations with elliptic coefficients, II"Tohoku Math. J.. (印刷中).

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Morosawa: "Julia sets of subhyperbolic rational functions"Complex Variables. 41. 151-162 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Morosawa: "The Caratheodory convergence of Fatou components of polynomials to Baker domains or wandering domains"Proc.of the 2nd ISAAC Congress. 1. 347-356 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Niizeki: "On a Generalization of Egorof's theorem"Mem.Fac.Sci.Kochi Univ. (Math.). 22. 103-107 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Kato: "On attractors of hyperbolic iterated function systems"Mem.Fac.Sci.Kochi Univ. (Math.). 22. 29-43 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] E.Fujikawa,H.Shiga & M.Taniguchi: "Discreteness of the mapping class group for Riemann surfaces of infinite analytic type"Proc.of the 2nd ISAAC Congress. 2. 1015-1023 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Ishizaki,I.Laine,S.Shimomura & K.Tohge: "Riccati differential equations with elliptic coefficients"Result.Math.. 38. 58-71 (2000)

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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