Studies on complex dynamics of transcendental entire functions and the value distribution theory
Project/Area Number 
12640182

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Basic analysis

Research Institution  KOCHI UNIVERSITY 
Principal Investigator 
MOROSAWA Shunsuke Faculty of Science, Kochi University, Associate Professor, 理学部, 助教授 (50220108)

CoInvestigator(Kenkyūbuntansha) 
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)

Keywords  complex 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 socalled 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
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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)
Research Products
(27 results)