Research on dynamics of transcendental entire functions and polynomial semigroups based on dynamics of polynomials

2022 Fiscal Year Final Research Report

• Project/Area Number: 17K14212
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Tohoku Gakuin University (2022); Ichinoseki National College of Technology (2017-2021)
• Principal Investigator: Katagata Koh 東北学院大学, 教養学部, 准教授 (10529598)
• Project Period (FY): 2017-04-01 – 2023-03-31
• Keywords: 複素力学系 / 超越整関数 / 擬多項式写像 / ジュリア集合

Outline of Final Research Achievements

The results of this research are as follows. (1) We constructed a transcendental entire function which has order zero containing dynamics of n quadratic polynomials locally, that is, containing n quadratic-like maps. (2) We constructed a transcendental entire function which has order zero containing dynamics of infinitely many quadratic polynomials locally, that is, containing infinitely many quadratic-like maps. (3) We constructed a transcendental entire function of arbitrarily slow growth containing dynamics of infinitely many polynomials of degree at least two locally, that is, containing infinitely many polynomial-like maps.

Free Research Field

複素力学系

Academic Significance and Societal Importance of the Research Achievements

本研究では多項式の力学系を出発点として超越整関数の力学系特有の現象，特にファトウ集合やジュリア集合などの不変集合の位相的性質の解明と，擬多項式写像をキーワードにした力学系的性質の一般論の展開を目指した．本研究で得られた超越整関数はその構成方法から有限個または無限個の多項式の力学系を部分力学系として持っており，そのジュリア集合は多項式のジュリア集合のコピーを部分集合に持っている．上記(3)の超越整関数については無限個の多項式を扱っていることから，これまでにない新たな具体例が構成できたと言える．