Diversity of the dynamics of polynomials and transcendental entire functions

2023 Fiscal Year Final Research Report

• Project/Area Number: 17K05296
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kyoto University
• Principal Investigator: KISAKA Masashi 京都大学, 人間・環境学研究科, 教授 (70244671)
• Project Period (FY): 2017-04-01 – 2024-03-31
• Keywords: 超越整関数 / Mandelbrot集合 / Julia集合 / Fatou集合 / 擬等角写像 / 中立サイクル / 多項式類似写像 / 構造有限超越整関数

Outline of Final Research Achievements

We investigated the dynamics of polynomials and transcendental entire functions by mainly using complex analytic methods. As a remarkable result, we proved that the following phenomena which can be observed in the Mandelbrot set by computer graphics actually exist by formulating them mathematically and proving the statements: Take a small Mandelbrot set in the Mandelbrot set, and choose a parameter from it which corresponds to a quadratic polynomial with either a parabolic periodic point or whose critical point 0 is preperiodic. By zooming in its neighborhood, we can see a quasiconformal image of a Cantor Julia set which is a perturbation of a parabolic or Misiurewicz Julia set. Furthermore, by zooming in its middle part, we can see a certain nested structure ("decoration") and finally another "smaller Mandelbrot set" appears.

Free Research Field

力学系

Academic Significance and Societal Importance of the Research Achievements

概要で述べた成果は「相空間上の対象物であるJulia集合がパラメータ空間であるMandelbrot集合内のあちこちに現れる」という驚くべき現象を数学的に解明するもので，そのインパクトと意義は大きい．また「Mandelbrot集合の境界のHausdorff次元は2である」という宍倉による有名な結果に対して「Mandelbrot集合の境界にはCantor型のJulia集合の擬等角写像による像で，Hausdorff次元が2にいくらでも近いものがあるから」という，非常にわかりやすい説明ができるようになったのは意義深い．これは社会に向かって力学系理論の面白さをアピールするための非常に強力な武器となる．