Research on the structures of parameter spaces by bifurcation in complex dynamics and its visualization

2023 Fiscal Year Final Research Report

• Project/Area Number: 18K03367
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Kyoto University
• Principal Investigator: Inou Hiroyuki 京都大学, 理学研究科, 准教授 (00362434)
• Project Period (FY): 2018-04-01 – 2024-03-31
• Keywords: 複素力学系 / 分岐 / くりこみ / ヴァーチャル・リアリティ

Outline of Final Research Achievements

We studied renormalization, which is closely related to various properties of the Mandelbrot set, including self-similarity, and constructed new examples of renormalizable polynomials and rational functions. Using them, we analyzed the complicated phenomena of bifurcation loci in higher-dimensional parameter spaces which do not appear in one dimension.
We also implemented an interactive visualization of point clouds in two complex dimensions (four real dimensions) with virtual reality devices, and by using them, we discovered a "hole" in the support of the bifurcation measure, and numerically verified its existence.

Free Research Field

複素力学系

Academic Significance and Societal Importance of the Research Achievements

Mandelbrot集合はその複雑な構造から多くの人を魅了している．この(境界の)一般化である複素力学系の族の分岐軌道は，更に複雑な性質を持つ．Mandelbrot集合は局所連結であると予想されているが，複素2次元(実4次元)以上の分岐軌道は局所連結とは限らない．本研究ではそのような更に複雑な集合を可視化し観察するための手法を構築し，また到達可能性や組み合わせ剛性といった局所連結性と関連した重要な性質に関して，高次元で初めて現れる複雑な現象について新たな知見を得ることに成功した．