Analysis of mu-conformal mappings and its application to complex dynamics

2023 Fiscal Year Final Research Report

• Project/Area Number: 20KK0310
• Research Category: Fund for the Promotion of Joint International Research (Fostering Joint International Research (A))
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Hitotsubashi University (2022-2023); Tokyo Institute of Technology (2020)
• Principal Investigator: KAWAHIRA Tomoki 一橋大学, 大学院経済学研究科, 教授 (50377975)
• Project Period (FY): 2022 – 2023
• Keywords: 複素力学系 / 放物的分岐 / 擬等角写像 / Beltrami方程式 / μ-等角写像 / 数値解析

Outline of Final Research Achievements

This research project aims to further develop the research subject, "Research on μ-quasiconformal perturbations toward solving the Goldberg-Milnor conjecture" (Kiban C, 19K03535), through numerical analysis. Specifically, it addresses the Goldberg-Milnor conjecture, which questions whether the chaotic parts of a dynamical system can be essentially preserved when "gently" perturbing a complex dynamical system that has parabolic periodic points, a state where multiple periodic points have degenerated. To tackle this, the project applies a class of homeomorphisms that includes quasiconformal mappings known as "μ-conformal mappings." In particular, the research aims to implement numerical computation methods with error estimates for "μ-conformal mappings" that exhibit significant degeneration and distortion.

Free Research Field

複素力学系理論

Academic Significance and Societal Importance of the Research Achievements

一般に時間発展するシステムを「力学系」とよぶが，力学系を決定するパラメーターは多くの場合振動や摂動にさらされており，ある範囲で絶え間なく揺らぎ続けていると考えるのが自然である．一方で，そのような力学系の振る舞いが将来にわたって予測可能であるためには，力学系全体がパラメーターの変化に対して「安定」している必要がある．本研究では，パラメーターの変化に対して「不安定」なシステムにむしろ着目し，パラメーターの変化の方向を限定することで，システムの変化を最小限に抑えるための研究を数値解析の観点から行った．