Theory of implosion in complex dynamics in dimensions one and higher and its applications

2022 Fiscal Year Final Research Report

• Project/Area Number: 16K05213
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Tokyo Polytechnic University
• Principal Investigator: Nakane Shizuo 東京工芸大学, 工学部, 名誉教授 (50172359)
• Project Period (FY): 2016-04-01 – 2023-03-31
• Keywords: parabolic implosion / skew product / サドル不動点 / fiber Julia 集合 / fiber Julia-Lavaurs 集合

Outline of Final Research Achievements

The behavior of the orbits obtained by iteration of maps can change drastically as the mappings are perturbed. In particular, discontinuous phenomenon called parabolic implosion appears when a fixed point of mappings bifurcates into two fixed points. An analogous phenomenon occurs when a skew product map in dimension two has an orbit connecting two saddle fixed points.
By linearizing the maps at saddle fixed points, we can control the orbits in a neighborhoods of the saddle fixed points. Then, by showing the local uniform convergence of the sequence of high iterates of the maps, we have clarifed the behavior of fiber Julia sets. We have also shown that fiber Julia sets converge to the fiber Julia-Lavaurs set.

Free Research Field

複素力学系

Academic Significance and Societal Importance of the Research Achievements

① 2つのサドル不動点をつなぐ軌道（ヘテロクリニック軌道）が存在するときに、fiber Julia 集合が不連続に振る舞うという、parabolic implosion と類似の現象を数値実験で見出した。
② 軌道の振る舞いから類推して、写像の反覆列が Lavaurs 写像に収束するという parabolic implosion と同様の構造が存在することを示し、fiber Julia 集合の不連続性を証明した。
③ 力学系の分岐現象は物理学における相転移現象に対応するものであるので、本研究の物理学等への寄与が期待される。