Theory of implosion in complex dynamics in dimensions one and higher and its applications
| Project/Area Number |
16K05213
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Tokyo Polytechnic University |
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Principal Investigator |
Nakane Shizuo 東京工芸大学, 工学部, 名誉教授 (50172359)
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| Project Period (FY) |
2016-04-01 – 2023-03-31
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| Project Status |
Completed (Fiscal Year 2022)
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| Budget Amount *help |
¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
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| Keywords | parabolic implosion / skew product / サドル不動点 / fiber Julia 集合 / fiber Julia-Lavaurs 集合 / スーパーサドル / implosion / characteristic direction / parabolic curve / 形式的共役 / tangent to the identity / Lavaurs 写像 / stretching ray / critical portrait / finger / Lavaurs map / fiber filled Julia 集合 / Lavaurs 写像 / stretching rays / 関数論 / 複素力学系 / 遊走領域 |
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| 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.
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| 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。
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Report
(8 results)
Research Products
(15 results)
