1998 Fiscal Year Final Research Report Summary
Geometric Study of Complex Dynamical Systems
| Project/Area Number |
09440029
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | The University of Tokyo |
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Principal Investigator |
SHISHIKURA Mitsuhiro The University of Tokyo, Department of Mathematical Sciences, Asociate Professor, 大学院・数理科学研究科, 助教授 (70192606)
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| Co-Investigator(Kenkyū-buntansha) |
TSUJII Masato Hokkaido Univ., Dept.of Math., Assoc.Prof., 大学院・理学研究科, 助教授 (20251598)
TANIGUCHI Masahiko Kyoto University, Depy.of Math., assoc.Prof., 大学院・理学研究科, 助教授 (50108974)
UEDA Tetsuo Kyoto University, Depy.of Integrated Human Sci., prof., 総合人間学部, 教授 (10127053)
USHIKI Shigehiro Kyoto University, Depy.of Human & Env., prof., 大学院・人間・環境学研究科, 教授 (10093197)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Cpmplex dynamical systems / Julis set / Mandelbrot set / Fractal / Chaos / Teichmuller space / Renormalization |
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| Research Abstract |
In this project, we studied theory of one-dimensional and higher dimensional complex dynamical systems and related real dynamical systems.
Results in one-dimensional dynamics : A new proof for the real quadratic polynomial family, by reducing it to the strong contraction of certain self-map of the universal Teichmuller space. A reproduction of the Abel functions for parabolic fixed points via Ecalle's theory of resurgent functions. The properties of eigenvalues and eigenfunctions of complexified Ruelle-Perron-Frobebius operator. Proof of the topological completeness of decorated exponential famillies. Proof of the monotonicity of the real polynomial family x^<2n>+c via an action onto the cotangent space of Teichmuller space.
Results in higher dimensional dynamics : We completely determined completely invariant subvarieties of holomorphic mappings of P^n. A construction of a new critically finite holomorphic mapping of P^n. The existence of absolutely continuous invariant measure (wrt Lebesgue measure) for certain real expanding map in higher dimension.
Numerical experiments : We studied the bifurcation of complex dynamical systems, Julia sets, renormalization and the universality phenomena, via various numerical experiments on computers.
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