2001 Fiscal Year Final Research Report Summary
Research on Complex Dynamics
| Project/Area Number |
11440053
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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 |
Global analysis
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| Research Institution | KYOTO UNIVERSITY (2001)
Hiroshima University (1999-2000) |
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Principal Investigator |
SHISHIKURA Mitsuhiro Kyoto University, Graduate School of Scienece, Professor, 大学院・理学研究科, 教授 (70192606)
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| Co-Investigator(Kenkyū-buntansha) |
TANIGUCHI Masahiko Kyoto University, Graduate School of Scienece, Assoc. professor, 大学院・理学研究科, 助教授 (50108974)
UEDA Tetsuo Kyoto University, Faculty of Integrated Human Studies, Professor, 総合人間学部, 教授 (10127053)
USHIKI Shigehiro Kyoto Univ., Graduate school of Human & Envi. Stud., Professor, 大学院・人間環境学研究科, 教授 (10093197)
TSUJII Masato Hokkaido University, Graduate School of Scienece, Assoc. professor, 大学院・理学研究科, 助教授 (20251598)
SHIGA Hiroshige Tokyo Inst. of Technology, Grad. School of Sci. & Tech., Professor, 大学院・理工学研究科, 教授 (10154189)
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| Project Period (FY) |
1999 – 2001
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| Keywords | Complex dynamics / Fractal / Chaos / Mandelbrot set / Julia set / renormalization / rigidity / bifurcation |
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| Research Abstract |
In this research project, we studied various problems in complex dynamics and related fields, such as real dynamics, function theory, Kleininan groups and Teichmuller spaces.
In previous research, it was proved that the boundary of the Mandelbrot set as well as quadratic Julia sets for generic parameters from the boundary have Hausdorff dimension two. The next natural question is whether these sets have Lebesgue measure zero. We were able to answer this question affirmatively when all periodic points are repelling and the map is not infinitely renormalizable. The main methods are Yoccoz puzzle partion, combinatorial analysis through tau-functions and the modulus-area inequality for annuli.
Using a similar idea, we also studied the rigidity problem of quadratic polynomials. We showed that the rigidity problem for real maps can be reduced to the problem of a self map of the universal Teichmuller space, and that the uniform contraction and apriori bound on the displacement of the base point
… More
for the self map is enough to ensure the rigidity for infinitely renormalizable real quadratic polynmials.
The monotonicity problem of real quadratic-like maps was also studied via comple point of view. The possibility of infinite oscillation was discussed in connection with Ruelle operators acting of holomorphic quadratic differentials and Fatou coordinates.
For holomorphic maps on protective spaces, the classification of totally invariant varieties is given for the case of dimension 2 and 3. A generalization of Lattes example to higher dimension was also constructed. This gives a new class of examples of critically finite maps.
With S. Matsumoto (Nihon Univ.), we studied a class of skew product map on an infinite annulus over an irrational rotation and characterized those maps which have minimal sets.
In order to have perspaectives on the research on higher dimensional complex dynamics, we invied Prof. Serge Cantat (Universte de Rennes, France) to give a series of talks on the dynamics on K3 surfaces.
Numerical experiments on varius complex dynamical systems including quadratic polynomials have been done, and this contributed the understanding of dynamics and visualization of Julia sets, Mandelbrot set and Yoccoz puzzles etc. Less
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