2006 Fiscal Year Final Research Report Summary
Research on Complex Dynamics
Grant-in-Aid for Scientific Research (B)
|Allocation Type||Single-year Grants |
|Research Institution||Kyoto University |
UEDA Tetsuo Kyoto University, Grad.School of Science, Professor -> 京都大学, 大学院理学研究科, 教授 (10127053)
USHIKI Shigehiro Kyoto University, Grad.School of Human and Environmental Science, Professor, 大学院人間・環境学研究科, 教授 (10093197)
TANIGUCHI Masahiko Nara Women's University, Faculty of Science, Professor, 理学部, 教授 (50108974)
NAKAI Isao Ochanomizu Women's University, Faculty of Scienec, Professor, 理学部, 教授 (90207704)
TSUJII Masato Kyushu University, Faculty of Mathematics, Professor, 数理学研究院, 教授 (20251598)
KISAKA Masashi Kyoto University, Grad.School of Human and Environmental Science, Associate Professor, 大学院人間・環境学研究科, 助教授 (70244671)
|Project Period (FY)
2003 – 2006
|Keywords||Complex Dynamics / Chaos / Fractal / Bifurcation / Julia set / Dynamical System / Renormalization / Teichmuller space|
Ueda studied the Fatou coordinate (solution to Abel's equation) for parabolic fixed points and the linearization function (solution to Schroeder equation) for attracting fixed points for complex analytic functions of one variable. He showed that The Fatou coordinate can be obtained as an appropriate limit of the linearization functions for the sequence of maps whose multiplier tends to 1. He also studied holomorphic mappings on complex projective spaces and characterized the condition for the analytic continuation of Fatou maps, which is a generalized notion of Fatou components.
Tsujii studied using functional analytic methods the ergodic theoretical properties of (partially) hyperbolic dynamical systems. For certain two dimensional partially hyperbolic systems, he showed under a genericity assumption that there exist a finite number of measure theoretic attractors and their basins coincide with the entire phase space modulo a set of Lebesgue measure zero. He also studied the dynamical
zeta function for hyperbolic dynamical systems and its analytic continuation.
In a joint work with Shishikura, Inou studied the parabolic renormalization of one dimensional complex dynamical systems. They showed the existence of an invariant space of function for the parabolic renormalization, and this implied that its perturbation leads to the hyperbolicity of the near-parabolic renormalization for irrationally indifferent fixed points. As an application they obtained the universal behavior of the multipliers of the small periodic cycles around irrationally indifferent fixed points. Buff and Cheritat also used the above result as a key step to show the existence of a quadratic polynomial with Julia set of positive Lebesgue measure. This became a counter-example to a long standing problem which is an analogy of Ahlfors conjecture for rational maps.
In a joint work with Shishikura, Kisaka developed the technique of quasiconformal surgery to show the existence of doubly connected wandering domains for transcendental entire functions.
Ushiki visualized higher dimensional Julia sets. Less
Research Products (8 results)