1993 Fiscal Year Final Research Report Summary
Analytic transformations of complex manifolds
| Project/Area Number |
04640154
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
解析学
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
UEDA Tetsuo Kyoto Univ.Faculty of Integrated Human Studies, AP., 総合人間学部, 助教授 (10127053)
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| Co-Investigator(Kenkyū-buntansha) |
MORIMOTO Yoshinori Kyoto Univ., Graduate School of Human & Environment Studies, Assist.Professor, 人間・環境学研究科, 助教授 (30115646)
USHIKI Shigehiro Kyoto Univ., Graduate School of Human & Environment Studies, Professor, 人間・環境学研究科, 教授 (10093197)
MIYAMOTO Munemi Kyoto Univ., Integrated Human Studies, Professor, 総合人間学部, 教授 (00026775)
TAKEUTI Akira Kyoto Univ., Integrated Human Studies, Professor, 総合人間学部, 教授 (40026761)
AKIBA Tomoharu Kyoto Univ., Integrated Human Studies, Professor, 総合人間学部, 教授 (60027670)
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| Project Period (FY) |
1992 – 1993
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| Keywords | Complex dynamics / Fatou set / Critically finite map / Kobayashi hyperbolicity |
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| Research Abstract |
We investigeted complex dynamical system defind by holo-morphic maps of a complex projective space onto itself, as a generalization of the iteration theory of rational function of one complex variable.
The Fatou set is defined to be the maximal open set on which the family of the iterates of such a holomorphic map constitute a normal family. This is considered as one of the most fundamental object in the theory. In our study we have proved that the Fatou set os pseudoconvex and hence a Stain open set, and further that follows that, every basin of attraction of an attracting periodic point or that of parabolic periodic point contains a critical point.
We have also given some examples of dynamical systems ori pro-jective planes for which the Fatou set can be concretely described using elliptic functions, and for which the Fatou set is empty.
Further we studied the ralation among the Fatou set, the forward orbit of the set of the critical points and its limit set set. In particular, we stydied the critically finite case, i.e., the case for which the orbit of the the set of the critical points is an algebraic set. For the case of dimension 2, we have given the condi-tion for the Fatou set is empty.
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