2014 Fiscal Year Final Research Report
Classification of invariant sets at an indeterminate point of rational mappings
Project/Area Number |
24540225
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Research Category |
Grant-in-Aid for Scientific Research (C)
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Allocation Type | Multi-year Fund |
Section | 一般 |
Research Field |
Global analysis
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Research Institution | Tokyo Metropolitan College of Industrial Technology |
Principal Investigator |
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Project Period (FY) |
2012-04-01 – 2015-03-31
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Keywords | 不定点 / ブローアップ / 不変多様体 / カントールブーケ / ニュートン法 |
Outline of Final Research Achievements |
In the study of dynamics of a rational mapping F on the 2-dimensional complex projective space, it has known that there exists some family of holomorphic curves at an indeterminate point, which is called a Cantor bouquet. In this research, on the n-dimensional complex projective space we tried to classify invariant sets at the set I of indeterminate points of a rational mapping F. In particular, we showed that if dim I = n-2, then there exists an invariant manifold V with dim V=n-1. On the 2-dimensional complex projective space, we can approximate a Cantor bouquet by some sequence of open sets which are defined by formal power series. Moreover, consider Newton’s method NF toward a multiple root, in the case of a polynomial mapping F of two variables. A multiple root of F is an indeterminate point of NF. Therefore, applying our results to Newton’s method NF, we give a geometric description of the basin of a multiple root.
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Free Research Field |
複素力学系
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