Study of higher dimensional complex dynamical systems
Project/Area Number  09640232 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
解析学

Research Institution  Osaka Medical College 
Principal Investigator 
NISHIMURA Yasuichiro Osaka Medical College Faculty of medicine, Associate Professor, 医学部, 助教授 (90156117)

CoInvestigator(Kenkyūbuntansha) 
UEDA Tetsuo Kyoto University Faculty of integrated human studies Professor, 総合人間学部, 教授 (10127053)
YASUDA Reiko Osaka Medical College Faculty of medicime, Associate Professor, 医学部, 助教授
安田 〓子 大阪医科大学, 医学部, 助教授
安田 苓子 大阪医科大学, 医学部, 助教授 (90084855)

Project Period (FY) 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥1,100,000 (Direct Cost : ¥1,100,000)
Fiscal Year 1998 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1997 : ¥600,000 (Direct Cost : ¥600,000)

Keywords  complex dynamics / iteration / Julia set / Fatou set / Henon map / iteration / Fatcu集合 / Heron写像 
Research Abstract 
Nishimura and Yasuda studied dynamical systems of birational maps of twodimensional complex projective space P^2. In the study of dynamics of rational maps, it appear some phen, omena which do not appeal in the study of dynamics of holomorphic maps, and these phenomena cause some problems, that is, tlie problem of distribution of inderterminate points and the problem of degree lowering of homogeneous polynomial representation. In order to understand these phenomena, we investigated in detail the family of birational polynomial quadratic maps psi and their inverse maps psi. In the case of maps psi, we expressed the mechanism of the way how the degree lowering occurs and we described concretely the distribution of inderminate points. We have successfully picked out the algebraic curves C_n of common divisers which appears when the degree lowering occurs at the ntimes iteration psi^n. We also proved that the set of all indeterminate points of the iterated maps psi^n coincides with the set of intersection points of all pairs of the algebraic curves C_m and C_n. Though our research is of purely mathrmatical, Nishimura implemented a computer program of two dimensional complex dynamical systems which help the pure mathematical research. The computation of iteration by using the homogeneous coordinate system of projective space was tried. Ueda studied the complex dynamics on the ndimensional projective space P^n. Specifically, the case of critically finite maps, that is the case when the orbits of the branch points constitute an algebraic sets, were studied deeply. In this case, he proved that the Fatou maps are constant maps, the Julia set coincides the whole space P^n and that the set of repelling periodic points is dense in the whole space P^n Furthermore, he classified the quadratic maps of P^2 and constructed some examples of critically finite maps of P^2.

Report
(3results)
Research Output
(3results)