Applications of fixed point theory to bifurcation problems
Project/Area Number 
15540081

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Geometry

Research Institution  Naruto University of Education 
Principal Investigator 
MATSUOKA Takashi Naruto University of Education, College of Education, Professor, 学校教育学部, 教授 (50127297)

CoInvestigator(Kenkyūbuntansha) 
NARUKAWA Kimiaki Naruto University of Education, College of Education, Professor, 学校教育学部, 教授 (60116639)
KOBAYASHI Shigeru Naruto University of Education, College of Education, Associate Professor, 学校教育学部, 助教授 (10195779)
HAYAKAWA Eijirou Toyama University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (50240776)
澤辺 正人 鳴門教育大学, 学校教育学部, 助手 (60346624)

Project Period (FY) 
2003 – 2004

Project Status 
Completed (Fiscal Year 2004)

Budget Amount *help 
¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2004: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2003: ¥900,000 (Direct Cost: ¥900,000)

Keywords  bifurcation of fixed points / family of continuous maps / branch / parametrized fixed point theory / linking number / braid / 不動点 / 周期点 / 分岐 / 位相的指数 
Research Abstract 
Bifurcation of fixed points was considered for a oneparameter family of homeomorphisms defined on a closed disk. Some results were proved on the topological property of branches under bifurcation. Generically, a branch of fixed points becomes a continuous curve. We studied a branch which is homeomorphic to a circle. We assume that the deformation starts from a map f having at least two fixed points, and ends at the same map f. Also, assume that under the deformation, every fixed point of the map f does not disappear, and that the braid of the branches containing the fixed points of f is trivial. Under these assumptions, we proved that the linking numbers of a circular branch with the branches containing the fixed points of f are equal to those of some fixed point of f. In particular, if the braid of the fixed point set of f is also trivial, then any circular branch has linking number zero with the branches containing the fixed points of f. The proof uses parametrized fixed point theory developed by Geoghegan and Nicas in the mid of 1990's. This theory treats a family of continuous selfmaps on a compact space, and defines a homotopy invariant in the 1dimensional homology group of the space. In our case, the space is a punctured disk which is not compact, and therefore a compactification is necessary. We analyzed the effect caused by this compactification by determining the linking numbers of new fixed points generated by this compactification. In this determination, we used a result proved by the head investigator on the relationship between the braid type and selflinking numbers of fixed points.

Report
(3 results)
Research Products
(6 results)