Applications of fixed point theory to bifurcation problems

研究了闭盘上定义的单参数同胚族不动点的分岔问题。证明了分支在分岔下的拓扑性质。一般来说，不动点的分支成为连续曲线。我们研究了一个与圆同胚的分支。我们假设变形从至少有两个不动点的映射f开始，并在同一个映射f结束。并且假设在变形下，映射f的每个不动点都不消失，并且包含f不动点的分支编织是平凡的。在这些假设下，我们证明了一个圆形分支与包含f不动点的分支的连接数等于f的某个不动点的连接数。特别是，如果f的不动点集的编织也是平凡的，则任何圆形分支与包含f不动点的分支的连接数为零。证明使用了Geoghegan和Nicas在20世纪90年代中期提出的参数化不动点理论。这个理论处理紧空间上的连续自映射族，并在空间的一维同调群中定义了一个同伦不变量。在我们的例子中，空间是一个不紧致的被刺穿的圆盘，因此需要紧致化。我们通过确定紧化所产生的新不动点的连接数来分析紧化所带来的影响。在此确定中，我们使用了由首席调查员证明的关于编织类型与固定点自连接数之间关系的结果。

Bifurcation of fixed points was considered for a one-parameter 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 self-maps on a compact space, and defines a homotopy invariant in the 1-dimensional 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 self-linking numbers of fixed points.