Applications of fixed point theory to bifurcation problems

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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.