2008/05/08 by Robbert Fokkink, Robbert J. Fokkink, John C. Mayer +7
Mathematics · #54F20 #FOS: Mathematics #Fixed Point Theorems Analysis #General Topology (math.GN) #Mathematical Dynamics and Fractals #Mathematics and Applications #math.GN #msc:54F20
paper · pdf · doi:10.48550/arxiv.0805.1184
50 pages, 4 figures Version 2: 52 pages 5 figures. Corrected authors, added picture, expanded and simplified some arguments
openalex publication_date 2008/05/08 · arxiv created 2008/10/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we present proofs of basic results, including those developed so far by H. Bell, for the plane fixed point problem. Some of these results had been announced much earlier by Bell but without accessible proofs. We define the concept of the variation of a map on a simple closed curve and relate it to the index of the map on that curve: Index = Variation + 1. We develop a prime end theory through hyperbolic chords in maximal round balls contained in the complement of a non-separating plane continuum X. We define the concept of an \em outchannel for a fixed point free map which carries the boundary of X minimally into itself and prove that such a map has a unique outchannel, and that outchannel must have variation =-1. We also extend Bell's linchpin theorem for a foliation of a simply connected domain, by closed convex subsets, to arbitrary domains in the sphere. We introduce the notion of an oriented map of the plane. We show that the perfect oriented maps of the plane coincide with confluent (that is composition of monotone and open) perfect maps of the plane. We obtain a fixed point theorem for positively oriented, perfect maps of the plane. This generalizes results announced by Bell in 1982 (see also \citeakis99). It follows that if X is invariant under an oriented map f, then f has a point of period at most two in X.