# Fixed Point Theorems for Plane Continua with Applications

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*Alexander M. Blokh; Robbert J. Fokkink; John C. Mayer; Lex G. Oversteegen; E. D. Tymchatyn*

In this memoir the authors present proofs of basic results,
including those developed so far by Harold Bell, for the plane fixed
point problem: Does every map of a non-separating plane continuum have
a fixed point? Some of these results had been announced much
earlier by Bell but without accessible proofs. The authors 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. A
prime end theory is developed through hyperbolic chords in maximal
round balls contained in the complement of a non-separating plane
continuum \(X\). They define the concept of an
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 \(-1\). Also Bell's Linchpin Theorem for a foliation
of a simply connected domain, by closed convex subsets, is extended to
arbitrary domains in the sphere.

The authors introduce the notion of an oriented map of the plane
and show that the perfect oriented maps of the plane coincide with
confluent (that is composition of monotone and open) perfect maps of
the plane. A fixed point theorem for positively oriented, perfect maps
of the plane is obtained. This generalizes results announced by Bell
in 1982.

#### Table of Contents

# Table of Contents

## Fixed Point Theorems for Plane Continua with Applications

- List of Figures ix10 free
- Preface xi12 free
- Chapter 1. Introduction 116 free
- Part 1 . Basic Theory 1126 free
- Part 2 . Applications of Basic Theory 4762