Fixed point theorems in plane continua with applications

TL;DR

This paper proves fixed point theorems for plane continua, introduces the variation concept, and applies these results to complex dynamics, showing certain invariant subcontinua must be degenerate under specific conditions.

Contribution

It develops new fixed point theorems for plane continua, relates variation to index, and applies these to complex dynamics, extending Bell's earlier results.

Findings

Fixed point theorem for positively oriented, perfect maps of the plane.

Invariant subcontinua in Julia sets are degenerate under certain conditions.

Impressions of external rays to polynomial Julia sets are degenerate.

Abstract

We present proofs of basic results, including those developed 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. 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. A fixed point theorem for positively oriented, perfect maps of the plane is obtained. This generalizes results announced by Bell in 1982. A continuous map of an interval to the real line which sends the endpoints in opposite directions has a fixed point. We generalize this to maps on non-invariant continua in the plane under positively oriented maps of the plane (with appropriate boundary conditions). These methods imply that in some cases non-invariant continua in the plane are degenerate. This has important applications in complex dynamics. E.g., a special case of our results shows that if $X$ is a non-separating invariant subcontinuum of the Julia set of a polynomial $P$ containing no fixed Cremer points and exhibiting no local rotation at all fixed points, then $X$ must be a point. It follows that impressions of some external rays to polynomial Julia sets are degenerate.