On Cartwright-Littlewood Fixed Point Theorem

TL;DR

This paper generalizes the Cartwright-Littlewood fixed point theorem for planar homeomorphisms, establishing conditions under which a component of the intersection of a continuum and its image contains a fixed point, extending previous results.

Contribution

It introduces a broader fixed point theorem for orientation-preserving planar homeomorphisms, generalizing earlier work and answering a question posed in 2017.

Findings

Proves a new fixed point theorem for acyclic continua under planar homeomorphisms.

Extends previous results by Ostrovski and Boroński.

Provides a simplified proof inspired by Hamilton's approach.

Abstract

We prove the following generalization of the Cartwright-Littlewood fixed point theorem. Suppose $ h\colon~{\mathbb R}^{2}\to{\mathbb R}^{2} $ is an orientation preserving planar homeomorphism, and $ X $ is an acyclic continuum. Let $ C $ be a component of $ X \cap h(X) $. If there is a $ c \in C $ such that $ {\mathcal O}_{+} (c) \subseteq C $ or $ {\mathcal O}_{-} (c) \subseteq C $ then $ C $ also contains a fixed point of $ h$. Our result also generalizes earlier results of Ostrovski and Boro\'nski, and answers the Question from Boro\'nski's work in 2017. The proof is inspired by a short proof of the result of Cartwright and Littlewood due to Hamilton.