On Cartwright-Littlewood Fixed Point Theorem

Abstract

We prove the following generalization of the Cartwright-Littlewood fixed point theorem. Suppose $h:{\mathbb{R}}^2\to {\mathbb{R}}^2$ 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 Boroński, and answers the Question from Boroński's work in 2017. The proof is inspired by a short proof of the result of Cartwright and Littlewood due to Hamilton.

corrections for readability