On a generalization of the Cartwright-Littlewood fixed point theorem for planar homeomorphisms

TL;DR

This paper generalizes the Cartwright-Littlewood fixed point theorem for planar homeomorphisms, establishing conditions under which a continuum contains fixed or periodic points, including orientation reversing cases, using topological methods.

Contribution

It extends the classical fixed point theorem to broader conditions and includes orientation reversing homeomorphisms with periodic orbit guarantees.

Findings

Fixed points exist in certain continua under specified conditions.

A counterpart for orientation reversing homeomorphisms guarantees 2-periodic orbits.

The approach simplifies proof using Morton Brown's method and linked periodic orbits theorem.

Abstract

We prove a generalization of the fixed point theorem of Cartwright and Littlewood. Namely, suppose $h : \mathbb{R}^2 \to\mathbb{R}^2$ is an orientation preserving planar homeomorphism, and let $C$ be a continuum such that $h^{-1}(C)\cup C$ is acyclic. If there is a $c\in C$ such that $\{h^{-i}(c):i\in\mathbb{N}\}\subseteq C$, or $\{h^i(c):i\in\mathbb{N}\}\subseteq C$, then $C$ also contains a fixed point of $h$. Our approach is based on Morton Brown's short proof of the result of Cartwright and Littlewood. In addition, making use of a linked periodic orbits theorem of Bonino we also prove a counterpart of the aforementioned result for orientation reversing homeomorphisms, that guarantees a $2$-periodic orbit in $C$ if it contains a $k$-periodic orbit ($k>1$).