On Cartwright-Littlewood Fixed Point Theorem

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.