On Cartwright-Littlewood Fixed Point Theorem

2021/08/05 by Przemysław Kucharski, Kucharski, Przemysław
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS

paper · pdf · doi:10.48550/arxiv.2108.02454

corrections for readability

arxiv created 2022/01/28 · arxiv updated 2022/01/31

Abstract

We prove the following generalization of the Cartwright-Littlewood fixed point theorem. Suppose h\colon~\mathbb R²→\mathbb R² is an orientation preserving planar homeomorphism, and X is an acyclic continuum. Let C be a component of X ∩ h(X) . If there is a c ∈ C such that \mathcal O₊ (c) ⊆ C or \mathcal O_- (c) ⊆ 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.

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