2015/10/22 by Jan P. Boroński, Boroński, Jan P.
Mathematics · #37C25 #37E30 #54H25 #55C20 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:37C25 #msc:37E30 #msc:54H25 #msc:55C20
paper · pdf · doi:10.48550/arxiv.1510.06663
Accepted to Ergodic Theory and Dynamical Systems
arxiv created 2015/10/22 · openalex publication_date 2015/10/22 · arxiv updated 2015/10/23 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
We prove a generalization of the fixed point theorem of Cartwright and Littlewood. Namely, suppose h : ℝ2 →ℝ2 is an orientation preserving planar homeomorphism, and let C be a continuum such that h-1(C)∪ C is acyclic. If there is a c∈ C such that \h-i(c):i∈ℕ\⊆ C, or \hi(c):i∈ℕ\⊆ 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).