Fixed points for nilpotent actions on the plane and the Cartwright-Littlewood theorem

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 Boroński, and answers the Question from Boroński's work in 2017. The proof is inspired by a short proof o…

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

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 co…

Cartwright-Littlewood 合作的遗产

The Legacy of the Cartwright-Littlewood Collaboration

本文回顾 Cartwright 与 Littlewood 对大参数受迫 van der Pol 方程的合作研究及其混沌发现的历史背景、数学遗产与跨领域影响。

来自固定素端的无椭圆点

No elliptic points from fixed prime ends

本文研究保面积曲面映射，通过固定素端不动点的非椭圆性，将 Mather 等结果从 Moser 稳定条件放宽到无理旋转导数，并应用于 standard map 同宿点及带边界曲面。

A Poincaré-Bendixson theorem for translation lines and applications to prime ends

For an orientation-preserving homeomorphism of the sphere, we prove that if a translation line does not accumulate in a fixed point, then it necessarily spirals towards a topological attractor. This is in analogy with the description of flow lines given by Poincaré-Bendixson theorem. We then apply this result to the study of invariant continua without fixed points, in particular to circloids and boundaries of simply connected open sets. Among the applications, we show that if the prime ends rotation number of such an open set $U$ vanishes, then either there is a fixed point in the boundary, or…