On Cartwright-Littlewood Fixed Point Theorem

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

The goal of this paper is proving the existence and then localizing global fixed points for nilpotent groups generated by homeomorphisms of the plane satisfying a certain Lipschitz condition. The condition is inspired in a classical result of Bonatti for commuting diffeomorphisms of the 2-sphere and in particular it is satisfied by diffeomorphisms, not necessarily of class $C^{1}$, whose linear part at every point is uniformly close to the identity. In this same setting we prove a version of the Cartwright-Littlewood theorem, obtaining fixed points in any continuum preserved by a nilpotent act…

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…

Horocyclic Orbits on $Γ(1)\frontslash\mathcal{H}$, \ Closed and Otherwise

This paper studies certain horocyclic orbits on $Γ(1)\frontslash\mathcal{H}$. In the first instance we examine horocycles defined using the pencil of circles whose common point (in the words of the Nielsen-Fenchel manuscript is $\infty$. The orbits involved in this case are closed and long - judged by arc length between two points compared to the hyperbolic distance between them. We are concerned with tracking the paths of individual horocycles. Using Ford circles of Farey sequences we find lifts to the Standard Fundamental Region (SFR) and find points of these lifts making given angles with a…