Linked orbits of homeomorphisms of the plane and Gambaudo-Kolev Theorem

TL;DR

This paper explores the relationship between bounded orbits and fixed points in plane homeomorphisms, establishing that each bounded orbit is linked to a fixed point in a topologically significant way, extending Gambaudo's concept.

Contribution

It proves that for any bounded orbit of an orientation-preserving plane homeomorphism, there exists a linked fixed point, generalizing Gambaudo's linking theorem.

Findings

Every bounded orbit is linked to a fixed point.

Linked fixed points cannot be separated from orbits by isotopic Jordan curves.

The result extends the understanding of orbit-fixed point relationships in plane dynamics.

Abstract

Let $h : \mathbb{R}^2 \to \mathbb{R}^2$ be an orientation preserving homeomorphism of the plane. For any bounded orbit $\mathcal{O}(x)=\{h^n(x):n\in\mathbb{Z}\}$ there exists a fixed point $x'\in\mathbb{R}^2$ of $h$ linked to $\mathcal{O}(x)$ in the sense of Gambaudo: one cannot find a Jordan curve $C\subseteq\mathbb{R}^2$ around $\mathcal{O}(x)$, separating it from $x'$, that is isotopic to $h(C)$ in $\mathbb{R}^2\setminus\left(\mathcal{O}(x)\cup\{x'\}\right)$.