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

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 $x^{\prime }\in {\mathbb{R}}^2$ 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 $x^{\prime }$, that is isotopic to $h(C)$ in ${\mathbb{R}}^2\setminus \left(\mathcal{O}(x)\cup \{x^{\prime }\}\right)$.

arXiv commentTo appear in MPCPS