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

J. P. Boronski

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 $\mathbb{R}^2\setminus\left(\mathcal{O}(x)\cup\{x'\}\right)$.

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

Abstract

Let be an orientation preserving homeomorphism of the plane. For any bounded orbit there exists a fixed point of linked to in the sense of Gambaudo: one cannot find a Jordan curve around , separating it from , that is isotopic to in .

Paper Structure

This paper contains 4 sections, 2 theorems, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main1']}
  3. Final Remarks
  4. Acknowledgments

Key Result

Theorem \oldthetheorem

Let $h:\mathbb{R}^2 \to \mathbb{R}^2$ be an orientation preserving homeomorphism. For any bounded orbit $\mathcal{O}(x)$ there exists a fixed point $p\in\mathbb{R}^2$ linked to $\mathcal{O}(x)$.

Figures (1)

  • Figure 1: Proof of Theorem \ref{['main1']}, CASE I: The component $U$ of $\mathbb{R}^2\setminus \operatorname{Fix}(h)$ containing $\operatorname{cl}(\mathcal{O}(x))$ and the covering spaces $\tilde{U}$ and $\bar{U}$.

Theorems & Definitions (6)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem: Brouwer Translation Theorem
  • proof
  • Claim \oldthetheorem
  • proof
  • Remark \oldthetheorem