Une nouvelle preuve du theoreme de point fixe de Handel

TL;DR

This paper provides a new proof of Handel's fixed point theorem for certain homeomorphisms of the disk, using Brouwer theory, and improves the theorem by establishing the existence of a simple closed curve with index 1.

Contribution

The authors offer a novel proof of Handel's fixed point theorem and enhance it by proving the existence of a simple closed curve of index 1, with implications for surface homeomorphisms.

Findings

New proof of Handel's fixed point theorem

Existence of a simple closed curve of index 1

Improved understanding of orbit linking properties

Abstract

M Handel has proved in [Topology 38 (1999) 235--264] a fixed point theorem for an orientation preserving homeomorphism of the open unit disk, that may be extended to the closed disk and that satisfies a linking property of orbits. We give here a new proof of Handel's fixed point theorem, based on Brouwer theory and some plane topology arguments. We will slightly improve the theorem by proving the existence of a simple closed curve of index 1. This index result was known to be true under an additional hypothesis and has been used by different authors (J Franks [NYJM 2 (1996) 1--19, Trans.AMS 348 (1996) 2637--2662] S Matsumoto [Topol. Appl. 104 (2000) 191--214]) to study homeomorphisms of surfaces.