Handel's fixed point theorem revisited

TL;DR

This paper presents a new, simplified proof of Handel's fixed point theorem for certain homeomorphisms of the disk, extending the result to non-oriented cycles of links at infinity.

Contribution

It provides a more straightforward proof and generalizes Handel's theorem to include non-oriented cycles of links at infinity.

Findings

Simplified proof of Handel's fixed point theorem.

Extension to non-oriented cycles of links at infinity.

Demonstrates existence of fixed points under broader conditions.

Abstract

Michael Handel proved in [7] the existence of a fixed point for an orientation preserving homeomorphism of the open unit disk that can be extended to the closed disk, provided that it has points whose orbits form an oriented cycle of links at infinity. Later, Patrice Le Calvez gave a different proof of this theorem based only on Brouwer theory and plane topology arguments [9]. These methods permitted to improve the result by proving the existence of a simple closed curve of index 1. We give a new, simpler proof of this improved version of the theorem and generalize it to non-oriented cycles of links at infinity