Cycles of links and fixed points for orientation preserving homeomorphisms of the open unit disk

TL;DR

This paper characterizes all cycles of links at infinity that guarantee fixed points for orientation-preserving homeomorphisms of the open disk, extending previous theorems and providing a complete classification.

Contribution

It provides a complete description of cycles of links that ensure fixed points, generalizing and completing earlier results by Handel and Xavier.

Findings

Identifies all cycles of links that force fixed points

Completes the classification of such cycles

Extends previous fixed point theorems

Abstract

Michael Handel proved in Handel (1999) 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. More recently, the author generalized Handel's theorem to a wider class of cycles of links (Xavier 2012). In this paper we complete this topic describing exactly which are all the cycles of links forcing the existence of a fixed point.