Handel's fixed point theorem: A Morse theoretical point of view

TL;DR

This paper offers a Morse theoretical perspective on Handel's fixed point theorem for surface homeomorphisms, providing a more conceptual proof using free brick decompositions and a new lemma about conditions at infinity.

Contribution

It introduces a Morse theoretical approach to Handel's fixed point theorem, simplifying the proof with a new lemma and conceptual framework.

Findings

A new Morse theoretical proof of Handel's fixed point theorem.

Introduction of a preliminary lemma providing conditions at infinity.

Enhanced understanding of fixed points in surface homeomorphisms.

Abstract

Michael Handel has proved in [Ha] a fixed point theorem for an orientation preserving homeomorphism of the open unit disk, that turned out to be an efficient tool in the study of the dynamics of surface homeomorphisms. The present article fits into a series of articles by the author [LeC2] and by Juliana Xavier [X1], [X2], where proofs were given, related to the classical Brouwer Theory, instead of the Homotopical Brouwer Theory used in the original article. Like in [LeC2], [X1] and [X2], we will use "free brick decompositions" but will present a more conceptual Morse theoretical argument. It is based on a new preliminary lemma, that gives a nice "condition at infinity" for our problem.