A foliated viewpoint on homotopy Brouwer theory

TL;DR

This paper unifies two major approaches to studying Brouwer homeomorphisms, using foliated methods to enhance and recover classical results in the dynamics of fixed-point-free plane homeomorphisms.

Contribution

It introduces a unified framework combining Handel's and Le Calvez's approaches, improving the understanding of Brouwer homeomorphisms through foliated techniques.

Findings

Le Calvez's foliated methods can recover classical Handel results

Unified framework enhances analysis of Brouwer homeomorphisms

Improves understanding of fixed-point-free plane dynamics

Abstract

Brouwer homeomorphisms are fixed-point-free, orientation-preserving homeomorphisms of the plane. In recent years, their dynamics have been mostly studied through two complementary approaches, one introduced by Handel and the other by Le Calvez, each offering a distinct perspective on the behavior of Brouwer homeomorphisms. In this work, we present a unified framework that allows Le Calvez's foliated methods to recover, and in some cases improve, the classical results of Handel's theory.