A foliated viewpoint on homotopy Brouwer theory

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.