arXiv subjects
Publications and source records attributed to Nelson Schuback.
In this work, we present an alternative definition of the Le Roux index, which generalizes the Poincar\'e-Hopf index for non-singular planar flows to the broader setting of Brouwer homeomorphisms. This new approach answers a question raised by Le Roux by establishing a connection between the index of a Brouwer homeomorphism and the structure of its transverse foliations, in the sense of Le Calvez.
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.
A Brouwer homeomorphism is a fixed-point free, orientation-preserving homeomorphism of the plane. A foundational result of Le Calvez establishes that every such homeomorphism $f$ admits an oriented planar foliation $\mathcal{F}$ such that every point $x \in \mathbb{R}^2$ can be connected to its image $f(x)$ by a path positively transverse to $\mathcal{F}$. This provides a powerful framework for analyzing the dynamics of $f$ by studying how its orbits cross the leaves of $\mathcal{F}$. In this article, we refine this framework by identifying additional qualitative dynamical information about $f$ that is encoded in $\mathcal{F}$, which can be systematically recovered through the concept of proper transverse trajectories. Later, we investigate the possible combinatorial configurations of these proper trajectories for finite collections of orbits and characterize their simplest forms. As a key application, this refined framework is used in a forthcoming work to offer a new perspective on Homotopy Brouwer Theory, originally introduced by Handel.