Refined methods in foliated Brouwer 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.