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 [A foliated version of the Brouwer translation theorem. Comment. Math. Helv. 79 (2) (2004), 229–259 (in French)] establishes that every such homeomorphism f admits an oriented planar foliation F F script upper F such that every point x ∈ R 2 x ∈ R² x element of double struck upper R squared can be connected to its image f ( x ) $f(x)$ f left parenthesis x right parenthesis by a path positively transverse to F F script upper F . This provides a powerful framework for analyzing the dynamics of f by studying how its orbits cross the leaves of F F script upper F . In this article, we refine this framework by identifying additional qualitative dynamical information about f that is encoded in F F script upper 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 subsequent work [A foliated viewpoint on homotopy Brouwer theory. Preprint, 2025, arXiv:2512.07510] to offer a new perspective on homotopy Brouwer theory, originally introduced by Handel [A fixed-point theorem for planar homeomorphisms. Topology 38 (2) (1999), 235–264].