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 ${\mathcal {F}}$ 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 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].

In this paper we study persistence features of topological entropy and periodic orbit growth of Hamiltonian diffeomorphisms on surfaces with respect to Hofer's metric. We exhibit stability of these dynamical quantities in a rather strong sense for a specific family of maps studied by Polterovich and Shelukhin. A crucial ingredient comes from enhancement of lower bounds for the topological entropy and orbit growth forced by a periodic point, formulated in terms of the geometric self-intersection number and a variant of Turaev's cobracket of the free homotopy class that it induces. Those bounds are obtained within the framework of Le Calvez and Tal's forcing theory.