Geometry, Dynamics, and Topology of the Thickness Landscape: A Morse-Theoretic Analysis of the Return Map in the Class $\mathcal{O}_C$

We study the geometry, dynamics, and topology of the thickness landscape associated with the geometric return map in the class $\mathcal O_C$. For a convex core $C$ and an admissible outer domain $Ω$, the fixed points of the return map $F:\partial C\to\partial C$ coincide with the critical points of