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 $\mathrm{\Omega }$, the fixed points of the return map $F:\mathrm{\partial }C\to \mathrm{\partial }C$ coincide with the critical points of the thickness function $d$. At any such critical point $c_{\ast }$, the exact linearization reads $$DF(c_{\ast })=I+B_{\ast }{H}_{\ast },$$ where $H_{\ast }={\mathrm{Hess}}_{\mathrm{\partial }C}d(c_{\ast })$ and $B_{\ast }=d_{\ast }(I-d_{\ast }{S}_{\ast }{)}^{-1}$ is positive definite. We show that the self-adjoint preconditioned Hessian $K_{\ast }=B_{\ast }^{1\mathrm{/}2}{H}_{\ast }{B}_{\ast }^{1\mathrm{/}2}$ admits an exact representation via the shape operators of the core and outer boundaries, implying that curvature modulates the return spectrum while preserving the inertia of $H_{\ast }$. In the strict thickness-ascent regime, every negative eigenvalue $\mu$ of $K_{\ast }$ satisfies $\mu \in (-2,0)$, yielding the stable/unstable dimension identities for any Morse equilibrium $c_{C}onfig$: $$\dim E^s(c_{\ast })={\mathrm{ind}}_M(c_{\ast }),\dim E^u(c_{\ast })=m-{\mathrm{ind}}_M(c_{\ast })\mathrm{.}$$ Since $\mathrm{\partial }C\cong {\mathbb{S}}^m$, Morse inequalities act as equilibrium-counting laws for the return dynamics, implying an even number of equilibria. We compute the local fixed-point index ${\mathrm{ind}}_{\mathrm{f}\mathrm{i}\mathrm{x}}(F,c_{\ast })=(-1{)}^{m+{\mathrm{ind}}_M(c_{\ast })}$, establish the global Euler balance with $\mathrm{deg}(F)=1$, and identify this index with the Euler-characteristic jump in the forward-invariant superlevel filtration. Finally, we discuss invariant manifolds, a conditional Morse–Smale structure, geometric examples, and the spectral extension to Morse–Bott manifolds, unifying boundary curvature, thickness geometry, discrete dynamics, and Morse topology.