The Return Map in the Class ${\mathcal{O}}_C$: Geometry, Dynamics, and Thickness Regularity

Abstract

We study a discrete dynamical system $F=\pi \circ \Phi$ on the boundary of a convex core $\partial C\subset {\mathbb{R}}^N$, generated by linking $C$ to an outer boundary $\partial \Omega$ via normal rays and a thickness function $d$. We derive the local expansion $F(c)-c=d(c)(I-d(c)S_c{)}^{-1}{\nabla }_{\partial C}d(c)+O(|{\nabla }_{\partial C}d(c)|{}^2)$, showing how curvature precalculates thickness gradients. Fixed points correspond to critical points of $d$, with linearization $DF(c_{\ast })=I+d_{\ast }(I-d_{\ast }{S}_{\ast }{)}^{-1}{H}_{\ast }$ yielding a stability classification driven by curvature and thickness variation. Results are validated on the unit circle, alongside regularity transfer and global bilipschitz bounds.