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

TL;DR

This paper applies Morse theory to analyze the geometric and dynamical properties of the return map in certain convex domains, revealing topological constraints and local behavior near equilibria.

Contribution

It introduces a Morse-theoretic framework to relate the return map's equilibria with the topology of the domain, providing bounds and local linearization results.

Findings

Equilibria correspond to critical points of the thickness function.

Lower bounds on the number of equilibria relate to Betti numbers of the boundary.

Near nondegenerate minima, the return map exhibits local linear convergence under spectral conditions.

Abstract

We study the geometric and dynamical structure induced by the return map associated with domains in the class \(\mathcal{O}_{C}\). This map, defined through a geometric round-trip between the convex core and the outer boundary, generates a discrete dynamical system on the boundary \(\partial C\). Building on previous results establishing global convergence of the return dynamics, we show that equilibria of the return map coincide with the critical points of the thickness function. This identification allows us to apply Morse-theoretic tools to derive global constraints on the dynamics. In particular, we obtain lower bounds on the number of equilibria in terms of the Betti numbers of \(\partial C\), as well as a global balance relation governed by the Euler characteristic. We further analyze the local behavior of the return map near equilibria. Using the differentiability of the return map inherited from the radial and reciprocal constructions, we derive a first-order expansion in which the linearization is governed by the Hessian of the thickness function and an operator arising from the geometry of the return map. This leads to an operator-valued generalization of the previously observed scalar structure, revealing that the dynamics behaves as an anisotropic gradient-like iteration rather than a purely isotropic descent. Near nondegenerate minima, we prove a quantitative descent estimate and local linear convergence under a spectral condition. Under aligned nonlocal geometry, the sign of the curvature gap between the convex core and the outer boundary determines whether the induced dynamics is contracting, neutral, or expanding in each principal direction. Finally, we discuss extensions beyond the Morse setting, including the Morse-Bott case, and highlight connections between the geometry of the domain, the topology of \(\partial C\), and the structure of the induced dynamics.