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

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.

• The paper introduces a Morse-theoretic framework linking the thickness function's critical points with return map dynamics and topological invariants.
• Using operator-valued local linearization, the study reveals anisotropic descent and stability analysis based on curvature differences.
• Morse inequalities constrain the number of fixed points, providing global topological insights into the dynamics of class O_C domains.

Morse-Theoretic Analysis of the Return-Map for Domains in Class $\mathcal{O}_C$

Introduction

The paper "Geometry, Dynamics and Topology of Thickness Landscape: A Morse-Theoretic Analysis of the Return-Map in the Class $\mathcal{O}_{C}$" (2603.30010) explores the intersection of geometric analysis, dynamical systems, and topology by studying the discrete return map associated with non-Lipschitz domains in the class $\mathcal{O}_C$. The central object of study is the return map generated by the geometric round-trip between the convex core $C$ and the outer boundary $\partial\Omega$ of $\Omega$, a domain containing $C$. This round-trip mechanism induces a discrete dynamical system on $\partial C$, entirely governed by the geometry of the thickness function $d$.

This work offers a Morse-theoretic framework to translate topological invariants of $\partial C$ into sharp dynamical and structural properties of the return map. The authors provide a detailed investigation of global and local dynamics, including Morse inequalities, Euler characteristic constraints, and fine-grained analysis of the return map's linearization near equilibria. The resulting perspective elucidates a deep interplay between geometry, dynamics, and topology for the class of non-Lipschitz domains.

Geometric and Dynamical Framework

The analysis is established for domains $\mathcal{O}_{C}$0 in $\mathcal{O}_{C}$1 such that their convex core $\mathcal{O}_{C}$2 is smooth and strictly convex ($\mathcal{O}_{C}$3 boundary). The class $\mathcal{O}_{C}$4 admits domains $\mathcal{O}_{C}$5 where, for almost every boundary point, the inward normal to $\mathcal{O}_{C}$6 intersects $\mathcal{O}_{C}$7, ensuring a geometric coupling between the inner and outer boundaries.

Central to the framework is the thickness function: $\mathcal{O}_{C}$8 where $\mathcal{O}_{C}$9 is the outward normal to $\mathcal{O}_C$0. The radial map $\mathcal{O}_C$1 parametrizes $\mathcal{O}_C$2 over $\mathcal{O}_C$3.

The return map is formulated as: $\mathcal{O}_C$4 where $\mathcal{O}_C$5 returns to $\mathcal{O}_C$6 by following the inward normal from $\mathcal{O}_C$7. The induced discrete dynamics is gradient-like with respect to $\mathcal{O}_C$8, and previous works demonstrated global convergence of forward orbits to the critical set of $\mathcal{O}_C$9 (Barkatou et al., 30 Mar 2026).

Morse Structure and Topological Constraints

A principal finding is the identification of equilibria of the return map $C$0 with critical points of $C$1. If the thickness function is Morse, the critical points are isolated and nondegenerate, permitting application of Morse theory.

Morse Inequalities and Betti Numbers

Classical Morse inequalities yield lower bounds on the number of critical points in terms of the topology of $C$2 via its Betti numbers: $C$3 This constrains the complexity of the boundary-induced dynamics by the underlying topology, regardless of the geometry of $C$4.

Euler Characteristic Balance

The critical points also satisfy the Euler characteristic formula: $C$5 This imposes global constraints on the Morse index distribution of equilibria and prohibits, for example, the existence of exclusively attracting fixed points when the Euler characteristic demands the presence of saddles or repellors. For $C$6, at least two fixed points always exist, typically corresponding to a minimum and a maximum.

Local Dynamics and Operator-Valued Linearization

Extending beyond the scalar and isotropic gradient-like view, the paper develops a local first-order expansion at nondegenerate equilibria: $C$7 with $C$8 a symmetric, positive-definite operator (under natural geometric conditions), dependent on the round-trip geometry and local curvature. This reveals that the dynamics is in general anisotropic—a preconditioned gradient descent on the manifold $C$9 where the preconditioner reflects the geometry of the round-trip between $\partial\Omega$0 and $\partial\Omega$1.

Under principal direction alignment and nonfocality, eigenanalyses show that contraction, expansion, or neutrality of the dynamics in each principal direction is dictated by the sign of the curvature gap $\partial\Omega$2. Explicitly, for eigenvalues $\partial\Omega$3:

• Contraction: $\partial\Omega$4 implies $\partial\Omega$5.
• Neutrality: $\partial\Omega$6 yields $\partial\Omega$7.
• Expansion: $\partial\Omega$8 yields $\partial\Omega$9.

Local descent estimates and linear convergence are shown under suitable spectral conditions, tightly paralleling results in Riemannian optimization.

Structural Properties: Gradient-Like Dynamics and Periodicity

The return map defines a discrete gradient-like system with Lyapunov function $\Omega$0 monotonically nonincreasing along orbits. The global Lyapunov structure precludes nontrivial periodic orbits: any periodic point must be a fixed point. Thus, the return dynamics, while generally nonlinear and anisotropic, retains substantial rigidity typical of gradient-like flows.

The setting extends to the Morse-Bott case when the critical set of $\Omega$1 forms embedded submanifolds, with the local normal dynamics around such manifolds continuing to exhibit operator-valued anisotropic descent.

Examples

The paper provides illustrative cases in low dimensions. For concentric spheres, the thickness landscape is degenerate and every point is fixed, representing the Morse-Bott scenario. Perturbations break symmetry, yielding isolated nondegenerate critical points and a Morse-theoretic structure. In general, the identification of explicit nontrivial dynamics requires breaking symmetry or considering non-aligned normal fields.

Implications and Future Directions

By forging a precise connection between the geometry of non-Lipschitz domains, the Morse-theoretic landscape of the thickness function, and the induced discrete dynamics, this work enables the transfer of topological invariants into quantitative and qualitative predictions about boundary-driven dynamics. The operator-valued linearization clarifies the role of anisotropy and geometric preconditioning in the stability and contraction of the return dynamics.

These insights suggest new lines of inquiry:

• Extension to degenerate and bifurcation scenarios in the thickness landscape.
• Global analysis of stable/unstable manifolds for the return map.
• Unified frameworks connecting discrete geometric flows and optimization algorithms on manifolds.
• Potential applications to inverse geometry, where boundary-encoded dynamics inform about domain structure.

Conclusion

This Morse-theoretic analysis situates the return map of class $\Omega$2 domains at the convergence of geometric analysis, dynamical systems, and algebraic topology. Equilibria and dynamics are precisely constrained by the topology of the convex core boundary, and local dynamics exhibit complex anisotropic structure governed by operator-valued preconditioners arising from the domain geometry. The framework delivers a rigorous foundation for studying intrinsic geometric dynamical systems and opens new perspectives for both theoretical development and applications in geometric control and shape optimization.