Global Convergence of the Return Dynamics in the Class $\mathcal{O}_C

TL;DR

This paper analyzes a geometric dynamical system within domains with a convex core, showing it behaves like gradient descent on the domain's thickness and converges to critical points.

Contribution

It introduces a return map mechanism linking boundary points and proves its dynamics align with gradient descent on thickness, connecting geometry and system behavior.

Findings

System behaves like an adaptive gradient descent on thickness.

Fixed points correspond to critical points of the thickness function.

Quantifies convergence rate and regularity of the thickness function.

Abstract

Here is an English summary of the abstract: This research investigates a geometric dynamical mechanism within a specific class of domains that contain a fixed convex core. By using a radial structure that links the boundaries of the core and the outer domain via a thickness function, the authors introduce a "return map." This map is constructed by projecting a point from the core to the outer boundary and then returning to the core by following the inward normals. The main results demonstrate that this motion behaves, to a first-order approximation, like an adaptive gradient descent for the domain's thickness. In other words, the system naturally evolves toward areas where the thickness is minimized. The study establishes that the fixed points of this dynamics coincide with the critical points of the thickness function. Additionally, the authors quantify the convergence rate, prove the regularity of the thickness function in relation to the boundary geometry, and establish a structural equivalence between the two surfaces under specific curvature conditions. Ultimately, this work links the dynamical properties of the system to the geometric smoothness of the studied shapes.