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

TL;DR

This paper studies a geometric return map in convex domains, revealing its behavior akin to gradient descent on a thickness function, and explores the regularity relationship between domain boundaries and this function.

Contribution

It introduces a new return map in the class O_C, derives its expansion, analyzes its fixed points, convergence, and links boundary regularity to the thickness function's smoothness.

Findings

Return map behaves like an adaptive gradient descent on thickness.

Fixed points of the map are critical points of the thickness function.

Thickness regularity is equivalent to boundary regularity under certain conditions.

Abstract

We investigate a geometric dynamical mechanism arising in the class $\mathcal{O}_C$ of domains containing a fixed convex set $C$ and satisfying two geometric normals properties introduced by Barkatou \cite{Barkatou2002}. The first property induces a radial structure linking the boundaries $\partial C$ and $\partial \Omega$ through a thickness function $d:\partial C\to \R_{+}$. Using this structure, we introduce a natural return map obtained by composing the radial projection from $\partial C$ to $\partial \Omega$ with the map that follows inward normals from $\partial \Omega$ back to $C$. This construction generates a discrete dynamical system on $\partial C$. We prove that the return map admits the first-order expansion \[ F(c) = c - 2d(c)\nablaTCd(c) + \text{higher order terms}, \] with explicit remainder estimates. This reveals that the induced dynamics behaves, to leading order, like an adaptive gradient descent for the thickness function. The expansion incorporates curvature corrections arising from the convex core $\partial C$ \cite{Schneider2014}. Consequently, the fixed points of the dynamics coincide with the critical points of $d$, and the iteration admits a natural Lyapunov structure \cite{Smale1961}. We further quantify the convergence rate, provide a rigorous error bound between the discrete and continuous gradient flows, and show that the product condition $d\kappa_i < 1$ can be relaxed. We then analyze the regularity of the thickness function and its relationship to the regularity of the outer boundary $\partial \Omega$. We show that the thickness function inherits the regularity of $\partial \Omega$ and vice versa, and we establish a bilipschitz equivalence between the two boundaries under a quantitative curvature condition. These results link the dynamical properties of the return map to the geometric smoothness of the admissible domains.