Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Mohammed Barkatou, Mohamed El Morsalani

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 $\partialΩ$ through a thickness function $d:\partial C\to\mathbb{R}_+$. Using this structure, we introduce a natural return map obtained by composing the radial projection from $\partial C$ to $\partialΩ$ with the map that follows inward normals from $\partialΩ$ back to $C$. This construction generates a discrete dynamical system on $\partial C$. We prove that the return map admits the first-order expansion \[ F(c)=c-2d(c)\nabla_{\partial C}d(c)+ \text{higher order terms}, \] which 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{doCarmo1976}. Consequently, the fixed points of the dynamics coincide with the critical points of $d$, and the iteration admits a natural Lyapunov structure \cite{nesterov2004}. The construction reveals a hidden geometric mechanism: a transformation acting on $\partial C$ emerges from a round-trip through the outer boundary $\partialΩ$, a phenomenon reminiscent of holonomy \cite{sharpe1997}. Numerical simulations illustrate convergence to fixed points, limit cycles, and chaotic behavior. Connections with variational problems (Cheeger, Faber-Krahn, Saint-Venant) within the class $\mathcal{O}_C$ are also explored \cite{henrot2018}.

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

Abstract

We investigate a geometric dynamical mechanism arising in the class of domains containing a fixed convex set and satisfying two geometric normals properties introduced by Barkatou \cite{barkatou2002}. The first property induces a radial structure linking the boundaries and through a thickness function . Using this structure, we introduce a natural return map obtained by composing the radial projection from to with the map that follows inward normals from back to . This construction generates a discrete dynamical system on . We prove that the return map admits the first-order expansion which 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 \cite{doCarmo1976}. Consequently, the fixed points of the dynamics coincide with the critical points of , and the iteration admits a natural Lyapunov structure \cite{nesterov2004}. The construction reveals a hidden geometric mechanism: a transformation acting on emerges from a round-trip through the outer boundary , a phenomenon reminiscent of holonomy \cite{sharpe1997}. Numerical simulations illustrate convergence to fixed points, limit cycles, and chaotic behavior. Connections with variational problems (Cheeger, Faber-Krahn, Saint-Venant) within the class are also explored \cite{henrot2018}.

Paper Structure

This paper contains 49 sections, 6 theorems, 27 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Geometric Setting and the Class $\mathcal{O}_C$
  3. Ambient Space and Notation
  4. The Convex Core $C$
  5. Geometric Normal Property
  6. The Class $\mathcal{O}_C$
  7. Radial Accessibility
  8. Thickness Function and Radial Parametrization
  9. Definition of the Thickness Function
  10. Radial Parametrization
  11. Regularity
  12. Reciprocal Map and Geometric Round-Trip
  13. Normal Rays from $\partial\Omega$
  14. Reciprocal Map
  15. Basic Properties
  16. ...and 34 more sections

Key Result

Proposition 3.2

For $\Omega\in\mathcal{O}_C$, the mapping is a bijection from $\partial C$ onto $\partial\Omega\setminus C$. Moreover,

Figures (1)

  • Figure 1: Convergence of the iterates $\theta_k$ to the attracting fixed point $\pi/2$ (local minimum of $d$).

Theorems & Definitions (13)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Proposition 3.2
  • Definition 4.1
  • Proposition 4.2
  • Proposition 5.1
  • proof
  • Lemma 6.1
  • Theorem 6.2: First-order expansion
  • ...and 3 more