A dynamical system using the Voronoi tessellation

TL;DR

This paper introduces a novel dynamical system based on Voronoi tessellations, analyzing its behavior on finite point sets and providing growth quantification, with open problems for future research.

Contribution

It presents a new dynamical system using Voronoi vertices, analyzing its properties and growth behavior for finite point sets.

Findings

Growth of point set sizes under iteration is quantified.

Behavior of the system for small point sets is characterized.

Open problems related to the system are proposed.

Abstract

We introduce a dynamical system based on the vertices of Voronoi tessellations. This dynamical system acts on finite or discrete point sets in the plane, taking a point set to the vertex set of its Voronoi tessellation. We explore the behavior of this system for small point sets, then prove a general result quantifying the growth of the sizes of the point sets under iteration. We conclude by giving the most interesting open problems.