  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.
