A natural rational map of configurations, and equidistribution of points on the sphere

We introduce a natural dynamical system on configurations of points on a sphere, based on projective geometry and represented by a rational map. In the

case S², the map is holomorphic. The map has interesting universality type properties which shed light on the structure of the space of rational maps, equivariance properties which relate it to moduli spaces and to metric properties such as equi-distribution (electrons on a sphere, but wrt log-potentials). Variations on the construction give interesting algorithms for dealing with rational maps. Computer experiments played a significant role in the initial stages of this work, and should be necessary in its subsequent development.