Dynamical moduli spaces and polynomial endomorphisms of configurations

TL;DR

This paper explores the geometric structure of moduli spaces associated with polynomial dynamical systems on finite sets, using computational surveys to investigate their intersections and properties.

Contribution

It introduces the study of portrait moduli spaces for polynomial dynamics and presents computational insights into their intersections in low degrees.

Findings

Identification of geometric properties of portrait moduli spaces

Computational results on intersections of these spaces in low degrees

Open questions for further research in dynamical moduli spaces

Abstract

A portrait is a combinatorial model for a discrete dynamical system on a finite set. We study the geometry of portrait moduli spaces, whose points correspond to equivalence classes of point configurations on the affine line for which there exist polynomials realizing the dynamics of a given portrait. We present results and pose questions inspired by a large-scale computational survey of intersections of portrait moduli spaces for polynomials in low degree.