Realizing polynomial portraits

TL;DR

This paper proves that every abstract polynomial portrait can be realized by a postcritically finite polynomial and classifies which portraits are realizable only by unobstructed maps.

Contribution

It establishes the realization of all abstract polynomial portraits as ramification portraits of postcritically finite polynomials and classifies unobstructed realizations.

Findings

Every abstract polynomial portrait is realizable.

Classification of portraits only realizable by unobstructed maps.

Provides a framework for understanding polynomial dynamics via portraits.

Abstract

It is well known that the dynamical behavior of a rational map $f:\widehat{\mathbb C}\to \widehat{\mathbb C}$ is governed by the forward orbits of the critical points of $f$. The map $f$ is said to be postcritically finite if every critical point has finite forward orbit, or equivalently, if every critical point eventually maps into a periodic cycle of $f$. We encode the orbits of the critical points of $f$ with a finite directed graph called a ramification portrait. In this article, we study which graphs arise as ramification portraits. We prove that every abstract polynomial portrait is realized as the ramification portrait of a postcritically finite polynomial, and classify which abstract polynomial portraits can only be realized by unobstructed maps.