Realizing polynomial portraits
William Floyd, Daniel Kim, Sarah Koch, Walter Parry, Edgar Saenz

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 is governed by the forward orbits of the critical points of . The map 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 . We encode the orbits of the critical points of 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.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Advanced Differential Equations and Dynamical Systems · Quantum chaos and dynamical systems
