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arXiv · 2105.10055

Realizing polynomial portraits

William Floyd·Daniel Kim·Sarah Koch·Walter Parry·Edgar Saenz
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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.

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William Floyd, Daniel Kim, Sarah Koch, Walter Parry, Edgar Saenz. 2021-05-20. Realizing polynomial portraits. https://arxiv.org/abs/2105.10055

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