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Realizing polynomial ramification portraits
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems , First View
- Published online by Cambridge University Press:
- 23 February 2026, pp. 1-25
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- Article
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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 ramification portrait is realized as the ramification portrait of a postcritically finite polynomial and classify which abstract polynomial ramification portraits can only be realized by unobstructed maps.
