Realizing polynomial ramification portraits

Authors

• William Floyd, Virginia Tech
• Daniel Kim, Virginia Tech
• Sarah Koch, University of Michigan - Ann Arbor
• Walter Parry, Eastern Michigan University
• Edgar Saenz, Virginia Tech

Document Type

Article

Publication Date

2026

Department/School

Mathematics

Publication Title

Ergodic Theory and Dynamical Systems

Abstract

It is well known that the dynamical behavior of a rational map 𝑓 :ˆℂ →ˆℂ 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.

Comments

W. Parry is a faculty member in EMU's Department of Mathematics and Statistics

Link to Published Version

DOI: 1017/etds.2026.10279

Recommended Citation

Floyd, W., Kim, D., Koch, S., Parry, W., & Saenz, E. (2026). Realizing polynomial ramification portraits. Ergodic Theory and Dynamical Systems, 46(7), 1710–1734. https://doi.org/10.1017/etds.2026.10279