2021/05/20 by William Floyd, Daniel Kim, Floyd, William +7
Mathematics · Physics and Astronomy · #Mathematical Dynamics and Fractals #Advanced Differential Equations and Dynamical Systems #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2105.10055
It is well known that the dynamical behavior of a rational map f:\widehat\mathbb C→ \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.