An Algebraic Proof of Thurston's Rigidity for a Polynomial

2012/01/10 by Alon Y. Levy, Levy, Alon · 1 citation
Mathematics · Physics and Astronomy · #14D10 #37P45 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1201.1969

openalex publication_date 2012/01/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study rational self-maps of ℙ¹ whose critical points all have finite forward orbit. Thurston's rigidity theorem states that outside a single well-understood family, there are finitely many such maps over ℂ of fixed degree and critical orbit length. We provide an algebraic proof of this fact for tamely ramified maps for which at least one of the critical points is periodic. We also produce wildly ramified counterexamples.

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