An Algebraic Proof of Thurston's Rigidity for a Polynomial

Abstract

We study rational self-maps of ${\mathbb{P}}^1$ 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 $\mathbb{C}$ 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.

11 pages; complete rewrite of the previous version

References in corpus (2)

• Attracting cycles in p-adic dynamics and height bounds for post-critically finite maps
• Multiplier Spectra and the Moduli Space of Degree 3 Morphisms on P1

Cited by in corpus (2)

• Rigidity and height bounds for certain post-critically finite endomorphisms of projective space
• The McMullen Map in Positive Characteristic