An Algebraic Proof of Thurston's Rigidity for a Polynomial

Details An Algebraic Proof of Thurston's Rigidity for a Polynomial An Algebraic Proof of Thurston's Rigidity for a Polynomial

2012-01-10

arxiv.org/abs/1201.1969v1

Mathematics

Algebraic Geometry

16 pages

Scientific paper

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 polynomial maps, valid over any field whose characteristic is zero or larger than the degree of the map. We also produce counterexamples when the characteristic of the field is positive and smaller than the degree.

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