Rational maps with a preperiodic critical point

2018/06/28 by Buff, Xavier, Epstein, Adam L., Koch, Sarah
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1806.11221

Abstract

We show that the set of conjugacy classes of cubic polynomials with a prefixed critical point, of preperiod k≥ 1, is an irreducible algebraic curve. We also establish an analogous result for quadratic rational maps. We then study a closely related question concerning the irreducibility (over \mathbb Q) of the set of conjugacy classes of unicritical polynomials, of degree D≥ 2, with a preperiodic critical point. Our proofs are purely algebraic.

Related

• Irreducibility of periodic curves in cubic polynomial moduli space
• The Complete Classification of Rational Preperiodic Points of Quadratic Polynomials over Q: A Refined Conjecture
• On irreducibility of prefixed algebraic sets in moduli spaces of prime degree polynomials
• Irreducibility of eventually 2-periodic curves in the moduli space of cubic polynomials
• Distribution of rational maps with a preperiodic critical point