Classification of special curves in the space of cubic polynomials

2016/03/16 by Favre, Charles, Gauthier, Thomas · 3 citations
#37F45 #37P45 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1603.05126

Abstract

We describe all special curves in the parameter space of complex cubic polynomials, that is all algebraic irreducible curves containing infinitely many post-critically finite polynomials. This solves in a strong form a conjecture by Baker and DeMarco for cubic polynomials. We also prove that an irreducible component of the algebraic curve consisting of those cubic polynomials that admit an orbit of any given period and given multiplier is special if and only if the multiplier is 0.

Cited by

• Rigidity in Complex Dynamics: Multiplier Spectrum and Dynamical André-Oort Conjecture
• On intertwined polynomials
• On simultaneously preperiodic points for one-parameter families of polynomials in characteristic p

Related

• The geometry of the critically-periodic curves in the space of cubic polynomials
• Irreducibility of periodic curves in cubic polynomial moduli space
• Special curves and postcritically-finite polynomials
• Irreducibility of eventually 2-periodic curves in the moduli space of cubic polynomials
• Relations between Escape Regions in the Parameter Space of Cubic Polynomials