Classification of Special Curves in the Space of Cubic Polynomials

Charles Favre (1, 2) , Thomas Gauthier (3)

1 CMLS - Centre de Mathématiques Laurent Schwartz
2 CNRS - Centre National de la Recherche Scientifique
3 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352

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 multiplier is special if and only if the multiplier is 0.

Domaines: Systèmes dynamiques [math.DS]; Variables complexes [math.CV]; Théorie des nombres [math.NT]

Charles Favre, Thomas Gauthier. Classification of Special Curves in the Space of Cubic Polynomials. International Mathematics Research Notices, 2018, ⟨10.1093/imrn/rnw245⟩. ⟨hal-01289586⟩