The dynatomic curves for unimodel polynomials are smooth and irreducible

Yan Gao (1) , Ya Fei Ou (1)

1 LAREMA - Laboratoire Angevin de Recherche en Mathématiques

We prove here the smoothness and the irreducibility of the periodic dynatomic curves $ (c,z)\in \C^2$ such that $z$ is $n$-periodic for $z^d+c$, where $d\geq2$. We use the method provided by Xavier Buff and Tan Lei in \cite{BT} where they prove the conclusion for $d=2$. The proof for smoothness is based on elementary calculations on the pushforwards of specific quadratic differentials, following Thurston and Epstein, while the proof for irreducibility is a simplified version of Lau-Schleicher's proof by using elementary arithmetic properties of kneading sequence instead of internal addresses.

Mots clés: dynatomic curve; irreducible; kneading sequence

Domaines: Systèmes dynamiques [math.DS]

Yan Gao, Ya Fei Ou. The dynatomic curves for unimodel polynomials are smooth and irreducible. 2012. ⟨hal-00814484⟩