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Preperiodic dynatomic curves for $z\mapsto z^d+c$

Volume 233 / 2016

Yan Gao Fundamenta Mathematicae 233 (2016), 37-69 MSC: 14H50, 37F45, 37F10. DOI: 10.4064/fm91-12-2015 Published online: 11 December 2015


The preperiodic dynatomic curve $\mathcal {X}_{n,p}$ is the closure in $\mathbb C^2$ of the set of $(c,z)$ such that $z$ is a preperiodic point of the polynomial $z \mapsto z^d+c$ with preperiod $n$ and period $p$ ($n,p\geq 1$). We prove that each $\mathcal {X}_{n,p}$ has exactly $d-1$ irreducible components, which are all smooth and have pairwise transverse intersections at the singular points of $\mathcal {X}_{n,p}$. We also compute the genus of each component and the Galois group of the defining polynomial of $\mathcal {X}_{n,p}$.


  • Yan GaoMathematical School of Sichuan University
    ChengDu, 610065, P.R. China

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