Preperiodic dynatomic curves for z-\textgreater{}z^d+c

Abstract

We study the preperiodic dynatomic curves $\mathcal{X}\_{n,p}$, the closure of set of $(c,z)\in \C^2$ such that $z$ is a preperiodic point of $f\_c$ with preperiod $n$ and period $p$ ($n,p\geq1$). We prove that each $\mathcal{X}\_{n,p}$ has exactly $d-1$ irreducible components, these components are all smooth and intersect pairwisely at the singular points of $\mathcal{X}\_{n,p}$. For each component, we calculate the genus of its compactification and then give a complete topological description of $\mathcal{X}\_{n,p}$. We also calculate the Galois group of the defining polynomial of $\mathcal{X}\_{n,p}$.