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

TL;DR

This paper investigates the structure and topology of preperiodic dynatomic curves for the map z→z^d+c, revealing their irreducible components, genus, singularities, and Galois groups, thus deepening understanding of complex dynamical systems.

Contribution

It provides a detailed analysis of the irreducible components, genus, and Galois groups of preperiodic dynatomic curves for polynomial maps, which was previously not fully understood.

Findings

Each dynatomic curve has exactly d-1 irreducible components.

All components are smooth and intersect at singular points.

The genus of each component's compactification is explicitly calculated.

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 $\mathcal{X}\_{n,p}$.