Cubic Polynomial Maps with Periodic Critical Orbit, Part II: Escape Regions

TL;DR

This paper studies the structure of parameter spaces for cubic polynomial maps with a critical point of a fixed period, focusing on the topology of escape regions and their role in the compactification of these spaces.

Contribution

It provides a detailed topological description of the parameter space and its compactification, emphasizing the role of escape regions characterized by Puiseux series.

Findings

Topology of escape regions described as punctured disks

Compactification of parameter space characterized by escape regions

Discussion of the real sub-locus of the parameter space

Abstract

The parameter space $\mathcal{S}_p$ for monic centered cubic polynomial maps with a marked critical point of period $p$ is a smooth affine algebraic curve whose genus increases rapidly with $p$. Each $\mathcal{S}_p$ consists of a compact connectedness locus together with finitely many escape regions, each of which is biholomorphic to a punctured disk and is characterized by an essentially unique Puiseux series. This note will describe the topology of $\mathcal{S}_p$, and of its smooth compactification, in terms of these escape regions. It concludes with a discussion of the real sub-locus of $\mathcal{S}_p$.