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

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 ${\mathcal{S}}_p$.

51 pages, 16 figures