Cubic polynomial maps with periodic critical orbit, Part II: Escape regions

Abstract

The parameter space S p \mathcal {S}_p for monic centered cubic polynomial maps with a marked critical point of period p p is a smooth affine algebraic curve whose genus increases rapidly with p p . Each S p \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 S p \mathcal {S}_p , and of its smooth compactification, in terms of these escape regions. In particular, it computes the Euler characteristic. It concludes with a discussion of the real sub-locus of S p \mathcal {S}_p .