The parameter space Spβ 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 Spβ 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
Spβ, and of its smooth compactification, in terms of these escape
regions. It concludes with a discussion of the real sub-locus of
Spβ.Comment: 51 pages, 16 figure