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

Abstract

The parameter space Sp for monic centered cubic polynomialmaps with a marked critical point of period p is a smooth affine algebraiccurve whose genus increases rapidly with p. Each Sp consists of a compactconnectedness locus together with finitely many escape regions, each of whichis biholomorphic to a punctured disk and is characterized by an essentiallyunique Puiseux series. This note will describe the topology of Sp, and of itssmooth compactification, in terms of these escape regions. In particular, itcomputes the Euler characteristic. It concludes with a discussion of the realsub-locus of Sp. © 2010 American Mathematical Society.