Cusp algebras

By a cusp V we shall mean the image of the unit disk D under a bounded injective holomorphic map h into C whose derivative vanishes at exactly one point. The simplest example is the Neil parabola, given by h(ζ) = (ζ, ζ). See [3, 4] for background and theory on the Neil parabola, which is pictured in Figure 1. A generalization of a cusp is a petal. A petal is the image of the unit disk D under a proper holomorphic map h from D into some bounded open set Ω, where, except for a finite set Eh, h is one-to-one and non-singular. If Eh is a singleton, then V is a cusp. Since the automorphism group of D is transitive, we may assume then that Eh = {0}. The function h : D → Ω induces a finite codimensional subalgebra Ah of the algebra O(D) of all analytic functions on D, namely