On the family of cubic parabolic polynomials

Abstract

For a sequence (an) of complex numbers we consider the cubic parabolic polynomials fn(z) = z 3 + anz 2 + z and the sequence (Fn) of iterates Fn = fn • • • • • f 1 . The Fatou set F 0 is the set of all z ∈Ĉ such that the sequence (Fn) is normal. The complement of the Fatou set is called the Julia set and denoted by J 0 . The aim of this paper is to study some properties of J 0 . As a particular case, when the sequence (an) is constant, an = a, then the iteration Fn becomes the classical iteration f n where f (z) = z 3 + az 2 + z. The connectedness locus, M , is the set of all a ∈ C such that the Julia set is connected. In this paper we investigate some symmetric properties of M as well.1. Introduction. For a sequence, (a n ) of complex numbers, we consider the following sequence of cubic parabolic polynomialsand their generalized iteration (F n ) which is the forward compositionsThe parabolic name is due to the fact that f n (0) = 0 and f n (0) = 1 for all n ∈ N.Note that for a constant sequence (a n ), this is, a n = c for all n, we have f n = f for all n ∈ N and the generalized iteration (F n ) is the classical Fatou-Julia iteration theory, withFor classical iteration theory of a fixed function we refer the reader to [10], [13] and [19]. Let (f n ) be a sequence of cubic parabolic polynomials associated with the sequence of complex numbers (a n ), and let (F m ) be the respective sequence of forward compositions. The Fatou set associated with the sequence (f n ), denoted here by F 0 , is defined as the set of all z ∈ C such that (F n ) is normal in some neighbourhood of z, that is, F 0 := {z ∈ C; (F m ) is normal in some neighbourhood of z} .