The Polynomials Associated With A Julia Set

We prove that, with two exceptions, the set of polynomials with Julia set J has the form {σ pn : n ∈ N , σ ∈ Σ} , where p is one of these polynomials and Σ is the symmetry group of J . The exceptions occur when J is a circle or a straight line segment. Several papers [1, 2, 3, 5] have appeared dealing with the relation between polynomials having the same Julia set J (for notation the reader is referred to [8]). This relation is very simple and by no means surprising: Theorem To any Julia set (of a polynomial) J , which is not a circle or a straight line segment, there exists a polynomial p such that any polynomial with Julia set J can be written in the form σ pn , where σ is a rotation mapping J onto itself, and n is a positive integer.