We show that if a polynomial filled Julia set has empty interior, then it is computable.

We discuss the connectivity properties of Julia sets of Weierstrass elliptic ℘ functions, accompanied by examples. We give sufficient conditions under which the Julia set is connected and show that triangular lattices satisfy this condition. We also give conditions under which the Fatou set of ℘ contains a toral band and provide an example of an order two elliptic function on a square lattice w...

In this paper we settle most of the open questions on algorithmic computability of Julia sets. In particular, we present an algorithm for constructing quadratics whose Julia sets are uncomputable. We also show that a filled Julia set of a polynomial is always computable.

In this article, we introduce the concept of normal families of bicomplex holomorphic functions to obtain a bicomplex Montel theorem. Moreover, we give a general definition of Fatou and Julia sets for bicomplex polynomials and we obtain a characterization of bicomplex Fatou and Julia sets in terms of Fatou set, Julia set and filled-in Julia set of one complex variable. Some 3D visual examples o...

Sequential rendering methods for the graphical representation of Julia sets are compared. Two groups of methods are presented. In the rst, the attractor of the Julia set is rendered and, in the second, the complement of the attractor is rendered. Examples of images obtained using these methods are also given.

The aim of this paper is to present an analytic characterization of the class of functions δ-spirallike with respect to a boundary point. The method of proof is based on Julia's lemma.

We construct a combinatorial model of the Julia set of the endomorphism f(z, p) = ((1 − 2z/p), (1− 2/p)) of PC.

We prove that the only possible biaccessible points in the Julia set of a Cremer quadraticpolynomialare the Cremer xed point and its preimages. This gives a partial answer to a question posed by C. McMullen on whether such a Julia set can contain any biaccessible point at all.

It is known that, for a transcendental entire function f, the Hausdorff dimension of J( f ) satisfies 1% dim J( f )% 2. For each d ` (1, 2), an example of a transcendental entire function f with dim J( f ) ̄ d is given. It is then indicated how this function can be modified to produce a transcendental meromorphic function F with one pole with dim J(F ) ̄ d. These appear to be the first examples o...

Let f be a transcendental entire function for which the set of critical and asymptotic values is bounded. The Denjoy-Carleman-Ahlfors theorem implies that if the set of all z for which |f(z)| > R has N components for some R > 0, then the order of f is at least N/2. More precisely, we have log logM(r, f) ≥ 1 2 N log r − O(1), where M(r, f) denotes the maximum modulus of f . We show that if f doe...