The Ahlfors five islands theorem has become an important tool in complex dynamics. We discuss its role there, describing how it can be used to deal with a variety of problems. This includes questions concerning the Hausdorff dimension of Julia sets, the existence of singleton components of Julia sets, and the existence of repelling periodic points. We point out that for many applications a simp...

Editor’s Note: This is the second installment of a two-part post by Amitai Etzioni examining the nation’s anti-obesity policies through the lens of a responsive communitarian philosophy. Yesterday, Etzioni laid out a responsive communitarian framework and used it to diagnose the problems with our current methods of fighting obesity. Today, Etzioni describes how these current policies should be ...

We show that the Hausdorff dimension of Julia sets in any analytic family of semihyperbolic generalized polynomial-like mappings (GPL) depends in a real-analytic manner on the parameter. For the proof we introduce abstract weakly regular analytic families of conformal graph directed Markov systems. We show that the Hausdorff dimension of limit sets in such families is real-analytic, and we asso...

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 ...

In this paper, we give a formula of the Hausdorff dimension of the boundary of the immediate basin of infinity of McMullen maps fp(z) = z Q + p/z, where Q ≥ 3 and p is small. This gives a lower bound of the Hausdorff dimension of the Julia sets of McMullen maps in the special cases.

A transcendental entire function f is called geometrically finite if the intersection of the set S(f) of singular values with the Fatou set F(f) is compact and the intersection of the postsingular set P (f) with the Julia set J (f) is finite. (In particular, this includes all entire functions with finite postsingular set.) If f is geometrically finite, then F(f) is either empty or consists of t...

For 0 < λ < 1/e the Julia set of λez is an uncountable union of pairwise disjoint simple curves tending to infinity [Devaney and Krych 1984], the Hausdorff dimension of this set is two [McMullen 1987], but the set of curves without endpoints has Hausdorff dimension one [Karpińska 1999]. We show that these results have three-dimensional analogues when the exponential function is replaced by a qu...

Let f be a transcendental entire function in the Eremenko-Lyubich class B. We give a lower bound for the Hausdorff dimension of the Julia set of f that depends on the growth of f . This estimate is best possible and is obtained by proving a more general result concerning the size of the escaping set of a function with a logarithmic tract.

In this paper we discuss the stability of Julia sets and lled{in Julia sets of functions meromorphic in the complex plane, i. e. rational or transcendental functions or polynomials. To this end, we illustrate relations to the appearence of repellors and attractors and study some chaotic features of Julia sets. The main results are: The Julia set is stable if it is a repellor and a lled{in Julia...

In a series of papers, Bandt and the author have given a symbolic and topological description of locally connected quadratic Julia sets by use of special closed equivalence relations on the circle called Julia equivalences. These equivalence relations reflect the landing behaviour of external rays in the case of local connectivity, and do not apply completely if a Julia set is connected but fai...