15 - Dimensions of Julia sets of transcendental meromorphic functions

Summary

Abstract. We look at the relationships between the different dimensions that can be used to describe the size of the Julia set of a transcendental meromorphic function and also look at the different values that these dimensions can take. We summarise the main results in this area and indicate some of the techniques that are used.

INTRODUCTION

Let f be a meromorphic function which is not rational of degree one and denote by fⁿ; n ∈ ℕ, the nth iterate of f. The Fatou set, F(f), is defined to be the set of points, z ∈ C, such that (fⁿ)_n∈N is well-defined, meromorphic and forms a normal family in some neighbourhood of z. The complement, J(f), of F(f) is called the Julia set of f. An introduction to the properties of these sets can be found in, for example, [2] for rational functions and in [3] for transcendental meromorphic functions.

Most Julia sets are very intricate and their size cannot be described well in terms of classical geometry. There are a number of definitions of dimension that are well suited to describing the size of fractals such as Julia sets and in Section 2 we discuss the advantages and disadvantages of these different dimensions.

In Section 3 we look at the relationships between these different dimensions — when they give the same value for the size of a Julia set and when they give different values.