In an article recently appearing in the Acta Mathematica Hungarica [15], J. P. R. Christensen and P. Fischer develop a method to produce invariant measures on the Julia set of any non-linear entire function f. Using Ahlfors' three domains theorem (Theorem 1.9), a powerful result from Ahlfors' theory of covering spaces, Christensen and Fischer give a conjugacy between the shift operator T on the shift space and the function F=f^N on a subset of the Julia set J(f), for some large value of N. This allows them to project shift-invariant measures on onto to produce invariant non-discrete probability measures on the Julia set J(f). Although never explicitly mentioned, the construction of these invariant measures on the Julia set parallels the construction of invariant measures on the attractor of a totally disconnected IFS.

This master's project is presented in response to questions motivated by Christensen and Fischer's article. In particular:

1. What precisely is the relationship between Julia sets and iterated function systems? That is, what are the qualities of the Julia set of an entire function f that allow it to be described as the attractor of an IFS?
2. Are there other examples of constructions that describe a subset of the Julia set as the attractor of a totally disconnected IFS?
3. Without delving too deeply into Ahlfors' theory of covering spaces, where does the three domains theorem fit into the theory of complex analysis? Where else is this theorem used in iteration theory?
4. Can the IFS machinery be used to construct some especially natural measures on Julia sets which may prove useful for the study of complex analytic dynamics?

The first three questions primarily provide a context for Christensen and Fischer's result --- developing some background and fleshing out the relationships between iterated function systems and complex iteration theory. The final question is of a more original nature. If we have this method for putting invariant measures on the Julia set, which measures should we choose? There are piles of invariant measures for the attractor of an IFS. Are there any that make especially good geometric sense?

To answer these questions, we begin with a development of the two major theories involved. First, iteration theory and the Julia set and then iterated function systems and attractors.

Much of the theory built around the iteration of entire functions was developed by Fatou, extending results obtained for rational functions by Fatou and Julia independently. Many good references for the iteration theory of rational functions are available. Blanchard [10] has an especially good survey of this subject and Brolin's article [14] written in the 1950's is also an excellent reference providing some measure-theoretic results. Two recent surveys which also include results for entire functions are Bergweiler [9] and Eremenko and Lyubich [19]. All of these papers develop the properties of Julia sets in a similar way --- we primarily follow the exposition of Blanchard and Bergweiler. The major difference between iteration theory for rational and entire functions is in the proof that, for a function f, the Julia set is the closure of the repelling periodic points. (See Proposition 1.8.) For entire functions, this theorem was not proved until the 1960's when Baker gave a proof using, interestingly enough, Ahlfors' three domains theorem.

Section 2 gives a general development of the theory of iterated function systems. The iterated function system machinery is fairly new --- it was primarily developed by Hutchinson [28]. Barnsley [7] has used and expanded this theory with applications to fractal image compression. We give their basic development and proofs of several fundamental properties.

To close this section we give the following sufficient condition for finding the attractor of an iterated function system within the Julia set of a non-linear entire function f.

Theorem 1: Let f be an entire function. Suppose that we can find a compact set and inverse branches of f such that each map is a contraction map taking K into K. Then the attractor of the IFS is a subset of the Julia set J(f) and is homeomorphic to the shift space . Moreover, is conjugate to the shift map, i.e., for

This result generalizes the relationship given by Christensen and Fischer and gives us the following goal --- we hope to find entire functions for which multiple contracting inverse branches can be found taking some compact set K into itself.

There are two basic examples in the literature of entire functions f for which J(f) (or some subset thereof) is given as the attractor of an iterated function system. We present these examples in Section 3. First is the family of quadratic functions . This example certainly predates the iterated function system framework, going back to Fatou and Julia's work on rational functions. The case is explicitly given by Falconer [20] --- and we give his proof, immediately extending this result to the case |c|>2 by incorporating the hyperbolic metric.

Figure 3 (from Section 3). Iteration of the IFS based on the quadratic function fc(z)=z2+c,c=-1+2i. The attractor of the IFS is the Julia set J(fc).

Figure 4 (from Section 3). Iteration of the IFS based on the exponential function fd(z)=dez, d =.2, with N=10. The attractor of the IFS is a proper subset of the Julia set J(fd).

For any entire function f, the construction of Christensen and Fischer produces a subset of the Julia set J(f) as the attractor of an IFS, in the manner of Theorem 1. In Section 5, we carry out this construction using the iterated function system framework. The general idea of this construction is demonstrated in a simple example.

Figure 10 (from Section 5). Iteration of an IFS based on f(z)=z2.

Two general measures for fractals are given in Section 6. The first is one that, although it seems very natural, we could not find explicitly in the literature (although it may be there). We call this measure the ``limiting Lebesgue measure''. This is a measure which we construct especially for the attractor of a totally disconnected IFS. This measure puts a mass distribution on the subsets of the attractor according to the normalized restriction of Lebesgue measure on successive approximations to . (See Section 6.1.) The second measure is the ubiquitous Hausdorff measure. Both of these measures are computed for the example of Section 5.

The framework of iterated function systems gives us an easy method for constructing invariant measures supported on the attractor . The general theory of an iterated function system with probabilities is developed in the final section, Section 7. We give the basic results of this theory as developed by Hutchinson [28] and Barnsley [7], and mention some further extensions of this theory and the related theory of g-measures. In particular, there is a whole family of IFS-invariant measures supported on the attractor of an IFS. This family of invariant measures for an IFS is especially useful in light of the following result:

Theorem 2: Let f be a non-linear entire function and suppose that is an IFS with place-dependent probabilities as in Theorem 1. Then if is an invariant measure for the IFS, is also an f-invariant measure.

So in order to specify an f-invariant measure on the Julia set, we need only specify the probability functions p_k(z) and if the functions p_k are sufficiently nice, the measure will be unique. Considering the two measures of the previous section, we give the following two ``naturally'' related invariant measures for any IFS. We will choose the functions p_k(z) to reflect the action of the function f (and its inverses) in a neighborhood of z. In particular, for a point z, we define p_k(z) to indicate how the particular inverse branch distorts volume near the point z, in relation to the other branches. So we set

This measure, although singular with respect to Lebesgue measure, somehow retains a faint memory of how the function f acts with respect to Lebesgue measure. In a similar manner, we can define an invariant measure to give an indication of how the different functions affect Hausdorff measure near the point z. If s_d represents the Hausdorff dimension of , we define

On the Julia set, these measures both are able to shadow, in some sense, the way that the function f distorts measure. We also give a method for constructing an f-invariant measure from the f^N-invariant measure of Christensen and Fischer's construction.