In this chapter we look at non-compact subdomains of the unit disk and show that, even if they are degenerate (Bloch), there are iterated function systems with more than one constant limit function.

We then show that, if the subdomain is non-Bloch, there are iterated function systems with non-constant limits and that, in fact, any function may be realized as the limit of an iterated function system.

By the Riemann mapping theorem and Theorem 12.3.1 the results in this chapter hold for subdomains of any simply connected hyperbolic domain.

Uniqueness of limits

In this section we show that for any integer n > 1, and any non-relatively-compact subdomain X ⊂ Δ, there is an iterated function system that has n arbitrarily chosen distinct accumulation points. We prove

Theorem 13.1.1Let X be any subdomain of Δ that is not relatively compact and let c0, c1, …, cn−1be n distinct points in X. There is an iterated function system that has at least n distinct accumulation points G0, G1, …, Gn−1 and Gi(0) = ci, i = 0, …, n − 1. If X is Bloch these accumulation points are constant and there are no other accumulation points.

The key lemma

In this section we prove a lemma that is the crux of the proof of Theorem 13.1.1.

Lemma 13.1.1Let X be any non-relatively-compact subset of Δ, and, for any fixed n, let a1, …, anbe any distinct points in Δ\{0}. Then there exist a degree-(n − 1) rational function f : Δ→Δ and points x1, …, xn ∈ X such that, for all i = 1, …, n, f(xi) = ai/xi.