COMPLETELY INVARIANT SETS OF NORMALITY FOR RATIONAL SEMIGROUPS

RICH STANKEWITZ

Abstract. Let G be a semigroup of rational functions of degree at least two where the semigroup operation is composition of functions. We prove that the largest open subset of the Riemann sphere on which the semigroup G is normal and is completely invariant under each element of G, can have only 0, 1, 2, or infinitely many components.

1. Introduction

It is well known in iteration theory that the set of normality of a rational function can have only 0, 1, 2, or infinitely many components (see [1], p. 94). In this paper we generalize this result by showing that the completely invariant set of normality of a rational semigroup can have only 0, 1, 2, or infinitely many components. The proof not only generalizes the iteration result, but it also provides an alternative proof for it.

The results of this paper appear as part of the author’s Ph.D. thesis, written under the supervision of Aimo Hinkkanen.

2. Definitions and basic facts

In what follows all notions of convergence will be with respect to the spherical metric on the Riemann sphere \overline{ℂ}.

A rational semigroup G is a semigroup of rational functions of degree greater than or equal to two defined on the Riemann sphere \overline{ℂ} with the semigroup operation being functional composition. When a semigroup G is generated by the functions {f_{1}, f_{2}, …, f_{n}, … }, we write this as

G = ⟨f_{1}, f_{2}, …, f_{n}, … ⟩.

In [3], p. 360 the definitions of the set of normality, often called the Fatou set, and the Julia set of a rational semigroup are as follows:

Definition 1. For a rational semigroup G we define the set of normality of G, N(G), by

N(G) = \{z ∈ \overline{ℂ} : there is a neighborhood of z on which G is a normal family\}

and define the Julia set of G, J(G), by

J(G) = \overline{ℂ} \ N(G).

1991 Mathematics Subject Classification: Primary 30D05, 58F23. Key words and phrases.

Rational semigroups, completely invariant sets, Julia sets. Research supported by a Department of Education GAANN fellowship and by the Research Board of the University of Illinois at Urbana-Champaign.

Clearly from these definitions we see that N(G) is an open set and therefore its complement J(G) is a compact set. These definitions generalize the case of iteration of a single rational function and we write N(⟨h⟩) = N_{h} and J(⟨h⟩) = J_{h}.

Note that J(G) contains the Julia set of each element of G. In fact, we have J(G) = \overline{∪_{f∈G}J_{f}} (see [3], p. 365).

Definition 2. If h is a map of a set Y into itself, a subset X of Y is:

i) forward invariant under h if h(X) ⊂ X;
ii) backward invariant under h if h^{−1}(X) ⊂ X;
iii) completely invariant under h if h(X) ⊂ X and h^{−1}(X) ⊂ X.

It is well known that the set of normality of h and the Julia set of h are completely invariant under h (see [1], p. 54), in fact,

h(N_{h}) = N_{h} = h^{−1}(N_{h}) and h(J_{h}) = J_{h} = h^{−1}(J_{h}). (2.1)

Further, we have the following result.

Property 1. The set J_{h} is the smallest closed completely invariant (under h) set which contains three or more points (see [1], p. 67).

In fact, this may be chosen as an alternate definition of J_{h}, equivalent to that given in Definition 1.

From Definition 1, we get that N(G) is forward invariant under each element of G and J(G) is backward invariant under each element of G (see [3], p. 360). The sets N(G) and J(G) are, however, not necessarily completely invariant under the elements of G. This is in contrast to the case of single function dynamics as noted in (2.1). The question then arises, what if we required the Julia set of the semigroup G to be completely invariant under each element of G? We consider in this paper the consequences of such an extension, given in the following definition.

Definition 3. For a rational semigroup G we define the completely invariant Julia set of G, E = E(G), by

E(G) = ⋂{S : S is closed, completely invariant under each g ∈ G, #(S) ≥ 3}

where #(S) denotes the cardinality of S.

We note that E(G) exists, is closed, is completely invariant under each element of G and contains the Julia set of each element of G by Property 1.

Definition 4. For a rational semigroup G we define the completely invariant set of normality of G, W = W (G), to be the complement of E(G), i.e.,

W (G) = \overline{ℂ} \ E(G).

Note that W (G) is open and it is also completely invariant under each element of G. The main result of this paper is the following theorem.

Theorem 1. For a rational semigroup G the set W (G) can have only 0, 1, 2, or infinitely many components.

3. Proof of the main result

Property 2. If E(G) has nonempty interior, then E(G) = \overline{ℂ}.

Proof. For a collection of sets A, and a function h, we denote new collections of sets by h(A) = {h(A) : A ∈ A} and h^{−1}(A) = {h^{−1}(A) : A ∈ A}.

Choose g ∈ G. Let us define the following countable collections of sets:

E_{0} = {J_{g}},

E_{1} = ⋃_{f∈G} f^{−1}(E_{0}) ∪ ⋃_{f∈G} f(E_{0}),

E_{n+1} = ⋃_{f∈G} f^{−1}(E_{n}) ∪ ⋃_{f∈G} f(E_{n}),

and E = ⋃_{n=0}^{∞} E_{n}.

In the same manner as in the proof of Lemma 1 in [7] we conclude E(G) = \overline{⋃_{A∈E} A}. Using this result we then finish the proof in the same manner as done in the proof of Lemma 2 in [7].

The remainder of this section will be devoted to the proof of Theorem 1.

Lemma 1. If W_{0} is a component of W, then f(W_{0}) is also a component of W for any f ∈ G.

Proof. Let W_{1} be the component of W that contains f(W_{0}). We show that f(W_{0}) = W_{1}. Suppose to the contrary that z ∈ W_{1} \ f(W_{0}). Since f is continuous on the compact set \overline{W_{0}} and an open map on W_{0}, we have ∂f(W_{0}) ⊂ f(∂W_{0}) ⊂ f(E) ⊂ E. Let γ be a path in W_{1} connecting z to a point w ∈ f(W_{0}). Hence γ must cross ∂f(W_{0}) ⊂ E. This contradicts the fact that γ ⊂ W_{1} and so we conclude that f(W_{0}) = W_{1}.

Since the remainder of this section will be devoted to the proof of Theorem 1, we will assume that W has L components where 2 ≤ L < +∞. We remark here that the strategy will be to show that each of the L components of W is simply connected and then the result will follow by an application of the Riemann-Hurwitz relation.

Definition 5. Let W have components W_{j} for j = 0, …, L − 1.

Remark 1. We see by Lemma 1 that each f ∈ G (and hence each f^{−1} as well) permutes the W_{j} for j = 0, …, L − 1 since f is a continuous map of W onto W.

We may assume that ∞ ∈ W_{0}, else we may impose this condition by conjugating each f ∈ G by the same rotation of the sphere.

Definition 6. For j = 1, …, L − 1, we define

K_{j} = {z ∉ W_{j} : there exists a simple closed curve γ ⊂ W_{j} such that Ind_{γ}(z) = 1}

where the winding number is given by Ind_{γ}(z) = (1/2πi) ∫_{γ} 1/(w−z) dw. If z ∈ K_{j} and the simple closed curve γ ⊂ W_{j} is such that Ind_{γ}(z) = 1, then we say that γ works for z ∈ K_{j}.

In order to properly define K_{0} we first need to move W_{0} so that it no longer contains ∞. Let ϕ be a rotation of the sphere so that ∞ ∈ ϕ(W_{1}) and denote W̃_{j} = ϕ(W_{j}) for j = 0, …, L − 1.

Definition 7. We define

K̃_{0} = {z ∉ W̃_{0} : there exists a simple closed curve γ ⊂ W̃_{0} such that Ind_{γ}(z) = 1}

and

K_{0} = ϕ^{−1}(K̃_{0}).

If z ∈ K_{0} and simple closed curve γ ⊂ W̃_{0} is such that Ind_{γ}(ϕ(z)) = 1, then we say that the simple closed curve ϕ^{−1}(γ) works for z ∈ K_{0}.

Remark 2. Note that saying ϕ^{−1}(γ) works for z ∈ K_{0} does not necessarily imply that Ind_{ϕ^{−1}(γ)}(z) = 1, since it may be the case that Ind_{γ}(ϕ(∞)) = 1 and hence Ind_{ϕ^{−1}(γ)}(z) = 0 since z lies in the unbounded component of \overline{ℂ} \ ϕ^{−1}(γ).

Definition 8. We define

K = ⋃_{j=0}^{L−1} K_{j}.

Definition 9. We define

W′_{j} = W_{j} ∪ K_{j}.

Lemma 2. For j = 0, …, L − 1, the set W′_{j} is open, connected and simply connected. Thus each K_{j} is the union of the “holes” in W_{j}.

Proof. Suppose that 1 ≤ j ≤ L−1, so that W_{j} is a bounded domain in the complex plane. Define A to be the unbounded component of \overline{ℂ} \ W_{j}. Hence B = \overline{ℂ} \ A is open, connected and simply connected.

Let F be a bounded component of \overline{ℂ}\W_{j}. Since A and F are each components of the closed set \overline{ℂ}\W_{j}, there exists a simple polygon γ ⊂ W_{j} which separates A from F (see [5], p. 134). Hence we see that F ⊂ K_{j}. Since F was an arbitrary bounded component of \overline{ℂ} \ W_{j}, we conclude that K_{j} contains all the bounded components of \overline{ℂ} \ W_{j}, i.e., the “holes” of W_{j}. Hence W′_{j} ⊃ B. Clearly K_{j} cannot contain any points of A since any simple closed path γ ⊂ W_{j} which would wind around such a point would have to necessarily wind around every point of A (since A is a component of the complement of W_{j}) including ∞ which cannot happen. Hence we conclude W′_{j} = B and is therefore open, connected and simply connected.

We show that ϕ(W′_{0}) is open, connected and simply connected using the same argument as above, and this implies that W′_{0} is open, connected and simply connected.

Definition 10. We define

W′ = ⋃_{j=0}^{L−1} W′_{j}.

Note that we have W′ = W ∪ K.

Lemma 3. If for some distinct r, s ∈ {0, …, L − 1}, we have W′_{r} ∩ W′_{s} ≠ ∅, then either \overline{W′_{r}} ⊂ W′_{s} or \overline{W′_{s}} ⊂ W′_{r}. In particular, if W_{r} ∩ W′_{s} ≠ ∅ for some distinct r, s ∈ \{0, …, L − 1\}, then \overline{W′_{r}} ⊂ W′_{s}.

Proof. Let z ∈ W′_{r} ∩ W′_{s}. Since W_{r} ∩ W_{s} = ∅, we may assume that z ∈ K_{s}, say. Let γ_{s} work for z ∈ K_{s}. Let I_{γ_{s}} be the component of \overline{ℂ} \ γ_{s} which contains z. Note that I_{γ_{s}} \ W_{s} = {z : γ_{s} works for z} whether or not s = 0 (see Definitions 6 and 7 and Remark 2). Since z ∈ W′_{r}, we have two cases, either z ∈ K_{r} or z ∈ W_{r}.

Suppose that z ∈ K_{r} and let γ_{r} work for z ∈ K_{r}. As γ_{s} ∩ γ_{r} = ∅ (since W_{r} ∩W_{s} = ∅) we see that either γ_{r} ⊂ I_{γ_{s}} or γ_{s} ⊂ I_{γ_{r}}, where I_{γ_{r}} is the component of \overline{ℂ} \ γ_{r} which contains z. By switching the roles of r and s, if necessary, we assume γ_{r} ⊂ I_{γ_{s}} and we note that this can be done since z ∈ K_{r} ∩ K_{s}. In particular, W_{r} ∩ I_{γ_{s}} ≠ ∅.

If z ∈ W_{r}, then we still get W_{r} ∩ I_{γ_{s}} ≠ ∅ since z ∈ I_{γ_{s}}.

Since W_{r} ∩ I_{γ_{s}} ≠ ∅, W_{r} ∩ W_{s} = ∅, \overline{W_{r}} is connected, and γ_{s} ⊂ W_{s}, we conclude that \overline{W_{r}} ⊂ I_{γ_{s}}. Hence \overline{W_{r}} ⊂ W′_{s} since γ_{s} then works for every z ∈ \overline{W_{r}}. Since W′_{s} is simply connected we see that \overline{W′_{r}} ⊂ W′_{s}.

Lemma 4. The boundary of W′_{0} is a nondegenerate continuum and as such contains more than three points.

Proof. We will first show that W′_{0} ∩ W′_{1} = ∅. The set W′_{1} cannot contain W′_{0} as ∞ ∈ W′_{0} and W′_{1} is a bounded subset of ℂ (since W_{1} is a bounded subset of ℂ). The same argument also shows that ϕ(W′_{0}) cannot contain ϕ(W′_{1}) where ϕ is as in Definition 7, and so we conclude that W′_{0} cannot contain W′_{1}. By Lemma 3 we conclude that W′_{0} ∩ W′_{1} = ∅.

Since W′_{0} is simply connected, ∂W′_{0} contains a nondegenerate continuum unless ∂W′_{0} consists of just a single point. If ∂W′_{0} consists of just a single point, then W′_{0} ∪ ∂W′_{0} = \overline{ℂ}, but this contradicts the fact that W′_{0} ∩ W′_{1} = ∅.

Lemma 5. For each j = 0, …, L − 1, we have J_{f} ⊂ ∂W_{j} for each f ∈ G. Since J(G) = \overline{∪_{f∈G}J_{f}}, we have J(G) ⊂ ∂W_{j} for each j = 0, …, L − 1.

Proof. Since f permutes the W_{j} by Remark 3, we may select a positive integer n so that f^{n}(W_{j}) = W_{j} = f^{−n}(W_{j}) for each j = 0, …, L − 1. Then we have \overline{⋃_{k=1}^{∞} f^{−kn}(W_{j})} ⊃ J_{f^{n}} = J_{f} (see [1], p.71 and p.51). But since \overline{⋃_{k=1}^{∞} f^{−kn}(W_{j})} = \overline{W_{j}} we see that ∂W_{j} ⊃ J_{f}, since W_{j} ∩ J_{f} = ∅.

Lemma 6. We have W_{r} ⊈ W′_{s} for distinct r, s ∈ {0, …, L − 1}, and therefore, by Lemma 3, the W′_{j} are disjoint for j = 0, …, L − 1.

Proof. If L = 2, then the proof of Lemma 4 shows that W′_{0} ∩ W′_{1} = ∅.

We assume now that L ≥ 3. We will first show that no bounded W′_{s} can contain any W_{r} with r ≠ s. Suppose that this does occur. Then there exists a simple closed curve γ_{s} ⊂ W_{s} such that \overline{W_{r}} ⊂ I_{γ_{s}} where I_{γ_{s}} is the component of \overline{ℂ} \ γ_{s} which contains the points z such that Ind_{γ_{s}}(z) = 1. Hence, by Lemma 5, J(G) ⊂ ∂W_{r} ⊂ \overline{W_{r}} ⊂ I_{γ_{s}}. But since \overline{W_{0}} ⊂ \overline{ℂ} \ I_{γ_{s}} we see that J(G) ⊂ ∂W_{0} ⊂ \overline{W_{0}} ⊂ \overline{ℂ} \ I_{γ_{s}}. This contradiction implies no bounded W′_{s} can contain any W_{r}.

We see that W′_{0} cannot contain any W_{r} with r ≥ 1 by the following similar argument. If \overline{W_{r}} ⊂ W′_{0}, then there exists a simple closed curve γ ⊂ W̃_{0} such that Ind_{γ}(z) = 1 for every z ∈ \overline{W̃_{r}}. Let I_{γ} be the component of \overline{ℂ}\γ which contains \overline{W̃_{r}}.

So ϕ(J(G)) ⊂ ϕ(∂W_{r}) = ∂ϕ(W_{r}) = ∂W̃_{r} ⊂ I_{γ}. Since \overline{W̃_{1}} ⊂ \overline{ℂ} \ I_{γ} (recall ∞ ∈ W̃_{1}), we see that ϕ(J(G)) ⊂ ϕ(∂W_{1}) = ∂ϕ(W_{1}) = ∂W̃_{1} ⊂ \overline{ℂ} \ I_{γ}. This contradiction implies W′_{0} cannot contain any W_{r} with r ≥ 1.

Corollary 1. The set K has no interior and therefore each K_{j} ⊂ ∂W_{j}.

Proof. By Lemma 6 we see that each K_{j} ⊂ E and hence K ⊂ E. The Corollary then follows from Property 2.

Corollary 2. We have ∂W_{j} = K_{j} ∪ ∂W′_{j}.

Proof. By Corollary 1 we get K_{j} ∪ ∂W′_{j} ⊂ ∂W_{j}. We also have ∂W_{j} = \overline{W_{j}} \ W_{j} ⊂ \overline{W′_{j}} \ W_{j} = (W′_{j} ∪ ∂W′_{j}) \ W_{j} = (W_{j} ∪ K_{j} ∪ ∂W′_{j}) \ W_{j} = K_{j} ∪ ∂W′_{j}.

Lemma 7. We have f(K) ⊂ K for all f ∈ G.

Proof. Let z ∈ K_{j} be such that γ ⊂ W_{j} works for z.

Suppose that W_{l} = f(W_{j}) ≠ W_{0}. So W′_{j} contains no poles of f, else such a pole would be in W_{j} (by the complete invariance of W under f since ∞ ∈ W_{0} ⊂ W and Lemma 6) and hence f(W_{j}) = W_{0}. By the argument principle, f(γ) ⊂ W_{l} winds around f(z), thus f(z) ∈ K_{l} as f(z) ∉ W_{l} by the complete invariance of W under the map f. Note that f(γ) might not work for f(z) ∈ K_{l} since it might not be simple, but f(z) ∈ K_{l} since it cannot be in the unbounded component of \overline{ℂ} \ W_{l} and have a curve in W_{l}, namely f(γ), wind around it.

Now suppose that f(W_{j}) = W_{0}. So (ϕ ∘ f)(W_{j}) = W̃_{0} is bounded and W′_{j} contains no poles of ϕ ∘ f (else f(W_{j}) = W_{1}). So (ϕ ∘ f)(γ) winds around (ϕ ∘ f)(z) and hence (ϕ ∘ f)(z) ∈ K̃_{0}, i.e., f(z) ∈ K_{0}.

So f(K_{j}) ⊂ K and hence we conclude f(K) ⊂ K.

Lemma 8. We have for all f ∈ G, f(W′) ∩ ∂W′_{0} = ∅. Also W′ ⊂ N(G) and in particular K ∩ J(G) = ∅.

Proof. We have f(W′) = f(W ∪ K) = f(W) ∪ f(K) ⊂ W ∪ K = W′. Since W′ ∩ ∂W′_{0} = ∅ (since W′ is open), Lemma 4 and Montel’s Theorem finish the proof.

Corollary 3. We have J(G) ⊂ ∂W′_{j} for each j = 0, …, L − 1.

Proof. This follows immediately from Lemma 5, Corollary 2 and Lemma 8.

Remark 3. It is of interest to note that for any positive integer n there exist disjoint simply connected domains D_{1}, …, D_{n} in \overline{ℂ} with ∂D_{1} = ∂D_{2} = ⋯ = ∂D_{n} (see [4], p. 143). Thus Corollary 3 does not imply that L < 3 from a purely topological perspective.

Lemma 9. We have f^{−1}(K) ⊂ K for all f ∈ G. Hence by Lemma 7, K is completely invariant under each f ∈ G.

Proof. Let z ∈ K_{j} ⊂ ∂W_{j} and say f(w) = z. Define W_{k} = f^{−1}(W_{j}) by Remark 3. We obtain sequences z_{n} ∈ W_{j} such that z_{n} → z, and w_{n} ∈ W_{k} such that w_{n} → w and f(w_{n}) = z_{n}. Hence we see that w ∈ ∂W_{k}, else w ∈ W_{k} and z = f(w) ∈ W_{j}. If w ∉ K_{k}, then w ∈ ∂W′_{k} by Corollary 2. Let Γ be the component of ∂W_{j} that contains f(∂W′_{k}). Since z ∈ Γ, the set Γ must be one of the components of K_{j}. By Corollary 3 we see that there exists a ζ ∈ ∂W′_{k} ∩ J_{f}. Hence f(ζ) ∈ K_{j} ∩ J_{f} which is a contradiction since we know by Lemma 8 that K is disjoint from J(G) ⊃ J_{f}. This contradiction implies w ∈ K_{k} and hence f^{−1}(K) ⊂ K.

Lemma 10. If W has L components where 2 ≤ L < +∞, then each is simply connected.

Proof. Since K and W are each completely invariant under each f ∈ G, so is W′ = W ∪ K. By Lemma 8 we see that \overline{ℂ} \ W′ is completely invariant under each f ∈ G, closed, and contains J(G). Hence E ⊂ \overline{ℂ} \ W′. This implies that W = W′ and hence each component of W is then simply connected.

We are now able to present the proof of Theorem 1.

Proof of Theorem 1. If W has L components where 2 ≤ L < +∞, then each is simply connected by Lemma 10. Select a map f ∈ G. Letting n ≥ 1 be selected so that each of the components W_{j} of W is completely invariant under f^{n}, we get by the Riemann-Hurwitz relation (see [8], p. 7)

δ_{f^{n}}(W_{j}) = deg(f^{n}) − 1

where we write δ_{g}(B) = ∑_{z∈B}[v_{g}(z) − 1] and v_{g}(z) is the valency of the map g at the point z.

Hence we obtain

L(deg(f^{n}) − 1) = ∑_{j=0}^{L−1} δ_{f^{n}}(W_{j}) ≤ δ_{f^{n}}(\overline{ℂ}) = 2(deg(f^{n}) − 1)

and so L ≤ 2. The last equality follows from Theorem 2.7.1 in [1].

Remark 4. Note that if L = 2, then each component of W is necessarily simply connected.

4. Conclusions

We know from iteration theory that each of the four possibilities (0, 1, 2, ∞) for the number of components of the set of normality can be achieved. So by constructing semigroups G such that all the elements have the same Julia set we know that the only four possibilities for the number of components of the completely invariant set of normality of G can also be achieved. However, it does not seem possible that all four possibilities can be achieved if we restrict ourselves to the cases where two elements of the semigroup G have nonequal Julia sets. For example, if G contains two polynomials with nonequal Julia sets then the completely invariant set of normality is necessarily empty (see [7], Theorem 1).

We do have the following examples however.

Example 1 Let f(z) = 2z − z^{−1} and g(z) = (z^{2} − 1)/2z. In this case J_{f} is a Cantor subset of the interval [−1, 1] and J_{g} = \overline{ℝ}, the extended real line. It is shown in [6] that J(⟨f, g⟩) = \overline{ℝ} = E(⟨f, g⟩).

Example 2 Let f(z) = 2z − z^{−1} and g(z) = f(z − 1) + 1. In this case J_{f} is a Cantor subset of the interval [−1, 1] and J_{g} = J_{f} + 1. It is shown in [6] that J(⟨f, g⟩) = [−1, 2] and E(⟨f, g⟩) = \overline{ℝ}.

So we see that it is possible for a completely invariant set of normality of a semigroup G which contains two elements with nonequal Julia sets, to have 0 or exactly 2 components. We feel that the interplay between functions with nonequal Julia sets and the fact that if E(G) has interior then E(G) = \overline{ℂ} demands that only under special circumstances can we have W (G) be nonempty, when two elements of the semigroup G have nonequal Julia sets.

We state the following conjectures which are due to Aimo Hinkkanen and Gaven Martin.

Conjecture 1. If G is a rational semigroup which contains two maps f and g such that J_{f} ≠ J_{g} and E(G) ≠ \overline{ℂ}, then W (G) has exactly two components, each of which is simply connected, and E(G) is equal to the boundary of each of these components.

Conjecture 2. If G is a rational semigroup which contains two maps f and g such that J_{f} ≠ J_{g} and E(G) ≠ \overline{ℂ}, then E(G) is a simple closed curve in \overline{ℂ}.

Of course Conjecture 1 would follow from Conjecture 2.

We finish by including some comments on the number of components of the set of normality N(G) of a rational semigroup G. It is not known if the set N(G) must have only 0, 1, 2, or infinitely many components when G is a finitely generated rational semigroup. However, for each positive integer n, an example of an infinitely generated polynomial semigroup G can be constructed with the property that N(G) has exactly n components. These examples were constructed by David Boyd in [2].

References

[1] Alan F. Beardon. Iterations of Rational Functions. Springer-Verlag, New York, 1991.

[2] David A. Boyd. Dynamics and measures for semigroups of rational functions. PhD thesis, University of Illinois, 1998.

[3] A. Hinkkanen and G.J. Martin. The dynamics of semigroups of rational functions I. Proc. London Math. Soc., 3:358–384, 1996.

[4] J.G Hocking and G.S. Young. Topology. Addison-Wesley, 1961.

[5] M. H. A. Newman. Elements of the Topology of Plane sets of points. Cambridge University Press, 1939.

[6] Rich Stankewitz. Completely invariant Julia sets of rational semigroups. PhD thesis, University of Illinois, 1998.

[7] Rich Stankewitz. Completely invariant Julia sets of polynomial semigroups. Proc. Amer. Math. Soc., To Appear.

[8] Norbert Steinmetz. Rational Iteration: Complex Analytic Dynamical Systems. de Gruyter, Berlin, 1993.

Department of Mathematics, University of Illinois, Urbana, Illinois 61801