Structure of Julia sets of polynomial semigroups with bounded finite postcritical set

Rich Stankewitz Department of Mathematical Sciences Ball State University Muncie, IN 47306, USA E-Mail: rstankewitz@bsu.edu

and

Hiroki Sumi

Department of Mathematics Graduate School of Science Osaka University Machikaneyama 1-1, Toyonaka Osaka, 560-0043, Japan E-Mail: sumi@math.sci.osaka-u.ac.jp

Abstract

Let G be a semigroup of complex polynomials (under the operation of composition of functions) such that there exists a bounded set in the plane which contains any finite critical value of any map g ∈ G. We discuss the dynamics of such polynomial semigroups as well the structure of the Julia set of G. In general, the Julia set of such a semigroup G may be disconnected, and each Fatou component of such G is either simply connected or doubly connected. In this paper, we show that for any two distinct Fatou components of certain types (e.g., two doubly connected components of the Fatou set), the boundaries are separated by a Cantor set of quasicircles (with uniform dilatation) inside the Julia set of G. Furthermore, we provide results concerning the (semi) hyperbolicity of such semigroups as well as discuss various topological consequences of the postcritically boundedness condition.

2000 Mathematics Subject Classification. Primary 37F10, 37F50, 30D05. Key Words and Phrases. Complex dynamics, Julia sets, Polynomials.

1 Introduction

A rational semigroup is a semigroup generated by non-constant rational maps on the Riemann sphere \overline{ℂ} with the semigroup operation being the composition of maps. We denote by ⟨h_{λ} : λ ∈ Λ⟩ the rational semigroup generated by the family of maps {h_{λ} : λ ∈ Λ}. A polynomial semigroup is a semigroup generated by non-constant polynomial maps. Research on the dynamics of rational semigroups was initiated by A. Hinkkanen and G.J. Martin in [2], who were interested in the role of the dynamics of polynomial semigroups while studying various one-complex-dimensional moduli spaces for discrete groups. Also, F. Ren, Z. Gong, and W. Zhou studied such semigroups from the perspective of random dynamical systems (see [18], [1]).

The polynomial maps f_{c}(z) = z^{2} + c for c in the Mandelbrot set are such that the orbit of the sole critical point {f_{c}^{n}(0)} is bounded, which in turn leads to many important dynamic and structural properties. It is then natural to look at the more general situation of polynomial semigroups with bounded postcritical set. We discuss the dynamics of such polynomial semigroups as well the structure of their Julia sets. For some properties of polynomial semigroups with bounded finite postcritical set, also see [12], [14].

Note that the research of polynomial semigroups is deeply related to the research of random dynamics of polynomials (See [16]).

Definition 1 Let G be a rational semigroup. We set

F (G) = \{z ∈ \overline{ℂ} | G is normal in a neighborhood of z\} and J(G) = \overline{ℂ} \ F (G).

We call F (G) the Fatou set of G and J(G) the Julia set of G. The Fatou set and Julia set of the semigroup generated by a single map g is denoted by F (g) and J(g), respectively.

From the definition, we get that F (G) is forward invariant under each element of G, i.e., g(F (G)) ⊂ F (G) for all g ∈ G, and thus J(G) is backward invariant under each element of G, i.e., g^{−1}(J(G)) ⊂ J(G) for all g ∈ G (see [2], p. 360). The sets F (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, i.e., the dynamics of semigroups generated by a single rational function. For a treatment of alternatively defined completely invariant Julia sets of rational semigroups the reader is referred to [4], [5], [6] and [7].

Note that J(G) contains the Julia set of each element of G. Moreover, the following result due to Hinkkanen and Martin holds.

Theorem 2 ([2], Corollary 3.1) For rational semigroups G with ♯(J(G)) ≥ 3, we have

J(G) = \overline{⋃_{f∈G} J(f)}.

The backward orbit of z is given by G^{−1}(z) = ∪_{g∈G}g^{−1}({z}) and the forward orbit of z is given by G(z) = ∪_{g∈G}g(\{z\}) . For any subset A of \overline{ℂ}, we set G^{−1}(A) = ∪_{g∈G}g^{−1}(A).

For any polynomial g, we set K(g) := {z ∈ ℂ | ∪_{n∈ℕ} g^{n}({z}) is bounded in ℂ}, which is known as the filled in Julia set of g. We note that J(g) = ∂K(g) and that K(g) is the polynomial hull of J(g). The appropriate extension (to our situation with polynomial semigroups) of the concept of filled in Julia set is as follows.

Definition 3 For a polynomial semigroup G, we set

K̂(G) := {z ∈ ℂ | G(z) is bounded in ℂ}.

Remark 4 We note that for all g ∈ G, we have K̂(G) ⊂ K(g) and g(K̂(G)) ⊂ K̂(G).

Definition 5 The postcritical set of a rational semigroup G is defined by

P (G) = \overline{⋃_{g∈G}\{all critical values of g : \overline{ℂ} → \overline{ℂ}\}} (⊂ \overline{ℂ}).

We say that G is hyperbolic if P (G) ⊂ F (G) and we say that G is subhyperbolic if both #{P (G) ∩ J(G)} < +∞ and P (G) ∩ F (G) is a compact set.

For research on (semi-)hyperbolicity and Hausdorff dimension of Julia sets of rational semigroups see [8, 9, 10, 15, 11].

Definition 6 The finite postcritical set (or, the planar postcritical set) of a polynomial semigroup G is defined by

P^{∗}(G) = P (G) \ {∞}.

Definition 7 Let G be the set of all polynomial semigroups G with the following properties:

• each element of G is of degree at least two, and

• P^{∗}(G) is bounded in ℂ.

Moreover, we set G_{dis} := {G ∈ G | J(G) is disconnected}.

Remark 8 Since P (G) is forward invariant under G, we see that G ∈ G implies P^{∗}(G) ⊂ K̂(G), and thus P^{∗}(G) ⊂ K(g) for all g ∈ G.

Remark 9 For a polynomial g of degree two or more, it is well known that ⟨g⟩ ∈ G implies J(g) is connected. (Hence, for any g ∈ G ∈ G, we have that J(g) is connected.) We note, however, that the analogous result for polynomial semigroups does not hold as there are many examples where G ∈ G, but J(G) is not connected (see [17]). See also [13] for an analysis of the number of connected components of J(G) involving the inverse limit of connected components of the realizations of the nerves of finite coverings U of J(G), where U consists of backward images of J(G) under maps in G.

The aim of this paper is to investigate what can be said about the structure of the Julia sets and the dynamics of semigroups G ∈ G? We begin by examining the structure of the Julia set and note that a natural order (that is respected by the backward action of the maps in G) can be placed on the components of J(G), which then leads to implications on the connectedness of Fatou components.

Definition 10 For a polynomial semigroup G ∈ G, we denote by J = J_{G} the set of all connected components of J(G) which do not include ∞.

Definition 11 We place a partial order on the space of all non-empty compact connected sets in ℂ as follows. For any connected compact sets K_{1} and K_{2} in ℂ, “K_{1} ≤ K_{2}” indicates that K_{1} = K_{2} or K_{1} is included in a bounded component of ℂ \ K_{2}. Also, “K_{1} < K_{2}” indicates K_{1} ≤ K_{2} and K_{1} ≠ K_{2}. We call ≤ the surrounding order and read K_{1} < K_{2} as “K_{1} is surrounded by K_{2}”.

Theorem 12 ([12]) Let G ∈ G (possibly infinitely generated). Then

1. (J , ≤) is totally ordered.

2. Each connected component of F (G) is either simply or doubly connected.

3. For any g ∈ G and any connected component J of J(G), we have that g^{−1}(J) is connected. Let g^{∗}(J) be the connected component of J(G) containing g^{−1}(J). If J ∈ J, then g^{∗}(J) ∈ J . If J_{1}, J_{2} ∈ J and J_{1} ≤ J_{2}, then both g^{−1}(J_{1}) ≤ g^{−1}(J_{2}) and g^{∗}(J_{1}) ≤ g^{∗}(J_{2}).

With this order and the following notation we will then be able to state our main results. Let h_{1}, ⋯, h_{m} be rational functions on \overline{ℂ}. Let Σ_{m} = \{1, ⋯, m\}^{ℕ} be the one-sided shift

space and let σ : Σ_{m} → Σ_{m} be the shift map, i.e., σ(x_{1}, x_{2}, ⋯) = (x_{2}, x_{3}, ⋯). Let f̃: Σ_{m} × \overline{ℂ} → Σ_{m} × \overline{ℂ} be the map defined by f̃(x, y) = (σ(x), h_{x_{1}}(y)), where x = (x_{1}, x_{2}, ⋯) ∈ Σ_{m}. This is called the skew product map associated with \{h_{1}, ⋯, h_{m}\}. Let π : Σ_{m}×\overline{ℂ} → Σ_{m} and π\overline{ℂ} : Σ_{m} × \overline{ℂ} → \overline{ℂ} be the natural projections. We set f̃_{x}^{n} := f̃^{n}|_{π^{−1}x} : π^{−1}x → π^{−1}σ^{n}(x) ⊂ Σ_{m} × \overline{ℂ} and we denote by F_{x}(f̃) the set of points y ∈ π^{−1}x which has a neighborhood U in π^{−1}x such that \{f̃_{x}^{n} : U → Σ_{m}×\overline{ℂ}\}_{n∈ℕ} is normal. Furthermore, we set J_{x}(f̃) := π^{−1}x\F_{x}(f̃).

Remark 13 Note that π\overline{ℂ}(J_{x}(f̃)) is equal to the set of points z ∈ \overline{ℂ} where the sequence of rational functions {h_{x_{n}} ∘ ⋯ ∘ h_{x_{1}}}_{n∈ℕ} is not normal. This is sometimes called the Julia set along the trajectory (sequence) x ∈ Σ_{m}.

Theorem 14 Let G ∈ G and let A and B be disjoint subsets of \overline{ℂ}. Suppose that we have one of the following conditions:

• A and B are doubly connected components of F (G).

• A is a doubly connected component of F (G) and B is a connected component of F (G) with ∞ ∈ B .

• A = K̂(G) and B is a doubly connected component of F (G).

Then ∂A ∩ ∂B = ∅. Furthermore, \overline{A} and \overline{B} are separated by a Cantor family of quasicircles with uniform dilatation which all lie in J(G). More precisely, there exist two elements α_{1}, α_{2} ∈ G satisfying all of the following.

1. There exists an open set U in \overline{ℂ} with α_{1}^{−1}(\overline{U}) ∩ α_{2}^{−1}(\overline{U}) = ∅ and α_{1}^{−1}(\overline{U}) ∪ α_{2}^{−1}(\overline{U}) ⊂ U.

2. H = ⟨α_{1}, α_{2}⟩ is hyperbolic.

3. Let f̃: Σ_{2} × \overline{ℂ} → Σ_{2} × \overline{ℂ} be the skew product map associated with \{α_{1}, α_{2}\}. Then

(a) J(H) = ⋃_{x∈Σ_{2}} π\overline{ℂ}(J_{x}(f̃)) (disjoint union),

(b) for any component J of J(H), there exists an x ∈ Σ_{2} with J = π\overline{ℂ}(J_{x}(f̃)) and

(c) there exists a constant K ≥ 1 such that any component J of J(H) is a Kquasicircle.

4. \{π\overline{ℂ}(J_{x}(f̃))\}_{x∈Σ_{2}} is totally ordered with ≤, consisting of mutually disjoint subsets of J(H). Furthermore, for each x ∈ Σ_{2}, the set π\overline{ℂ}(J_{x}(f̃)) separates \overline{A} and \overline{B}.

Remark 15 It should be noted that in the above theorem, the quasicircles π\overline{ℂ}(J_{x}(f̃)) are all disjoint components of J(H), but may all lie in the same component of J(G).

Example 16 We give an example of a semigroup G ∈ G such that J(G) is a Cantor set of round circles. Let f_{1}(z) = az^{k} and f_{2}(z) = bz^{j} for some positive integers k and j. Then J(f_{1}) and J(f_{2}) are both circles centered at the origin. Let A denote the closed annulus between J(f_{1}) and J(f_{2}). For positive integers m_{1} and m_{2} large enough, we see that the iterates g_{1} = f_{1}^{m_{1}} and g_{2} = f_{2}^{m_{2}} will yield A_{1} = g_{1}^{−1}(A) ∪ g_{2}^{−1}(A) ⊂ A where g_{1}^{−1}(A) ∩ g_{2}^{−1}(A) = ∅. Now iteratively define A_{n+1} = g_{1}^{−1}(A_{n}) ∪ g_{2}^{−1}(A_{n}) and note that for G = ⟨g_{1}, g_{2}⟩ we have that J(G) = ∩_{n=1}^{∞}A_{n}, since J(G) is the smallest closed backward invariant (under each element of G) set which contains three or more points.

The next results concern the (semi-)hyperbolicity of polynomial semigroups in G, and in particular show how one can build larger (semi-)hyperbolic polynomial semigroups in G from smaller ones by including maps with certain properties. For this result we need to note the existence of a minimal element in J and state a few of its properties.

Theorem 17 ([12]) Let G be a polynomial semigroup in G_{dis}. Then there is a unique element J_{min} ∈ J such that J_{min} meets (and therefore contains) ∂K̂. Furthermore, we have the following

• J_{min} ≤ J for all J ∈ J.

• P^{∗}(G) is contained in the polynomial hull of J_{min}.

Definition 18 A rational semigroup H is semi-hyperbolic if for each z ∈ J(H) there exists a neighborhood U of z and a number N ∈ ℕ such that for each g ∈ H we have deg(g : V → U) ≤ N for each connected component V of g^{−1}(U).

Theorem 19 Let H ∈ G and let G = ⟨H, h_{1}, …, h_{n}⟩ be a polynomial semigroup generated by H and h_{1}, ⋯, h_{n}. Suppose

(1) G ∈ G_{dis},

(2) J(h_{j}) ∩ J_{min}(G) = ∅ for each j = 1, ⋯, n, and

(3) H is semi-hyperbolic. Then, G is semi-hyperbolic.

Remark 20 Theorem 19 would not hold if we were to replace both instances of the word semi-hyperbolic with the word hyperbolic (see Example 37). However, with an additional hypothesis we do get the following:

Theorem 21 Let H ∈ G and let G = ⟨H, h_{1}, …, h_{n}⟩ be a polynomial semigroup generated by H and h_{1}, ⋯, h_{n}. Suppose

(1) G ∈ G_{dis},

(2) J(h_{j}) ∩ J_{min}(G) = ∅ for each j = 1, ⋯, n,

(3) P^{∗}(H) ∩ J(H) = ∅. and

(4) For each j = 1, …, n, the critical values of h_{j} do not meet J_{min}(G). Then, P^{∗}(G) ∩ J(G) = ∅.

Remark 22 We note that if in Theorem 21 we replace hypothesis (3) with the hypothesis “H is hyperbolic”, then the conclusion becomes “G is hyperbolic”. This follows immediately as one can show that ∞ ∈ F (H) implies ∞ ∈ F (G) (see the proof of Theorem 19 for more details).

The rest of this paper is organized as follows. In Section 2 we give the necessary background and tools required, in Section 3 we give the proof of Theorem 14, and in Section 4 we give the proofs of Theorems 19 and 21 along with Example 37.

2 Background and Tools

Although not all connected compact sets in ℂ are comparable in the surrounding order, we do have the following lemma whose proof we leave to the reader.

Lemma 23 Given two connected compact sets A and B in ℂ we must have exactly one of the following:

1. A < B

2. B < A 3. A ∩ B ≠ ∅

4. A and B are outside of each other, i.e., A is a subset of the unbounded component of ℂ \ B and B is a subset of the unbounded component of ℂ \ A.

For various sets of interest in this paper the last case listed above is not possible (see Corollary 33). We now proceed to show this through a series of results which we will also find useful later in the paper.

Lemma 24 Let A and B be compact connected subsets of ℂ and suppose there exists a point p ∈ ℂ such that one of the following holds

(i) {p} < A and {p} < B

(ii) p ∈ A and {p} < B

(iii) {p} < A and p ∈ B

(iv) p ∈ A and p ∈ B.

Then exactly one of the following holds:

1. A < B

2. B < A

3. A ∩ B ≠ ∅.

Proof: Cases (ii), (iii), and (iv) are trivial, so we detail only case (i). Suppose A∩B = ∅. Call V_{A} and V_{B} the bounded components of ℂ \ A and ℂ \ B, respectively, which contain p. Since V_{A} ≠ V_{B} (else A and B are not disjoint) we must have ∂V_{B} ∩ V_{A} ≠ ∅ or ∂V_{A} ∩ V_{B} ≠ ∅. But ∂V_{B} ∩ V_{A} ≠ ∅ implies B ∩ V_{A} ≠ ∅ which implies B ⊂ V_{A} (since B is connected), and thus B < A. Similarly, ∂V_{A} ∩ V_{B} ≠ ∅ implies A < B. QED

Definition 25 For a compact set A ⊂ ℂ we define the polynomial hull P H(A) of A to be the union of A and all bounded components of ℂ \ A.

Remark 26 With regards to the Lemma 24, we note that if P H(A) and P H(B) share a common point p, then one of the cases (i) − (iv) must hold, and therefore so does the conclusion.

Lemma 27 Let g ∈ G ∈ G and suppose P^{∗}(G) ⊂ P H(A) where A ⊂ ℂ is compact. Then P^{∗}(G) ⊂ P H(g^{−1}(A)).

Proof: Suppose z_{0} ∈ P^{∗}(G) is in the unbounded component U of \overline{ℂ} \ g^{−1}(A). Let γ be a curve in U connecting z_{0} to ∞. Then Γ = g ∘ γ is a curve in \overline{ℂ} \ A which connects g(z_{0}) to ∞ which shows that g(z_{0}) ∉ P H(A). Since P^{∗}(G) is forward invariant we have that g(z_{0}) ∈ P^{∗}(G) \ P H(A) which contradicts our hypothesis. QED Corollary 28 Let f, g ∈ G ∈ G. If A is of the form J ∈ J , J(f), g^{−1}(J), or g^{−1}(J(f)), then P^{∗}(G) ⊂ P H(A).

Proof: Since by Theorem 17 we have P^{∗}(G) ⊂ P H(J_{min}) and J_{min} ≤ J, we must also have P^{∗}(G) ⊂ P H(J). By Remark 8 any point p ∈ P^{∗}(G) must lie in K(f) = P H(J(f)). The other cases then follow from Lemma 27. QED

An important consequence of Lemma 3.1 in [12] is the following.

Corollary 29 ([12]) Let G ∈ G. If ϵ > 0 and J ∈ J, then there exists g ∈ G such that J(g) ⊂ B(J, ϵ), where B(J, ϵ) = ∪_{z∈J}B(z, ϵ) denotes the ϵ neighborhood of J.

Lemma 30 ([3]) Let X be a compact metric space and let f : X → X be a continuous open map. Let K be a compact connected subset of X. Then for each connected component B of f^{−1}(K), we have f(B) = K.

Lemma 31 Let g be a polynomial with d = deg(g) ≥ 1 and let K ⊂ ℂ be a connected compact set such that the unbounded component U of \overline{ℂ} \ K contains no critical values of g other than ∞. Then g^{−1}(K) is connected. Further, if K_{1} is a connected compact set such K < K_{1}, then g^{−1}(K) < g^{−1}(K_{1}).

Proof: Set V = g^{−1}(U) and note that V contains no finite critical points of g. Thus by the Riemann-Hurwitz relation we have χ(V) + δ_{g}(V) = dχ(U), where χ(·) denotes the Euler characteristic and δ(·) is the deficiency. Since the hypotheses on U imply δ_{g}(V) = d − 1 and χ(U) = 1, we see that χ(V) = 1. Hence the open and connected set V is simply connected and thus ∂V is connected. Let A = \overline{ℂ} \ \overline{V} and note that ∂A ⊂ ∂V. Suppose B is a component of g^{−1}(K) which does not meet ∂V. Then B ⊂ A. We note that by Lemma 30 we have g(B) = K ⊃ ∂U. Since g(B) ⊃ ∂U and B ⊂ A we see that g(A) ∩ ∂U ≠ ∅. Thus g(A) ∩ U ≠ ∅ by the Open Mapping Theorem, which contradicts the fact that A and V = g^{−1}(U) are disjoint. Thus, since the connected set ∂V ⊂ g^{−1}(K) meets all the connected components of g^{−1}(K), we conclude that g^{−1}(K) is connected.

Let V_{1} = g^{−1}(U_{1}) where U_{1} is the unbounded component of \overline{ℂ} \ K_{1}. Since K < K_{1} implies K ∩ K_{1} = ∅, we get that g^{−1}(K) ∩ g^{−1}(K_{1}) = ∅ which implies ∂V ∩ ∂V_{1} = ∅ since ∂V ⊂ g^{−1}(K) and ∂V_{1} ⊂ g^{−1}(K_{1}). Since ∞ ∈ V ∩V_{1}, it is easy to show then that ∂V < ∂V_{1}. Since g^{−1}(K) ⊃ ∂V and g^{−1}(K_{1}) ⊃ ∂V_{1}, it quickly follows that g^{−1}(K) < g^{−1}(K_{1}). QED

Corollary 32 Let g, h ∈ G ∈ G and J ∈ J. Then g^{−1}(J) and g^{−1}(J(h)) are connected. Furthermore, J_{1} < J_{2} for J_{1}, J_{2} ∈ J implies g^{−1}(J_{1}) < g^{−1}(J_{2}), and J(h_{1}) < J(h_{2}) for h_{1}, h_{2} ∈ G implies g^{−1}(J(h_{1})) < g^{−1}(J(h_{2})).

Proof: The result follows immediately from Lemma 31 once it is shown that the unbounded components U and U′ of \overline{ℂ} \ J and \overline{ℂ} \ J(h), respectively, contain no critical values of g other than ∞. Suppose U contains a finite critical value z_{0} of g. Let γ be a path in U connecting z_{0} to ∞ and let ϵ = dist(γ, J). By Corollary 29 we see that B(J, ϵ/2) contains J(f) for some f ∈ G. Since J(f) cannot meet γ, the point z_{0} lies in the unbounded component of ℂ \ J(f) and thus f^{n}(z_{0}) → ∞, which contradicts the hypothesis that P^{∗}(G) is bounded. Hence U contains no critical values of g other than ∞. The condition on U′ follows from the fact that P^{∗}(g) ⊂ P^{∗}(G) ⊂ K(h). QED

Corollary 33 Let f, g ∈ G ∈ G. For any two sets A and B of the form J ∈ J , J(f), g^{−1}(J), or g^{−1}(J(f)), exactly one of the following must hold:

1. A < B

2. B < A

3. A ∩ B ≠ ∅.

Proof: This follows immediately from Corollary 32, Corollary 28 and Remark 26. QED

3 Proof of Theorem 14

Definition 34 For compact connected sets K_{1} and K_{2} in ℂ such that K_{1} < K_{2} we define Ann(K_{1}, K_{2}) = U ∩ V where U is the bounded component of ℂ \ K_{2} which contains K_{1}, and V is the unbounded component of \overline{ℂ} \ K_{1}. Thus Ann(K_{1}, K_{2}) is the open doubly connected region “between” K_{1} and K_{2}.

Remark 35 For any connected compact set A ⊂ Ann(K_{1}, K_{2}) we immediately see that A < K_{2} and, by Lemma 23, either K_{1} and A are outside of each other or K_{1} < A.

Lemma 36 Let f, g ∈ G ∈ G be such that J(f) and J(g) lie in different components of J(G) with J(f) < J(g). Then for any fixed n, m ∈ ℕ there exists h, k ∈ G such that f^{−(n+1)}(J(g)) < J(h) < f^{−n}(J(g)) and g^{−m}(J(f)) < J(k) < g^{−(m+1)}(J(f)).

Proof: We claim that g^{−1}(J(f)) > J(f). If g^{−1}(J(f)) ∩ J(f) ≠ ∅, then the connected set \overline{∪_{n=0}^{∞}g^{−n}(J(f))}, which lies in J(G), meets both J(g) and J(f), thus contradicting the hypothesis that J(f) and J(g) lie in different components of J(G). If g^{−1}(J(f)) < J(f), then one could easily show that g^{−1}(K(f)) ⊂ K(f) which would imply that J(g) ⊂ K(f), thus contradicting the hypothesis that J(f) < J(g). Hence, we conclude by Corollary 33 that g^{−1}(J(f)) > J(f).

Let X = g^{−1}(J(f)), A = g^{−m}(J(f)) and B = g^{−(m+1)}(J(f)) and note that J(f) < A < B from Lemma 31. Choose ℓ ∈ ℕ large enough so that f^{−ℓ}(B) ⊂ Ann(J(f), X). Then g^{−m}(f^{−ℓ}(\overline{Ann(A, B)})) ⊂ g^{−m}(Ann(J(f), X)) ⊂ Ann(A, B) ⊂ \overline{Ann(A, B)} which implies that k = f^{ℓ} ∘ g^{m} ∈ G is such that J(k) ∈ Ann(A, B). Since by Corollary 33 we must have either J(k) < A or A < J(k), we see by construction that A < J(k) must hold.

The other result is proved similarly. QED

Proof of Theorem 14: We give a proof in the case that A and B are doubly connected components of F (G). We assume without loss of generality that A is contained in the bounded component of \overline{ℂ}\B. Let Γ_{A} denote the component of ∂A which meets the bounded component of \overline{ℂ} \ A and let Γ denote the other component of ∂A. Let Γ_{B} denote the component of ∂B which meets the unbounded component of \overline{ℂ} \ B. Let J ∈ J be such that Γ ⊂ J. Let ϵ = min{dist(Γ_{A}, J), dist(Γ_{B}, J)}. By Corollary 29 we see that B(J, ϵ/2) contains J(g) for some g ∈ G and thus, using Corollary 33, Γ_{B} > J(g) > Γ_{A}.

Let J_{A} ∈ J be such that Γ_{A} ⊂ J_{A} and set ϵ′ = dist(J_{A}, J(g)). By the Corollary 29 we see that B(J_{A}, ϵ′/2) contains J(f) for some f ∈ G and thus J(g) > J(f). Choose m large enough so that g^{−m}(J(f)) > Γ_{A}. By Lemma 36 there exists k ∈ G such that g^{−m}(J(f)) < J(k) < g^{−(m+1)}(J(f)) < J(g). By choosing m_{1}, m_{2} ∈ ℕ large (as in Example 16), we will find that α_{1} = k^{m_{1}} and α_{2} = g^{m_{2}} generate a subsemigroup H of G where J(H), by Theorem 3.5 in [12], is a Cantor set of topological circles each of which separate A from B and also satisfy the conclusion of the theorem. QED

4 Proof of Theorems 19 and 21

Example 37 Let f_{1}(z) = z^{2} + c where c > 0 is small (thus J(f_{1}) is a quasi-circle). Let z_{0} ∈ ℝ denoted the finite attracting fixed point of f_{1}. Note that f_{1}^{k}(0) increases to z_{0}. Choose f_{2}(z) = \frac{(z−z_{0})^{2}}{(c−z_{0})} + z_{0} and note that J(f_{2}) = C(z_{0}, |c − z_{0}|). For m_{1}, m_{2} ∈ ℕ large h_{1} = f_{1}^{m_{1}} and h_{2} = f_{2}^{m_{2}} each map B(z_{0}, |c − z_{0}|) into itself and J(G) is disconnected for G = ⟨h_{1}, h_{2}⟩. Note that P^{∗}(G) ⊂ \overline{B(z_{0}, |c − z_{0}|)} and so G ∈ G. We have H = ⟨h_{2}⟩ is hyperbolic, but since f_{1}(0) = c ∈ J(h_{2}) ⊂ J(G), G is not hyperbolic.

By conjugating h_{2} by a rotation we may assume that {h_{2}^{k}(c) : k ∈ ℕ} is dense in J(h_{2}) and therefore we see that H can be hyperbolic and have G fail to even be sub-hyperbolic. However, Theorem 19 implies that G is semi-hyperbolic.

Lemma 38 Let H_{1} be a polynomial semigroup in G. Let H_{2} = ⟨H_{1}, h_{1}, …, h_{n}⟩ be the semigroup generated by H_{1} and h_{1}, ⋯, h_{n}. Suppose

(1) H_{2} ∈ G_{dis}, and

(2) J(h_{j}) ∩ J_{min}(H_{2}) = ∅ for j = 1, …, n. Then intK̂(H_{1}) = intK̂(H_{2}), which then implies J_{min}(H_{1}) meets J_{min}(H_{2}) since ∂K̂(H_{1}) ⊂ J_{min}(H_{1}) and ∂K̂(H_{2}) ⊂ J_{min}(H_{2}).

Remark 39 We recall the facts given in [12] that for any G ∈ G we have intK̂(G) = K̂(G)∩F (G). Moreover, for any G ∈ G_{dis}, if intK̂(G) ≠ ∅, then we have g(K̂(G)) ⊂ intK̂(G) for any g ∈ G such that J(g) ∩ J_{min}(G) = ∅.

Proof: First note that since H_{1} ⊂ H_{2} we have K̂(H_{1}) ⊃ K̂(H_{2}). Supposing intK̂(H_{1}) = ∅, yields intK̂(H_{2}) = ∅ also. Then by Theorem 1.9 in [12] each g ∈ H_{2} is of the form g(z) = a(z − z_{0})^{m} + z_{0} for some z_{0} ∈ ℂ. Thus, since intK̂(H_{1}) = intK̂(H_{2}) = ∅, we see that K̂(H_{1}) = {z_{0}} = K̂(H_{2}) and thus the lemma holds.

Now suppose that intK̂(H_{1}) ≠ ∅ and that the lemma does not hold. Thus there exists a point η ∈ ∂K̂(H_{2}) ∩ intK̂(H_{1}).

We first consider the case that intK̂(H_{2}) ≠ ∅. By hypothesis (2) and Remark 39 we see that h_{j}(K̂(H_{2})) ⊂ intK̂(H_{2}) for all j = 1, …, n. Thus there exists ϵ > 0 such that w ∈ K̂(H_{2}) implies h_{j}(B(w, ϵ)) ⊂ intK̂(H_{2}) for all j = 1, …, n. Since intK̂(H_{1}) = K̂(H_{1}) ∩ F (H_{1}) by Remark 39, H_{1} is normal at η and so there exists δ > 0 such that diamf(B(η, δ)) < ϵ for all f ∈ H_{1}. We assume that δ < ϵ and B = B(η, δ) ⊂ intK̂(H_{1}).

We now show that g ∈ H_{2} implies g(B) lies in the bounded set intK̂(H_{1}) which gives the contradiction that η ∈ B ⊂ intK̂(H_{2}). If g ∈ H_{1}, then g(B) ⊂ g(intK̂(H_{1})) ⊂ intK̂(H_{1}). If g ∉ H_{1}, then we may write g = k_{2}h_{j}k_{1} where k_{1} ∈ H_{1} ∪ {id}, k_{2} ∈ H_{2} ∪ {id} and j ∈ {1, …, n}. Then k_{1}(B) ⊂ intK̂(H_{1}) with diam k_{1}(B) < ϵ and k_{1}(η) ∈ K̂(H_{2}) (since η ∈ K̂(H_{2})). Then h_{j}(k_{1}(B)) ⊂ h_{j}(B(k_{1}(η), ϵ)) ⊂ intK̂(H_{2}) and so g(B) = k_{2}(h_{j}(k_{1}(B))) ⊂ k_{2}(intK̂(H_{2})) ⊂ intK̂(H_{2}). This concludes the proof in the case that intK̂(H_{2}) ≠ ∅.

If intK̂(H_{2}) = ∅, then by Theorem 1.9 in [12] we have K̂(H_{2}) = {z_{0}} and each g ∈ H_{2} is of the form g(z) = a(z −z_{0})^{m} +z_{0} which implies K̂(H_{1}) = \{z_{0}\} or K̂(H_{1}) = \overline{B(z_{0}, ρ)} for some ρ > 0. If K̂(H_{1}) = {z_{0}}, then we are done. Otherwise, the proof of the lemma then follows by using the above argument with B = B(z_{0}, r) where r = min{ρ, min_{j=1,...,n} dist(z_{0}, J(h_{j}))}. QED

Definition 40 Let G be a rational semigroup and let N be a positive integer. We define SH_{N}(G) to be the set of all z ∈ \overline{ℂ} such that there exists a neighborhood U of z such that for all g ∈ G we have deg(g : V → U) ≤ N for each connected component V of g^{−1}(U).

Definition 41 Let G be a rational semigroup. We define UH(G) = \overline{ℂ} \ ∪_{N=1}^{∞}SH_{N}(G).

Remark 42 For a rational semigroup G we note that each SH_{N}(G) is open and thus UH(G) is closed.

Remark 43 For a rational semigroup G we see that UH(G) ⊂ P (G). This holds since for z ∉ P (G) and U = B(z, δ) such that U ∩ P (G) = ∅ it must be the case (by an application of the Riemann Hurwitz relation) that deg(g : V → U) = 1 for each connected component V of g^{−1}(U).

Remark 44 We note from Lemma 1.14 in [10] that, the attracting cycles of g, parabolic cycles of g, and the boundary of every Siegel disk of g are contained in UH(⟨g⟩), for any polynomial g with deg(g) ≥ 2. Hence we may conclude that such points are also in UH(G) for any G containing g.

Theorem 45 Let H ∈ G and let G = ⟨H, h_{1}, …, h_{m}⟩ be a polynomial semigroup generated by H and h_{1}, ⋯, h_{m}. Suppose

(1) G ∈ G_{dis},

(2) J(h_{j}) ∩ J_{min}(G) = ∅ for each j = 1, ⋯, m, and

(3) ℂ ∩ J(H) ∩ UH(H) = ∅. Then, ℂ ∩ J(G) ∩ UH(G) = ∅.

Remark 46 This theorem does not require that H or G be finitely generated.

Proof: Assume the conditions stated in the hypotheses. Since UH(G) ⊂ P (G) and P^{∗}(G) ∩ J(G) ⊂ J_{min}(G), we have only to show J_{min}(G) ⊂ ℂ \ UH(G).

If #J_{min}(G) = 1, then Theorem 1.9 in [12] implies that there exists z_{0} ∈ ℂ such that each map in G is of the form a(z −z_{0})^{n} +z_{0} and J_{min}(G) = {z_{0}}. We also then see that UH(H) = P (H) = {z_{0}, ∞}. Letting r = min{dist(J(H), z_{0}), min_{j=1,...,m} dist(J(h_{j}), z_{0})} > 0, one can easily show that B(z_{0}, r) is forward invariant under each map in G and thus z_{0} ∉ J(G). This contradiction shows that #J_{min}(G) > 1. This, together with hypothesis (2), implies h_{j}^{−1}(J_{min}(G)) ∩ J_{min}(G) = ∅ for j = 1, …, m (else \overline{∪_{n=1}^{∞}h_{j}^{−1}(J_{min}(G))} would be a connected set in J(G) meeting both J_{min}(G) and J(h_{j}) contradicting hypothesis (3)), which in turn implies

h_{j}^{−1}(J(G)) ∩ A = ∅  (I)

where A = P H(J_{min}).

Let d = min_{j=1,...,m} dist(h_{j}^{−1}(J(G)), A) > 0. By (I) there exists d_{1} > 0 such that for all j = 1, …, m, for all z ∈ J(G), and all components U of h_{j}^{−1}(B(z, d_{1})) we have

U ∩ B(A, d/2) = ∅.  (II)

Now by Lemma 38 and by hypothesis (3) we have UH(H) ∩ ℂ ⊂ P^{∗}(H) ∩ F (H) ⊂ K̂(H) ∩ F (H) = intK̂(H) = intK̂(G) ⊂ F (G) and so, taking compliments, J_{min}(G) ⊂ ℂ \ UH(H).

Claim: There exists b ∈ UH(H) ∩ intK̂(H). Proof of claim: Note that intK̂(H) = intK̂(G) ≠ ∅. Let g_{0} ∈ H and consider the iterates {g_{0}^{n}} at any w ∈ intK̂(H) ⊂ F (H). Hypothesis (3) implies UH(H) ∩ ℂ ⊂ F (H) which implies that g_{0} cannot have a cycle of Siegel disks nor a parabolic cycle (see Remark 44). Thus by Sullivan’s No Wandering Domains Theorem the orbit {g_{0}^{n}(w)} must be drawn toward an attracting cycle. By replacing, if necessary, g_{0} by an iterate we may assume that g_{0}^{n}(w) approaches a fixed point b of g_{0}. Thus b ∈ UH(H) ∩ ℂ ⊂ P^{∗}(H) ∩ F (H) ⊂ K̂(H) ∩ F (H) = intK̂(H) which completes the proof of the claim.

Now let z ∈ J_{min}(G) ⊂ ℂ \ UH(H). Then there exists δ > 0 such that B(z, 2δ) ⊂ ℂ \ UH(H). Since g(UH(H)) ⊂ UH(H) for each g ∈ H, we must have g(b) ∉ B(z, 2δ). Since H is normal at b, there exists ϵ_{1} > 0 such that g ∈ H gives g(B(b, ϵ_{1})) ∩ B(z, δ) = ∅, which implies g^{−1}(B(z, δ)) ∩ B(b, ϵ_{1}) = ∅. Since z ∈ ℂ \ UH(H) there exists δ_{1} < δ and N ∈ ℕ such that for all h ∈ H and for all components V of h^{−1}(B(z, δ_{1})) we have deg(h : V → B(z, δ_{1})) ≤ N.

Fix h ∈ H and consider a component V of h^{−1}(B(z, δ_{1})) and note that the maximum principle implies that V is simply connected. Let ϕ_{V,h} : B(0, 1) → V be the Riemann map chosen such that h∘ϕ_{V,h}(0) = z. By applying the distortion Theorem 1.10 in [10], there exists 0 < δ_{2} < δ_{1} such that any component W of (h ∘ ϕ_{V,h})^{−1}(B(z, δ_{2})) is such that diamW ≤ c where c > 0 is a small number independent of h, to be specified later.

The family {ϕ_{V,h}} is normal on B(0, 1) since ϕ_{V,h}(B(0, 1)) = V does not meet B(b, ϵ_{1}). Thus

diamϕ_{V,h}(W) < d_{1}/10  (III)

when c is sufficiently small.

Let g ∈ G. If g ∈ H, then deg(g : V → B(z, δ_{2})) ≤ N where V is any component of g^{−1}(B(z, δ_{2})). If g ∉ H, then we write g = hh_{j}g_{1} where g_{1} ∈ G ∪ {id}, h ∈ H ∪ {id} and j ∈ {1, …, m}. Let V_{0} be a component of h_{j}^{−1}h^{−1}(B(z, δ_{2})). Thus we have deg(hh_{j} : V_{0} → B(z, δ_{2})) ≤ NM where M = max_{j=1,...,m}{deg h_{j}}. By (III) we have diamh_{j}(V_{0}) < d_{1}/10. By the definition of d_{1} we have V_{0} ∩ B(A, d/2) = ∅ and thus V_{0} ∩ P (G) = ∅. Using the maximum principle applied to the polynomial hh_{j} implies V_{0} is simply connected and hence each branch of g_{1}^{−1} is well defined on V_{0}. So for all components V_{1} of g^{−1}(B(z, δ_{2})) we have deg(g : V_{1} → B(z, δ_{2})) ≤ NM.

In the above, N depends on z, but what we have shown is that z ∈ J_{min}(G) implies z ∈ J_{min}(G) ∩ SH_{N}(H) for some N, which in turn implies z ∈ J_{min}(G) ∩ SH_{NM}(G), thus giving z ∉ UH(G). QED

Proof of Theorem 19: Assume the hypotheses hold. Since H is semi-hyperbolic, the point ∞, which is an attracting fixed point for every map in H and therefore in UH(H), must lie in F (H). From this it follows easily that ∞ ∈ F (G) since there must then be a neighborhood of ∞ which is forward invariant under the finite number of maps h_{j} as well as each map in H. Applying Theorem 45 now gives the desired conclusion. QED

Proof of Theorem 21: The proof follows the same line at the proof of Theorem 45. We note that the usual Koebe Distortion Theorem applies, and on the domains of interest in the proof each h_{j} is one-to-one by hypothesis (4) and h ∈ H is one-to-one by hypothesis (3). We omit the details. QED

References

[1] Z. Gong and F. Ren. A random dynamical system formed by infinitely many functions. Journal of Fudan University, 35:387–392, 1996.

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

[3] S.B.Nadler. Continuum Theory: An introduction. Marcel Dekker, 1992.

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

[5] Rich Stankewitz. Completely invariant Julia sets of polynomial semigroups. Proc. Amer. Math. Soc., 127(10):2889–2898, 1999.

[6] Rich Stankewitz. Completely invariant sets of normality for rational semigroups. Complex Variables Theory Appl., 40(3):199–210, 2000.

[7] Rich Stankewitz, Toshiyuki Sugawa, and Hiroki Sumi. Some counterexamples in dynamics of rational semigroups. Ann. Acad. Sci. Fenn. Math., 29(2):357–366, 2004.

[8] Hiroki Sumi. On hausdorff dimension of Julia sets of hyperbolic rational semigroups. Kodai. Math.J., 21(1):10–28, 1998.

[9] Hiroki Sumi. Skew product maps related to finitely generated rational semigroups. Nonlinearity, 13:995–1019, 2000.

[10] Hiroki Sumi. Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products. Ergod.Th.& Dynam. Sys., 21:563–603, 2001.

[11] Hiroki Sumi. Dimensions of Julia sets of expanding rational semigroups. Kodai Mathematical Journal, 28(2):390–422, 2005. (See also http://arxiv.org/abs/math.DS/0405522.).

[12] Hiroki Sumi. Dynamics of polynomial semigroups with bounded postcritical set in the plane. RIMS Kokyuroku, 1447:198–215, 2005.

[13] Hiroki Sumi. Dynamics of postcritically bounded polynomial semigroups and interaction cohomology. RIMS Kokyuroku, 1447:227–238, 2005.

[14] Hiroki Sumi. Dynamics of postcritically bounded polynomial semigroups. In preparation.

[15] Hiroki Sumi. Semi-hyperbolic fibered rational maps and rational semigroups. To appear in Ergod.Th.& Dynam. Sys. (See also http://arxiv.org/abs/math.DS/0509719).

[16] Hiroki Sumi. Random dynamics of polynomials and devil’s staircase-like functions in the complex plane. To appear in this volume.

[17] Y.Sun and C-C.Yang. On the connectivity of the Julia set of a finitely generated rational semigroup. Proc. Amer.Math.Soc., 130(1):49–52, 2001.

[18] W. Zhou and F. Ren. The Julia sets of the random iteration of rational functions. Chinese Bulletin, 37(12):969–971, 1992.