DYNAMICAL PROPERTIES AND STRUCTURE OF JULIA SETS OF POSTCRITICALLY BOUNDED POLYNOMIAL SEMIGROUPS

RICH STANKEWITZ AND HIROKI SUMI

Abstract. We discuss the dynamic and structural properties of polynomial semigroups, a natural extension of iteration theory to random (walk) dynamics, where the semigroup G of complex polynomials (under the operation of composition of functions) is such that there exists a bounded set in the plane which contains any finite critical value of any map g ∈ 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 ([29, 33]). 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. Important in this theory is the understanding of various situations which can and cannot occur with respect to how the Julia sets of the maps g ∈ G are distributed within the Julia set of the entire semigroup G. We give several results in this direction and show how such results are used to generate (semi) hyperbolic semigroups possessing this postcritically boundedness condition.

1. Introduction

The dynamics of iteration of a complex analytic map has been studied quite deeply and in various contexts, e.g., rational, entire, and meromorphic maps. It is natural then to consider the generalization of this theory to the setting where the map may be changed at each point of the orbit, exactly as in a random walk. Instead of repeatedly applying the same map over and over again, one may start with a family of maps {h_{λ} : λ ∈ Λ}, and consider the dynamics of any iteratively defined composition sequence of maps, that is, any sequence h_{λ_{n}} ∘ ⋯ ∘ h_{λ_{1}} where each λ_{k} ∈ Λ. Assigning probabilities to the choice of map at each stage is the setting for research of random dynamics (see [9, 4, 6, 7, 5, 28, 30, 31, 32, 33] for previous work related to such dynamics). In this paper, however, we will concern ourselves with questions of dynamic stability, not just along such composition sequences one at a time, but rather we will study when such stability exists no matter which composition sequence is chosen. Restricting one’s attention to the case where all h_{λ} are rational, one is lead to study the dynamics of rational semigroups.

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

Date: May 9, 2011. To appear in Trans. Amer. Math. Soc. The first author was partially supported by the BSU Lilly V grant. He would also like to

thank Osaka University for their hospitality during his stay there while this work was begun.

2000 Mathematics Subject Classification: Primary 37F10, 37F50, 30D05. Key words and phrases. Complex dynamics, Julia sets, polynomial semigroups, random iteration, random complex dynamics.

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 [11], who were interested in the role of the dynamics of polynomial semigroups while studying various onecomplex-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 [38, 10]). Note that there is a strong connection between the study of dynamics of rational semigroups and that of random complex dynamics (see [29, 30, 31, 32, 33]). For example, for a random dynamical system generated by a family of polynomials, we consider the function T_{∞} : \overline{ℂ} → [0, 1] of the probability of tending to ∞. It turns out that under certain conditions, the function T_{∞} is continuous on \overline{ℂ} and varies only on the Julia set J of the associated polynomial semigroup G, and J is a thin fractal set. Moreover, T_{∞} respects the surrounding order (see Definition 1.13). The function T_{∞} is a complex analogue of the devil’s staircase or Lebesgue’s singular functions. For the detail of these results, see the second author’s works [32, 28, 39].

As is well known, the iteration of polynomial maps f_{c}(z) = z^{2} + c for c in the Mandelbrot set (where the orbit of the sole critical point {f_{c}^{n}(0)} is bounded in ℂ), provides a rich class of maps with many interesting properties. Many of these dynamic and structural properties are direct consequences the boundedness of the critical orbit. 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 see also [26, 27, 29, 30, 31, 25, 33]. This paper is a continuation of the program initiated in [29, 30, 31, 33]. One may also see the paper [19], which is based on conference talks by the authors, where preliminary (weaker) versions of a portion of the results contained herein were announced and proven.

Definition 1.1. 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.

We quote the following results from [11]. The Fatou set 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. Furthermore, when the cardinality #J(G) is three or more, J(G) is the smallest closed subset of \overline{ℂ} which contains three or more points and is backward invariant. Letting the backward orbit of z be denoted by G^{−1}(z) = ∪_{g∈G}g^{−1}({z}), we have that J(G) = \overline{G^{−1}(z)} for any z ∈ J(G) whose backward orbit contains three or more points.

We should take a moment to note that 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 iteration 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 [13, 14, 15, 18].

Although the Julia set of a rational semigroup G may not be completely invariant, J(G) has an interesting property. Namely, if G is generated by a compact family {h_{λ} : λ ∈ Λ} of rational maps, then J(G) = ⋃_{λ∈Λ} h_{λ}^{−1}(J(G)). This property is called the backward self-similarity. In particular, if G = ⟨h_{1}, …, h_{m}⟩, then J(G) = ⋃_{j=1}^{m} h_{j}^{−1}(J(G)). From this property, the dynamics of rational semigroups can be regarded as “backward iterated function systems,” and in the study of rational semigroups, we sometimes borrow and further develop techniques from iterated function systems and fractal geometry. For these things, see the second author’s works [20] – [33] and [35, 36].

Note that J(G) contains the Julia set of each element of G. Moreover, the following critically important result first due to Hinkkanen and Martin holds (see also [16]).

Theorem 1.2 ([11], Corollary 3.1). For rational semigroups G with ♯(J(G)) ≥ 3, we have

J(G) = \overline{⋃_{f∈G} J(f)}.

Remark 1.3. Theorem 1.2 can be used to easily show that F (⟨h_{λ} : λ ∈ Λ⟩) is precisely the set of z ∈ \overline{ℂ} which has a neighborhood on which every composition sequence generated by {h_{λ} : λ ∈ Λ} is normal, i.e., has stable dynamics (see [38]).

In what follows we employ the following notation. 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 denote the filled-in Julia set of g by K(g) := {z ∈ ℂ | ∪_{n∈ℕ} g^{n}({z}) is bounded in ℂ}. We note that for a polynomial g with deg(g) ≥ 2, J(g) = ∂K(g) and K(g) is the polynomial hull of J(g). The appropriate extension (to our situation with polynomial semigroups) of the concept of the filled-in Julia set is as follows. (See [11, 3] for other kinds of filled-in Julia sets.)

Definition 1.4. For a polynomial semigroup G, we set

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

and call K̂(G) the smallest filled-in Julia set.

Remark 1.5. We note that for all g ∈ G, we have K̂(G) ⊂ K(g) and g(K̂(G)) ⊂ K̂(G).

Definition 1.6. 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 subset of F (G). For research on (semi-)hyperbolicity and Hausdorff dimension of Julia sets of rational semigroups see [20, 21, 22, 23, 24, 29, 30, 31, 35, 33].

Remark 1.7. It is clear that if rational semigroup G is hyperbolic, then each g ∈ G is hyperbolic. However, the converse is not true. See Remark 5.2.

Definition 1.8. The planar postcritical set (or, the finite postcritical set) of a polynomial semigroup G is defined by

P^{∗}(G) = P (G) \ {∞}.

We say that a polynomial semigroup G is postcritically bounded if P^{∗}(G) is bounded in ℂ.

Definition 1.9. 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, i.e., G is postcritically bounded. Furthermore, we set G_{con} = {G ∈ G | J(G) is connected} and G_{dis} = {G ∈ G | J(G) is disconnected}.

Remark 1.10. If G = ⟨h_{λ} : λ ∈ Λ⟩, then P (G) = \overline{∪_{λ∈Λ} ∪_{z∈CV (h_{λ})} (G(z) ∪ \{z\})} where CV (h) denotes the critical values of h. From this one may, in the finitely generated case, use a computer to see if G ∈ G much in the same way as one verifies the boundedness of the critical orbit for the maps f_{c}(z) = z^{2} +c. The freely available software [17] can be used for this purpose.

Remark 1.11. 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 1.12. For a polynomial g of degree two or more, it is well known that ⟨g⟩ ∈ G if and only if J(g) is connected (see [2], Theorem 9.5.1). 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 [37, 29, 30, 31, 33]).

See also [25] for an analysis of the number of connected components of J(G) involving the inverse limit of the spaces 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 finite word maps in G. In fact, the number of connected components of the Julia set of a finitely generated rational semigroup is deeply related to a new kind of cohomology (so called the “interaction cohomology”), which has been introduced by the second author of this paper. Using this cohomology, one can also investigate the number of connected components of the Fatou set of a finitely generated rational semigroup.

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.

Notation: 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 1.13. We place a partial order on the space of all non-empty connected sets in ℂ as follows. For connected 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}”.

Convention: When a set K_{1} is contained in the unbounded component of \overline{ℂ} \ K_{2} we say that K_{1} is “outside” K_{2}.

Theorem 1.14 ([29, 33]). 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}).

Remark 1.15. We note that under the hypothesis of the above theorem J_{1} < J_{2} for J_{1}, J_{2} ∈ J does not necessarily imply g^{∗}(J_{1}) < g^{∗}(J_{2}), but only that g^{∗}(J_{1}) ≤ g^{∗}(J_{2}) as can be seen in Example 4.7. The Julia set of a G ∈ G_{dis} is shown in Figure 1.

Figure 1. The Julia set of G = ⟨h_{1}, h_{2}⟩, where we set g_{1}(z) := z^{2} − 1, g_{2}(z) := z^{2}/4, h_{1} := g_{1}^{2}, h_{2} := g_{2}^{2}. G ∈ G_{dis} and G is hyperbolic.

We now present the main results of this paper, first giving some notation that will be needed to state our result on the existence of quasicircles in J(G).

Notation: Given polynomials α_{1} and α_{2}, we set Σ_{2} = {x = (γ_{1}, γ_{2}, …) : γ_{k} ∈ {α_{1}, α_{2}}}. Then, for any x = (γ_{1}, γ_{2}, …) ∈ Σ_{2}, we set J_{x} equal to the set of points z ∈ \overline{ℂ} where the sequence of functions \{γ_{n} ∘ ⋯ ∘ γ_{1}\}_{n∈ℕ} is not normal. This is sometimes called the Julia set along the trajectory (sequence) x ∈ Σ_{2}. See [21, 22, 24, 30, 31, 33] for much more on such fiberwise dynamics.

Theorem 1.16. Let G ∈ G_{dis} and let A and B be disjoint subsets of \overline{ℂ}. Suppose that A is a doubly connected component of F (G) and B satisfies one of the following conditions:

• B is a doubly connected component of F (G),

• B is the connected component of F (G) with ∞ ∈ B,

• B = K̂(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 a non-empty 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) Letting Σ_{2} denote the sequence space associated with {α_{1}, α_{2}}, we have (a) J(H) = ⋃_{x∈Σ_{2}} J_{x} (disjoint union),

(b) for any component J of J(H) there exists a unique element x ∈ Σ_{2} with J = J_{x}, and

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

a K-quasicircle.

(4) {J_{x}}_{x∈Σ_{2}} is totally ordered with ≤, consisting of mutually disjoint subsets of J(H).

(5) For each x ∈ Σ_{2}, the set J_{x} separates \overline{A} from \overline{B}.

Remark 1.17. It should be noted that in the above theorem, the quasicircles J_{x} are all disjoint components of J(H), but may all lie in the same component of J(G). See the proof of Theorem 1.23, where a semigroup is constructed such that there exist only a finite number of components of the Julia set.

Remark 1.18. There are many hyperbolic polynomial semigroups G = ⟨α_{1}, α_{2}⟩ ∈ G_{dis} such that for a generic x ∈ Σ_{2}, the fiberwise Julia set J_{x} is a Jordan curve but not quasicircle, the unbounded component of \overline{ℂ} \ J_{x} is a John domain, and the bounded component of \overline{ℂ} \ J_{x} is not a John domain (see [31, 33, 30]). See Figure 1. This phenomenon does not occur in the usual iteration dynamics of a single polynomial.

Example 1.19. 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, j ≥ 2. Then, for |a|^{k−1} ≠ |b|^{j−1}, the sets J(f_{1}) and J(f_{2}) are disjoint 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} each greater than or equal to 2 (if k and j are not both equal to 2 then m_{1} = m_{2} = 1 will also suffice), 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 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.

For our remaining results we need to note the existence of both a minimal element and a maximal element in J and state a few of their properties.

Theorem 1.20 ([29, 33]). Let G be a polynomial semigroup in G_{dis}. Then there is a unique element J_{min}(G) (abbreviated by J_{min} ) ∈ J such that J_{min} meets (and therefore contains) ∂K̂(G). Also, ∞ ∈ F (G) and there exists a unique element J_{max}(G) (abbreviated by J_{max}) ∈ J such that J_{max} meets (and therefore contains) ∂U_{∞}, where U_{∞} is the simply connected component of F (G) which contains ∞. Moreover, intK̂(G) ≠ ∅. Furthermore, we have the following

• J_{min} ≤ J for all J ∈ J,

• J_{max} ≥ J for all J ∈ J,

• K̂(G), and therefore P^{∗}(G), is contained in the polynomial hull of each J ∈ J.

Remark 1.21. We see that ∂K̂(G) ⊂ J(G) when G ∈ G, but, in general, we do not have ∂K̂(G) = J(G), unlike in iteration theory where ∂K(g) = J(g) for polynomials g of degree two or more. In fact, ∂K̂(G) might not even equal J_{min}(G) either (see Example 4.24).

Remark 1.22. When G ∈ G_{con} we will use the convention that J_{min} = J_{max} = J(G) and note that it is still the case that J_{min} meets ∂K̂ and P^{∗}(G) is contained in the polynomial hull of each J ∈ J. However, it is not necessarily the case that ∞ ∈ F (G), as exhibited by the example ⟨z^{2}/n : n ∈ ℕ⟩.

In the proofs of many results concerning postcritically bounded polynomial semigroups, it is critical to understand the distribution of the sets J(g) where g ∈ G, especially when g is a generator of G. In particular, it is important to understand the relationship between such J(g) and the special components J_{min} and J_{max} of J(G). In Section 4 we investigate such matters carefully providing several results including Theorem 1.23 below.

In [29], it was shown that, for each positive integer k, there exists a semigroup G ∈ G_{dis} with 2k generators such that J(G) has exactly k components. Furthermore, in [25] it was shown that any semigroup in G generated by exactly three elements will have a Julia set with either one component or infinitely many components (examples where the number of components is one, ℵ_{0} or uncountable were given). Hence we have the following question: For fixed integer k ≥ 3, what is the fewest number of generators that can produce a semigroup G ∈ G_{dis} with ♯J = k? The answer to this question is four as stated in Theorem 1.23 below.

Theorem 1.23. For any k ∈ ℕ, there exists a 4-generator polynomial semigroup H ∈ G such that #J_{H} = k. Furthermore, H can be chosen so that no J ∈ J_{H} \ {J_{min}, J_{max}} meets the Julia set of any generator of H.

The next two results, whose proofs depend on understanding the distribution of the J(g) within J(G), concern the (semi-)hyperbolicity of polynomial semigroups in G. In particular, they show how one can build larger (semi-)hyperbolic polynomial semigroups in G from smaller ones by including maps with certain properties. We first state two definitions.

Definition 1.24. We define Poly = \{h : \overline{ℂ} → \overline{ℂ} | h is a non-constant polynomial\}, endowed with the topology of uniform convergence on \overline{ℂ} with respect to the spherical metric.

Remark 1.25. For use later we note that given integer d ≥ 1, a sequence p_{n} of polynomials of degree d converges to a polynomial p in Poly if and only if the coefficients converge appropriately and p is of degree d.

Definition 1.26. 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 1.27. Let H ∈ G, Γ be a compact family in Poly, and let G = ⟨H, Γ⟩ be the polynomial semigroup generated by H and Γ. Suppose

(1) G ∈ G_{dis},

(2) J(γ) ∩ J_{min}(G) = ∅ for each γ ∈ Γ, and

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

Remark 1.28. Theorem 1.27 would not hold if we were to replace both instances of the word semi-hyperbolic with the word hyperbolic (see Example 5.1). However, with an additional hypothesis we do get the following result.

Theorem 1.29. Let H ∈ G, Γ be a compact family in Poly, and let G = ⟨H, Γ⟩ be the polynomial semigroup generated by H and Γ. Suppose

(1) G ∈ G_{dis},

(2) J(γ) ∩ J_{min}(G) = ∅ for each γ ∈ Γ,

(3) H is hyperbolic, and

(4) for each γ ∈ Γ, the critical values of γ do not meet J_{min}(G). Then, G is hyperbolic.

Remark 1.30. We note that hypothesis (3) can be replaced by the slightly weaker hypothesis that P^{∗}(H) ∩ J(H) = ∅ since if ∞ ∈ J(H) ⊂ J(G), then J(G) is connected by Theorem 1.20 and so hypothesis (1) fails to hold. A similar remark can be made about hypothesis (3) in Theorem 1.27.

Remark 1.31. Theorems 1.27 and 1.29 do not require that H or G be generated by a finite, or even compact, subset of Poly.

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 1.16. In Section 4 we provide a more detailed look at the distribution of J(g) within J(G), in particular, proving Theorem 1.23. In Section 5 we give the proofs of Theorems 1.27 and 1.29 along with Example 5.1.

2. Background and Tools

We first state some notation to be used later.

Notation: Given any set A ⊂ \overline{ℂ} we denote by \overline{A} the closure of A in \overline{ℂ}. For z_{0} ∈ ℂ and r, R > 0 we set B(z_{0}, r) = {z ∈ ℂ : |z − z_{0}| < r}, C(z_{0}, r) = {z ∈ ℂ : |z − z_{0}| = r}, and Ann(z_{0}; r, R) = {z ∈ ℂ : r < |z − z_{0}| < R}. Furthermore, given any set C ⊂ ℂ we denote the ϵ−neighborhood of C by B(C, ϵ) = ∪_{z∈C}B(z, ϵ).

Most often our understanding of the surrounding order ≤ given in Definition 1.13 will be applied to compact connected sets in ℂ and so in this section we state many results which we will need later. Although not all connected compact sets in ℂ are comparable in the surrounding order, we do have the following two lemmas whose proofs we leave to the reader.

Lemma 2.1. 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.

Definition 2.2. 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.

Lemma 2.3. Let A and B be compact connected subsets of ℂ such that P H(A) ∩ P H(B) ≠ ∅ Then exactly one of the following holds:

(1) A < B

(2) B < A

(3) A ∩ B ≠ ∅.

Remark 2.4. We note that for compact connected sets A and B in ℂ, it follows that A < B if and only if P H(A) < B since the set P H(A) is also compact and connected.

Lemma 2.5. Let g be a polynomial of degree at least one and suppose B ⊂ P H(A) where g(B) ⊂ B and A ⊂ ℂ is compact. Then B ⊂ P H(g^{−1}(A)). In particular, if g ∈ G ∈ G and P^{∗}(G) ⊂ P H(A) where A ⊂ ℂ is compact, then P^{∗}(G) ⊂ P H(g^{−1}(A)).

Proof. Suppose B ⊈ P H(g^{−1}(A)). Thus there exists z_{0} ∈ B 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 B is forward invariant we have that g(z_{0}) ∈ B \ P H(A) which contradicts our hypothesis. □

Corollary 2.6. 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. By Theorem 1.20 we have P^{∗}(G) ⊂ P H(J) for all J ∈ J. By Remark 1.11 P^{∗}(G) ⊂ K(f) = P H(J(f)). The other cases then follow from Lemma 2.5. □

Lemma 2.7 ([12]). 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 2.8. 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 ∞, i.e., the finite critical values of g lie in P H(K). Then g^{−1}(K) is connected. Further, if K_{1} is a connected compact set such that 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_{g}(·) 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.

Suppose that g^{−1}(K) is not connected. Then there exists a bounded component V_{0} of ℂ \ g^{−1}(K) which is not simply connected (see [2], Proposition 5.1.5). Thus there exists a Jordan curve γ ⊂ V_{0} such that the bounded component B of ℂ \ γ contains some component E of g^{−1}(K). Hence V_{0} ∪ B is open and does not meet V. By Lemma 2.7 we have g(E) = K. Hence g(V_{0} ∪ B) ⊃ K ⊃ ∂U, which, by the Open Mapping Theorem, implies g(V_{0} ∪ B) meets U, and thus V_{0} ∪ B meets V. This contradiction implies that V_{0} is simply connected and hence g^{−1}(K) is connected.

Now suppose K < K_{1}. Let U_{0} be the bounded component of ℂ \ K_{1} such that U_{0} ⊃ K and let U_{1} be the unbounded component of ℂ \ K_{1}. Hence g^{−1}(U_{0}) does not meet V_{1} = g^{−1}(U_{1}). Hence P H(g^{−1}(K_{1})) = ℂ\V_{1} ⊃ g^{−1}(U_{0}) ⊃ g^{−1}(K). Since g^{−1}(K_{1}) ∩ g^{−1}(K) = ∅ (which follows from the fact that K_{1} ∩ K = ∅), we conclude that g^{−1}(K_{1}) > g^{−1}(K). □

Corollary 2.9. 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. Any finite critical value of g must lie in P^{∗}(G) ⊂ P H(J) ∩ P H(J(h)) by Corollary 2.6. The result then follows immediately from Lemma 2.8. □

Corollary 2.10. 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 is immediate from Corollary 2.9, Corollary 2.6 and Lemma 2.3. □

The following lemma will allow one to understand the surrounding order through an imbedding, of sorts, into the real numbers.

Lemma 2.11. Suppose z_{0} ∈ P H(C_{1}) ∩ P H(C_{2}) where C_{1} and C_{2} are disjoint compact connected sets in ℂ. Then dist(z_{0}, C_{1}) < dist(z_{0}, C_{2}) if and only if C_{1} < C_{2} in the surrounding order.

Proof. First suppose that d_{2} = dist(z_{0}, C_{2}) > dist(z_{0}, C_{1}). Then we have B(z_{0}, d_{2}) ⊂ P H(C_{2}). Since B(z_{0}, d_{2})∩C_{1} ≠ 0 and C_{1} ∩C_{2} = ∅, we must have that the bounded component of \overline{ℂ} \ C_{2} which contains the connected set B(z_{0}, d_{2}) also contains the connected set C_{1}. Thus C_{2} > C_{1}.

Suppose C_{1} < C_{2}. Letting d_{1} = dist(z_{0}, C_{1}) we see that \overline{B(z_{0}, d_{1})} ⊂ P H(C_{1}) < C_{2} implies \overline{B(z_{0}, d_{1})} must not meet C_{2}, i.e., d_{1} < dist(z_{0}, C_{2}). □

Lemma 2.12. Let {C_{α}}_{α∈A} be a collection of non-empty compact connected sets in ℂ that are linearly ordered by the surrounding order ≤. Suppose {C_{β}}_{β∈B} is a sub-collection of \{C_{α}\}_{α∈A} such that \overline{∪_{β∈B}C_{β}} ⊂ ∪_{α∈A}C_{α}. Then both inf_{β∈B} C_{β} and sup_{β∈B} C_{β} exist and are in {C_{α}}_{α∈A}.

Proof. By compactness and the linear ordering on {C_{α}}_{α∈A}, one can quickly show that the collection {P H(C_{α})}_{α∈A} satisfies the finite intersection property. Thus there exists some z_{0} ∈ ∩_{α∈A}P H(C_{α}). For each β ∈ B, let r_{β} = dist(z_{0}, C_{β}) and consider r_{0} = inf r_{β}.

We only need to consider the case where r_{0} < r_{β} for all β ∈ B, since if r_{β_{0}} = r_{0}, then clearly, by Lemma 2.11, C_{β_{0}} = inf_{β∈B} C_{β}. Select a strictly decreasing sequence r_{β_{n}} → r_{0}. By Lemma 2.11, we have that C_{β_{1}} > C_{β_{2}} > . . .. Let z_{β_{n}} ∈ C_{β_{n}} be arbitrary. Without loss of generality we may assume that z_{β_{n}} → a_{0} ∈ ℂ. By hypothesis there exists C_{α_{0}} which contains a_{0}. We will now show that C_{α_{0}} = inf_{β∈B} C_{β}.

Fixing β ∈ B and applying Lemma 2.11, we see that {a_{0}} < C_{β}, since the sequence (z_{β_{k}})_{k≥n} must lie in P H(C_{β_{n}}) < C_{β} for large n (whenever r_{β_{n}} < r_{β}). Thus we must have C_{α_{0}} < C_{β} for all β ∈ B. Hence C_{α_{0}} is a lower bound for {C_{β}}_{β∈B}. Suppose that C_{α_{1}} > C_{α_{0}}. It must then be the case that {a_{0}} < C_{α_{1}} and so it follows that {z_{β_{n}}} < C_{α_{1}} for large n. Thus C_{β_{n}} < C_{α_{1}} for large n, implying that C_{α_{1}} is not a lower bound for {C_{β}}_{β∈B}. We conclude that C_{α_{0}} = inf_{β∈B} C_{β}.

The proof that sup_{β∈B} C_{β} exist in {C_{α}}_{α∈A} follows a similar argument using sup r_{β} and Lemma 2.11. We omit the details. □

By the proof of the above lemma we see that if \overline{∪_{β∈B}C_{β}} = ∪_{β∈B}C_{β}, then both inf_{β∈B} C_{β} and sup_{β∈B} C_{β} are in {C_{β}}_{β∈B}. Thus we have the following.

Lemma 2.13. Let {C_{β}}_{β∈B} be a collection of compact connected sets in ℂ that are linearly ordered by the surrounding order ≤. If \overline{∪_{β∈B}C_{β}} = ∪_{β∈B}C_{β}, then we can conclude that both min_{β∈B} C_{β} and max_{β∈B} C_{β} exist.

Lemma 2.14. Let f ∈ G ∈ G. Let K be a connected compact set in ℂ such that P H(K) ⊃ P^{∗}(f).

(a) Let J(f) > K. Then J(f) > f^{−1}(K). Also, f^{−1}(K) > K or f^{−1}(K) ∩ K ≠ ∅.

(b) Let J(f) < K. Then J(f) < f^{−1}(K). Also, f^{−1}(K) < K or f^{−1}(K) ∩ K ≠ ∅.

Proof. We now prove (a). We first note that J(f) = f^{−1}(J(f)) > f^{−1}(K) follows immediately from Lemma 2.8. Since P^{∗}(f) ⊂ P H(K) < J(f) we see that f cannot have a Siegel disk or parabolic fixed point. Hence, f must have a finite attracting fixed point z_{0}. Furthermore, since P H(K) is connected and P^{∗}(f) ⊂ P H(K) < J(f), it is clear that there can be only one attracting fixed point for f and K must lie in the immediate attracting basin A_{f}(z_{0}). Since P H(f^{−1}(K)) ⊃ P^{∗}(f) by Lemma 2.5, we see that f^{−1}(K) also lies in A_{f}(z_{0}). Hence A_{f}(z_{0}) must be completely invariant under f. This implies F (f) has only two components A_{f}(∞) and A_{f}(z_{0}), each which are simply connected (see [2], Theorem 5.6.1).

Letting φ : A_{f}(z_{0}) → B(0, 1) be the Riemann map such that z_{0} ↦ 0, then one may apply Schwarz’s Lemma to the degree greater than or equal to two (finite Blaschke product) map B = φ ∘ f ∘ φ^{−1} to show that any point mapped to a point of maximum modulus of φ(K) must lie outside of φ(K). Thus either f^{−1}(K) > K or f^{−1}(K) ∩ K ≠ ∅.

Part (b) is proved more easily than (a) since it is already known that A_{f}(∞) is simply connected. Then one can similarly examine the Riemann map from A_{f}(∞) to B(0, 1) such that ∞ ↦ 0. □

We note that Theorem 1.20 along with the proof of part (a) above, with K = J_{min}, proves the following (which has been already shown in [29, 33]).

Lemma 2.15. Let f ∈ G ∈ G be such that J(f) > J_{min}, i.e., J(f) ∩ J_{min} = ∅. Then f has an attracting fixed point z_{0} ∈ ℂ and F (f) consists of just two simply connected immediate attracting basins A_{f}(∞) and A_{f}(z_{0}).

We note that the maps f = h_{1}^{2} and g = h_{2}^{2} where h_{1}(z) = z^{2}−1 and h_{2}(z) = z^{2}/4 generate G = ⟨f, g⟩ ∈ G_{dis} where f has two finite attracting fixed points (see [31, 33]). Thus we see that the condition J(f) ∩ J_{min} = ∅ in the above lemma is indeed necessary. We also note that for this G, the phenomena in Remark 1.18 holds. See Figure 1.

Lemma 2.16. Let f ∈ G ∈ G. Let K be a connected set in J(G) containing three or more points such that f^{−n}(K) is also connected for each n ∈ ℕ. If f^{−1}(K)∩K ≠ ∅, then J(f) and K are contained in the same component J ∈ J.

Proof. The lemma follows from the fact that the connected set \overline{∪_{n=0}^{∞}f^{−n}(K)} in J(G) must meet J(f). □

We now present a general topological lemma that will be used to justify a corollary which will be needed later.

Lemma 2.17. Let \{C_{α}\}_{α∈A} be a collection of compact connected sets in \overline{ℂ}. Let ϵ > 0 and let C be any connected component of \overline{∪_{α∈A}C_{α}}. Then there exists α ∈ A such that C_{α} ⊂ B(C, ϵ).

Proof. Choose any z ∈ C and let α_{n} ∈ A be such that dist(z, C_{α_{n}}) → 0. By compactness in the topology generated by the Hausdorff metric on the space of non-empty compact subsets of \overline{ℂ}, we then may conclude (by passing to subsequence if necessary) that C_{α_{n}} → K for some non-empty connected compact set K, which therefore must contain z and hence be contained in C. Thus for large n we have B(C, ϵ) ⊃ B(K, ϵ) ⊃ C_{α_{n}}. □

Using the fact that J(g) is connected whenever g ∈ G ∈ G we clearly obtain the following slight generalization of Lemma 4.2 in [29].

Corollary 2.18. Let {g_{λ}}_{λ∈Λ} ⊂ G ∈ G. Let ϵ > 0 and let C be any connected component of \overline{∪_{λ∈Λ}J(g_{λ})}. Then there exists λ ∈ Λ such that J(g_{λ}) ⊂ B(C, ϵ).

In particular, we apply Theorem 1.2 to obtain that if {g_{λ}}_{λ∈Λ} = G ∈ G and J ∈ J, then for every ϵ > 0 there exists g ∈ G such that J(g) ⊂ B(J, ϵ).

3. Proof of Theorem 1.16

We first present a definition and a lemma that will assist in the proof of Theorem 1.16.

Definition 3.1. 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 \ P H(K_{1}) where U is the bounded component of ℂ \ K_{2} which contains K_{1}. Thus Ann(K_{1}, K_{2}) is the open doubly connected region “between” K_{1} and K_{2}.

Remark 3.2. For any compact connected set A ⊂ Ann(K_{1}, K_{2}) we immediately see that A < K_{2} and, by Lemma 2.1, either K_{1} and A are outside of each other or K_{1} < A.

Lemma 3.3. 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. Corollary 2.6, Lemma 2.14(a), and Lemma 2.16 show that g^{−1}(J(f)) > J(f). Set X = g^{−1}(J(f)), A = g^{−m}(J(f)) and B = g^{−(m+1)}(J(f)) and note that J(f) < A < B by Lemma 2.8. Keeping Lemma 2.15 in mind, we may 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 2.10 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. □

We will require the following result which was proved via fiberwise quasiconformal surgery by the second author.

Proposition 3.4 ([31], Proposition 2.25). Let G = ⟨α_{1}, α_{2}⟩ ∈ G be hyperbolic such that P^{∗}(G) is contained in a single component of int(K̂(G)). Then there exists K ≥ 1 such that for all sequences x ∈ Σ_{2}, the set J_{x} is a K-quasicircle.

Remark 3.5. Under the hypotheses above we know that for each g ∈ G, the set J(g) is a quasicircle (see [8], p. 102). But the above result shows much more as it shows that the Julia sets along sequences are also all quasicircles, and that all such quasicircles have uniform dilations.

We now can present the proof of Theorem 1.16.

Proof of Theorem 1.16. We first give a proof in the case that A and B are doubly connected components of F (G). Since the doubly connected components of F (G) are linearly ordered by ≤, we may assume without loss of generality that A < B.

Let γ_{A} be a non-trivial curve in A (i.e., γ_{A} separates the components of ℂ \ A) and let γ_{B} be a non-trivial curve in B. Since J(G) = \overline{∪_{g∈G}J(g)}, the bounded component of ℂ \ γ_{A} and Ann(γ_{A}, γ_{B}) both meet J(G), and both A and B do not meet J(G), there must exists maps f, g ∈ G such that J(f) < γ_{A} and γ_{A} < J(g) < γ_{B}. Note then that J(g) < B since J(g) ∩ B = ∅. Since J(f) and J(g) lie indifferent components of J(G) (separated by A), J(g) ⊂ \overline{∪_{n∈ℕ}g^{−n}(J(f))}, and each g^{−n}(J(f)) ∩ A = ∅, there exists n_{0} ∈ ℕ such that g^{−n_{0}}(J(f)) > γ_{A} and thus g^{−n_{0}}(J(f)) > A. By Lemma 3.3 there exists k ∈ G such that A < g^{−n_{0}}(J(f)) < J(k) < g^{−(n_{0}+1)}(J(f)) < J(g).

We now find a sub-semigroup H′ that satisfies conclusions (1) - (4) of the theorem. Keeping Lemma 2.15 in mind, we see that we may choose m_{1}, m_{2} ∈ ℕ large (as in Example 1.19), such that β_{1} = k^{m_{1}} and β_{2} = g^{m_{2}} generate a sub-semigroup H′ of G where J(H′) is disconnected and contained in \overline{Ann(J(k), J(g))}. Further, H′ is hyperbolic since P^{∗}(H′) ⊂ P^{∗}(G) ⊂ K(f) < J(k). By choosing U to be a suitable open set containing \overline{Ann(J(k), J(g))} we see that H′ satisfies parts (1) and (2) of the theorem.

By Theorem 2.14(2) in [22], the hyperbolicity of H′ implies J(H′) = ∪_{x∈Σ′_{2}}J_{x}, where Σ′_{2} is the sequence space corresponding to the maps β_{1} and β_{2}. The fact that J_{x_{1}} ≠ J_{x_{2}} when x_{1} ≠ x_{2} follows in much the same way as the proof that the standard middle-third Cantor set is totally disconnected. We present the details now. First we define σ to be the shift map on Σ′_{2} given by σ(γ_{1}, γ_{2}, …) = (γ_{2}, γ_{3}, …). Then, for x = (γ_{1}, γ_{2}, …), one can show by using the definition of normality J_{x} = γ_{1}^{−1}(J_{σ(x)}) and thus by induction J_{x} = γ_{1}^{−1}(J_{σ(x)}) = ⋯ = (γ_{n}∘⋯∘γ_{1})^{−1}(J_{σ^{n}(x)}) ⊂ (γ_{n} ∘ ⋯ ∘ γ_{1})^{−1}(J(H′)). Thus J_{x} ⊂ ∩_{n=1}^{∞}(γ_{n} ∘ ⋯ ∘ γ_{1})^{−1}(J(H′)). But by (induction on) condition (1) we can see that this intersection will produce distinct sets for distinct sequences in Σ′. Thus we have shown that J_{x_{1}} ≠ J_{x_{2}} when x_{1} ≠ x_{2}

Each J_{x} is connected by Lemma 3.6 in [31]. Hence we have shown parts 3(a) and 3(b). Now part (4) is then clear by 3(a), 3(b), and Theorem 1.14(1). Part 3(c) now follows directly from Proposition 3.4.

We have thus shown that H′ = ⟨β_{1}, β_{2}⟩ satisfies items (1) - (4) of the theorem, but it is not certain that J(g) does not meet \overline{B}, and so (5) remains in question. However, letting α_{1} = β_{1} and α_{2} = β_{1} ∘ β_{2} (note J(α_{2}) < J(β_{2}) < B), we see that H = ⟨α_{1}, α_{2}⟩ will satisfy (1) - (5). We have thus proved the result in the case that A and B are both doubly connected Fatou components.

Consider the case where B is the unbounded component of F (G) containing ∞. As above we obtain f, g ∈ G such that J(f) < γ_{A} < J(g) where γ_{A} is a non-trivial curve in A. We then follow the above method to complete the proof.

Finally, we consider the case where B = K̂. As above we obtain f, g ∈ G such that J(g) < γ_{A} < J(f) where γ_{A} is a non-trivial curve in A. We then follow the above method, noting that the surrounding order inequalities are now reversed from above, to complete the proof. □

4. Structural properties of J

In this section we discuss issues related to the topological nature of J as well as discuss issues related to the question of where the “small” Julia sets J(g) for g ∈ G reside inside of the larger Julia set J(G). In particular, we investigate the question of when it is the case that a given J ∈ J must contain J(g) for some g ∈ G. Since J_{min} and J_{max} play special roles we will be particularly interested in when these components of J(G) have this property. When G = ⟨g_{λ} : λ ∈ Λ⟩, it is of particular interest to know which J ∈ J meet J(g_{λ}) for some λ ∈ Λ. The first result in this direction is the following, which appears as Proposition 2.24 in [29].

Proposition 4.1. If G ∈ G is generated by a compact family in Poly, then both J_{min} and J_{max} must contain the Julia set of one of the generating maps of G.

In order to succinctly discuss such issues we make the following definitions.

Definition 4.2. Let G = ⟨h_{λ} : λ ∈ Λ⟩ ∈ G. We say that J ∈ J has property (⋆) if J contains J(g) for some g ∈ G. We say that J ∈ J has property (⋆λ) if J contains J(h_{λ}) for some generator h_{λ} ∈ G.

Remark 4.3. A given rational semigroup G may have multiple generating sets. For example, the whole semigroup itself can always be taken as a generating set. However, in this paper when it is written that G = ⟨h_{λ} : λ ∈ Λ⟩, it is assumed that this generating set is fixed and thus the property (⋆λ) is always in relation to this given generating set.

Lemma 4.4. Let G = ⟨h_{λ} : λ ∈ Λ⟩ ∈ G.

a) If J_{min} has property (⋆), then J_{min} has property (⋆λ).

b) If J_{max} has property (⋆), then J_{max} has property (⋆λ).

Remark 4.5. Lemma 4.4 does not apply to a general J ∈ J. Indeed, in Example 1.19 we see that only J_{min} and J_{max} have property (⋆λ), although infinitely many other J ∈ J have property (⋆).

Proof. Suppose J(g) ⊂ J_{min} for g = h_{λ_{1}} ∘⋯∘h_{λ_{k}} and J_{min}∩J(h_{λ}) = ∅ for all λ ∈ Λ. By Corollary 2.10 we have J_{min} < J(h_{λ}) for all λ ∈ Λ, and thus by Lemma 2.14, Lemma 2.16 and Theorem 1.20 we have J_{min} < h_{λ}^{−1}(J_{min}). So it also follows from Corollary 2.10 that J(g) < h_{λ}^{−1}(J(g)) for all λ ∈ Λ. Thus J(g) < h_{λ_{1}}^{−1}(J(g)) and by Lemma 2.8 J(g) < h_{λ_{2}}^{−1}(J(g)) < h_{λ_{2}}^{−1}h_{λ_{1}}^{−1}(J(g)). By repeated application of this argument we then get that J(g) < h_{λ_{k}}^{−1}(J(g)) < ⋯ < h_{λ_{k}}^{−1} … h_{λ_{1}}^{−1}(J(g)) = g^{−1}(J(g)) = J(g), which is a contradiction. From this part (a) follows.

Part (b) follows in a similar manner. □

Corollary 4.6. If J_{min} (respectively, J_{max}) has non-empty interior, then J_{min} (respectively, J_{max}) has property (⋆λ).

Proof. Suppose int(J_{min}) ≠ ∅. Since J(G) = \overline{∪_{g∈G}J(g)} some J(g) must meet int(J_{min}). Thus it follows from Lemma 4.4 that J_{min} has property (⋆λ). □

It is not always the case, however, that J_{min} and J_{max} have property (⋆λ).

Example 4.7. We will give an example of an infinitely generated G ∈ G_{dis} such that (1) J_{max} does not have property (⋆λ),

(2) #J = ℵ_{0}, and

(3) there exists J′, J′′ ∈ J and g ∈ G such that J′ < J′′ and g^{∗}(J′) = g^{∗}(J′′).

Set b_{n} = 2 − 1/n for n ∈ ℕ and ϵ_{n} = min\{\frac{b_{n+1}−b_{n}}{10} , \frac{b_{n}−b_{n−1}}{10} \}. Set A = \overline{Ann(0; 1/2, 2)}, A_{n} = Ann(0; b_{n} − ϵ_{n}, b_{n} + ϵ_{n}) and A′_{n} = Ann(ϵ_{n}; b_{n} − ϵ_{n}, b_{n} + ϵ_{n}). Note that the set Ã_{n} := \overline{A_{n} ∪ A′_{n}} ⊂ \overline{Ann(0; b_{n} − 2ϵ_{n}, b_{n} + 2ϵ_{n})} and so, by the choice of the ϵ_{n}, the Ã_{n} are disjoint. Choose polynomials f_{n} and g_{n} such that J(f_{n}) = C(0, b_{n}) and J(g_{n}) = C(ϵ_{n}, b_{n}). Choose m_{n} ∈ ℕ large enough so that h_{n} = f_{n}^{m_{n}} yields h_{n}^{−1}(A) ⊂ A_{n}, h_{n}(\overline{B(0, 1/2)}) ⊂ B(0, 1/2) and h_{n}(\{|z| ≥ 2\}) ⊂ \{|z| > 5\}. Choose j_{n} ∈ ℕ large enough so that k_{n} = g_{n}^{j_{n}} yields k_{n}^{−1}(A) ⊂ A′_{n}, k_{n}(\overline{B(0, 1/2)}) ⊂ B(0, 1/2) and k_{n}(\{|z| ≥ 2\}) ⊂ \{|z| > 5\}. Note that Ã := \overline{∪_{n=1}^{∞}Ã_{n}} ⊂ A. Let G = ⟨h_{n}, k_{n} : n ∈ ℕ⟩ and note that P^{∗}(G) ⊂ B(0, 1/2) and G^{−1}(Ã) ⊂ G^{−1}(A) ⊂ Ã which implies J(G) ⊂ \overline{G^{−1}(Ã)} ⊂ Ã.

We see by forward invariance that {|z| > 2} ⊂ F (G), but note that C(0, 2) ∈ J (since the open sets Ann(Ã_{n}, Ã_{n+1}) are all in F (G)). Also, for no g ∈ G does J(g) meet C(0, 2) else there would exist z_{0} ∈ C(0, 2) ∩ J(g) such that |g(z_{0})| ≤ 2, contradicting the fact that each g ∈ G maps C(0, 2) into {|z| > 5}. Thus J_{max} = C(0, 2) fails to have property (⋆).

We now show that #J = ℵ_{0}. Letting J_{n} ∈ J be such that J_{n} contains the overlapping sets J(h_{n}) and J(k_{n}) we note that, since J_{n} ⊂ Ã_{n} and the Ã_{n} are separated from each other, each J_{n} is isolated from the other J_{m}, i.e., for each n there exists ϵ_{n} > 0 such that the ϵ_{n}-neighborhood B(J_{n}, ϵ_{n}) does not meet any other J_{m} ∈ J.

Let C = C(0, 2). Since C ⊂ J(G) we see that J(G) = \overline{G^{−1}(C)}. We now show that for each g ∈ G, the set g^{−1}(C) ⊂ J_{n} for some n. Write g = g_{i_{1}} ∘ … g_{i_{j}} where each g_{i_{ℓ}} is a generator for G. Suppose that g_{i_{j}} = h_{n} for some n. Then, by the backward invariance of A under each map in G, we have that g^{−1}(C) ⊂ g^{−1}(A) ⊂ h_{n}^{−1}(A) ⊂ A_{n}. Since g^{−1}(C) is connected, g^{−1}(C) > B(0, 1/2), and J(k_{n}) ∪ J(h_{n}) meets both the inner boundary and outer boundary of A_{n}, we must have that g^{−1}(C) meets J(k_{n}) ∪ J(h_{n}) and thus g^{−1}(C) ⊂ J_{n}. Note that the same argument (using A′_{n}) holds if g_{i_{j}} = k_{n}. Thus we have shown that G^{−1}(C) ⊂ ∪_{n∈ℕ}J_{n}. Since the J_{n} are isolated from each other and accumulate only to C, it follows that J(G) = \overline{G^{−1}(C)} ⊂ \overline{∪_{n∈ℕ}J_{n}} ⊂ C ∪ ⋃_{n∈ℕ} J_{n} and so #J = ℵ_{0}. Note also then that we must have J_{min} = J_{1} and so J_{min} does have property (⋆λ).

We now show that J′ < J′′ for J′, J′′ ∈ J does not necessarily imply g^{∗}(J′) < g^{∗}(J′′). Indeed, we see that h_{n}^{∗}(J) = J_{n} for all J ∈ J.

We note that we could easily adapt this example (by letting b_{n} = 0.5 + 1/n) to produce G_{1} ∈ G such that J_{min}(G_{1}) does not have property (⋆λ), but J_{max} does. Or we could produce G_{2} = ⟨G, G_{1}⟩ ∈ G such that neither J_{min} nor J_{max} has property (⋆λ).

Note that in the above example(s) where J_{min} (respectively J_{max}) did not meet ∪_{λ∈Λ}J(g_{λ}), it was true that J_{min} (respectively J_{max}) was contained in \overline{∪_{λ∈Λ}J(g_{λ})}. We will prove in Theorem 4.9 that this is indeed always the case. First we need to prove the following lemma.

Lemma 4.8. Let {g_{λ}}_{λ∈Λ} ⊂ G ∈ G_{dis}. Let C denote the connected components of \overline{∪_{λ∈Λ}J(g_{λ})}. Then both M′ = min_{C∈C} C and M′′ = max_{C∈C} C exist (with respect to the surrounding order ≤). Also, P H(C) ⊃ K̂(G) ⊃ P^{∗}(G) for each C ∈ C.

Proof. First we note that ∞ ∈ F (G) by Theorem 1.20 and so all sets in C are contained in ℂ. Let C ∈ C. Suppose that z_{0} ∈ K̂(G) \ P H(C). Let γ be a curve in \overline{ℂ} \ P H(C) connecting z_{0} to ∞ and set ϵ = dist(γ, P H(C)). By Corollary 2.18, there exists λ ∈ Λ such that J(g_{λ}) ⊂ B(C, ϵ). Hence, we see that γ is outside J(g_{λ}) implying that g_{λ}^{n}(z_{0}) → ∞ and thus contradicting the fact that z_{0} ∈ K̂(G).

Lemma 2.3 shows that the compact connected sets in C are linearly ordered with respect to the surrounding order. The existence of M′ and M′′ then follows directly from Lemma 2.13. □

Theorem 4.9. Consider G = ⟨g_{λ} : λ ∈ Λ⟩ ∈ G_{dis}. Let A = ∪_{λ∈Λ}J(g_{λ}) and denote by M′ and M′′ the minimal and maximal connected components of \overline{A}, respectively. Then both J_{min} ⊃ M′ and J_{max} ⊃ M′′ and, in particular, both J_{min} ∩ \overline{A} ≠ ∅ and J_{max} ∩ \overline{A} ≠ ∅. Furthermore, we have the following.

(1) If J_{min} ∩ A = ∅ (i.e., J_{min} does not have property (⋆λ)), then J_{min} = M′ and J_{min} is the boundary of the unbounded component of ℂ \ J_{min}.

(2) If J_{max} ∩ A = ∅ (i.e., J_{max} does not have property (⋆λ)), then J_{max} = M′′ and J_{max} is the boundary of the bounded component of ℂ \ J_{max} which contains J_{min}.

Remark 4.10. In the above theorem, if J(G) is connected , then J_{min} = J(G) = J_{max} meets all J(g) such g ∈ G and thus meets A.

Open Question: We notice in Example 4.7 that J_{max} ∩ A = ∅ and J_{max} is a simple closed curve. However, it is not clear, in general, whether the hypothesis J_{max} ∩ A = ∅ for G = ⟨g_{λ} : λ ∈ Λ⟩ ∈ G_{dis}, must necessarily lead to the conclusion that J_{max} is a simple closed curve. It is also not clear under this hypothesis whether J_{max} must be the common boundary of exactly two complementary domains. So we state these as open questions (noting the corresponding questions regarding J_{min} are also open).

Remark 4.11. We note that we could also use Theorem 4.9 to see that J_{min} in Example 4.7 has property (⋆λ), since J_{min} must meet \overline{∪_{j=1}^{∞}J(k_{j}) ∪ J(h_{j})} = C(0, 2)∪ ∪_{j=1}^{∞}(J(k_{j}) ∪ J(h_{j})), but does not meet C(0, 2).

Proof. Let J′, J′′ ∈ J be such that M′ ⊂ J′ and M′′ ⊂ J′′. Fix λ ∈ Λ. By the minimality of M′, we have either J(g_{λ}) ⊂ J′ or J(g_{λ}) > J′. Then g_{λ}^{∗}(J′) ≥ J′ by Lemmas 2.14 and 2.16 and Theorem 1.20. Since g_{λ}^{∗}(J′) ≥ J′ for all λ ∈ Λ we must have that J′ = J_{min} (because the closed set ∪_{{J∈J:J≥J′}}J is then backward invariant under each g ∈ G). Similarly we see that J′′ = J_{max}. Thus both J_{min}∩\overline{A} ≠ ∅ and J_{max} ∩ \overline{A} ≠ ∅.

We now prove (2) by first showing that J(G) ⊂ J_{min} ∪M′′ ∪Ann(J_{min}, M′′). Fix λ ∈ Λ. Since P H(M′′) ⊃ P^{∗}(G) by Lemma 4.8, we see that g_{λ}^{−1}(M′′) is connected using Lemma 2.8. Thus g_{λ}^{−1}(M′′) ∩ M′′ = ∅, else J(g_{λ}) meets J_{max} by Lemma 2.16 which violates our hypothesis that J_{max} ∩ A = ∅. Hence g_{λ}^{−1}(M′′) < M′′ by Lemma 2.14(b).

From the facts that g_{λ}^{−1}(J_{min}) ≥ J_{min} and g_{λ}^{−1}(M′′) < M′′ for all λ ∈ Λ, we deduce from Lemma 2.8 that the closed annulus-type region A_{1} = J_{min} ∪ M′′ ∪ Ann(J_{min}, M′′) is backward invariant under each generator (and thus under each g ∈ G). Hence J(G) ⊂ J_{min} ∪ M′′ ∪ Ann(J_{min}, M′′) as desired.

We now suppose there exists w ∈ J_{max} \ M′′. Such a point w must necessarily then lie in Ann(J_{min}, M′′) (since w ∈ J_{min} would imply J_{min} = J_{max} = J(G) and thus J_{max} clearly meets A). Let U be the connected component of ℂ \ M′′ which contains J_{min}. Note that w ∈ U by definition of Ann(J_{min}, M′′). Recall that ∂K̂(G) ⊂ J_{min}. Let γ be a curve in U which connects w to some z_{0} ∈ K̂(G) and set ϵ = dist(γ, M′′) > 0. By Corollary 2.18 there exists a generator g_{λ_{0}} ∈ G such that J(g_{λ_{0}}) ⊂ B(M′′, ϵ). Thus γ ∩ J(g_{λ_{0}}) = ∅. Since K̂(G) ⊂ P H(J(g_{λ_{0}})), we see that z_{0} ∈ γ ∩ P H(J(g_{λ_{0}})) and so γ < J(g_{λ_{0}}). Hence {w} < J(g_{λ_{0}}) which implies (by Corollary 2.10) either J_{max} < J(g_{λ_{0}}) or J_{max} ∩ J(g_{λ_{0}}) ≠ ∅. Since neither of these can occur we conclude that no such w exists and thus J_{max} = M′′.

Recall that U is the bounded component of ℂ \ J_{max} which contains J_{min}. Since J_{max} ∩ A = ∅, we have that for every λ ∈ Λ, the set J(g_{λ}) is contained in U. Hence \overline{A} ⊂ \overline{U} and so J_{max} = M′′ ⊂ \overline{A} ⊂ \overline{U}, which implies J_{max} = ∂U.

The proof for case (1) is similar, but simpler. In this case the point ∞ can play the role of z_{0} in order to help demonstrate that any point in J_{min} \ M′ must lie “outside” of some J(g_{λ}) (which is a contradiction). We omit the details. □

Corollary 4.12. When G = ⟨g_{λ} : λ ∈ Λ⟩ ∈ G with \overline{∪_{λ∈Λ}J(g_{λ})} = ∪_{λ∈Λ}J(g_{λ}), then both J_{min} and J_{max} must have property (⋆λ).

Remark 4.13. The above corollary applies, for example, when G = ⟨g_{λ} : λ ∈ Λ⟩ ∈ G_{dis} has ∪_{λ∈Λ}J(g_{λ}) = ∪_{k=1...n}J(g_{λ_{k}}). Such non compactly generated examples can easily be constructed. Other more “exotic” examples can also be constructed to satisfy the hypotheses of the corollary.

Example 4.14. We note that without the hypothesis that J_{min} ∩ A = ∅ in Theorem 4.9(1), the conclusion that J_{min} = M′ might not hold. Set f_{1}(z) = z^{2}, f_{2}(z) = (z − ϵ)^{2} + ϵ, and f_{3}(z) = z^{2}/4. For ϵ > 0 small and m_{1}, m_{2}, m_{3} all large we set g_{1} = f_{1}^{m_{1}}, g_{2} = f_{2}^{m_{2}}, and g_{3} = f_{3}^{m_{3}} and note that G = ⟨g_{1}, g_{2}, g_{3}⟩ ∈ G_{dis}. Thus M′ = J(f_{1}) ∪ J(f_{2}) = C(0, 1) ∪ C(ϵ, 1). However, the real point in g_{1}^{−1}({1 + ϵ}) is clearly in g_{1}^{−1}(J(f_{2})) ⊂ J_{min}, but not in M′.

Theorem 4.15. Let G = ⟨g_{λ} : λ ∈ Λ⟩ ∈ G_{dis} and suppose J_{min} ∩ ∪_{λ∈Λ}J(g_{λ}) = ∅. Then ∂K̂(G) ⊂ M′ ⊂ \overline{∪_{λ∈Λ}J(g_{λ})}, where M′ is the minimal connected component of \overline{∪_{λ∈Λ}J(g_{λ})}

Proof. By Theorem 4.9(1), we see that ∂K̂(G) ⊂ J(G) ⊂ M′∪J_{max}∪Ann(M′, J_{max}). Moreover, by Lemma 4.8, we have ∂K̂(G) ⊂ P H(M′). Hence ∂K̂(G) ⊂ M′. □

Having discussed properties (⋆) and (⋆λ) with respect to J_{min} and J_{max} we now turn our attention to a general J ∈ J. In particular, we investigate what can be said about which J ∈ J have property (⋆) or (⋆λ). We also concern ourselves with the question of when does every J ∈ J have property (⋆) or (⋆λ). We begin with the following definition.

Definition 4.16. Let G ∈ G. We say that J ∈ J is isolated in J if there exists ϵ > 0 such that B(J, ϵ) does not meet any other set in J.

Lemma 4.17. Let G ∈ G with J ∈ J isolated in J. Then J has property (⋆).

Proof. Assume that ϵ > 0 is such that B(J, ϵ) does not meet any other set in J. Since J(G) = \overline{∪_{g∈G}J(g)} by Theorem 1.2, we see that any point in J must have, within a distance ϵ, a point in some J(g), where g ∈ G. It must then be the case that J(g), which lies in some set in J, must lie entirely in J. □

Remark 4.18. If G ∈ G is such that #J < +∞, then clearly each J ∈ J is isolated in J and so each J ∈ J has property (⋆). We note, however, that if each J ∈ J is isolated in J, then it is not necessarily the case that each J ∈ J has property (⋆λ). See the proof of Theorem 1.23 where, for any positive integer k, a semigroup G′ ∈ G is constructed such that #J = k, but only J_{min} and J_{max} have properly (⋆λ).

Remark 4.19. If G = ⟨h_{λ} : λ ∈ Λ⟩ ∈ G where #Λ ≤ ℵ_{0} and #J is uncountable, then since #G = ℵ_{0} we see that some J ∈ J must fail to have property (⋆). An example of this is the Cantor set of circles in Example 1.19.

Example 4.20. Suppose G ∈ G and #J = ℵ_{0}. Then it is possible that not all J ∈ J have property (⋆) as in Example 4.7. But it is also possible that all J ∈ J do have property (⋆) as in [29] Theorem 2.26 where both J_{min} and J_{max} have property (⋆λ) and each other component of J(G) is isolated in J.

We saw above that isolated J ∈ J have property (⋆). We now show that this is also true for the components of J(G) which contain the pre-image of an isolated J ∈ J.

Claim 4.21. Let G ∈ G. If J_{1} ∈ J is isolated in J and h^{−1}(J_{1}) ⊂ J ∈ J for some h ∈ G, then J has property (⋆).

Proof. Since J_{1} is isolated in J, Lemma 4.17 implies there exists g ∈ G such that J(g) ⊂ J_{1}. Thus, since J_{1} is isolated in J, for large n ∈ ℕ we have g^{−n}(J) ⊂ J_{1}. Hence h^{−1}(g^{−n}(J)) ⊂ h^{−1}(J_{1}) ⊂ J which implies J(g^{n} ∘ h) ⊂ J. □

Open Question: If #J = ℵ_{0} and G ∈ G is finitely generated, then must every J ∈ J have property (⋆)? Note that the finitely generated condition is required by Example 4.7. Also, Example 1.19 shows that if #J is uncountable, J can have (uncountably many) J which fail to have property (⋆).

We now turn our attention to considering those semigroups where J_{min} has property (⋆λ). In particular, we examine the generating maps whose Julia sets meet J_{min} as well as the sub-semigroup generated by just these special maps.

Definition 4.22. Let G = ⟨h_{λ} : λ ∈ Λ⟩ ∈ G_{dis}. We set

B_{min} = B_{min}(G) := {λ ∈ Λ : J(h_{λ}) ⊂ J_{min}(G)}.

If B_{min} ≠ ∅, let H_{min}(G) be the sub-semigroup of G which is generated by {h_{λ} : λ ∈ B_{min}}.

The following proposition has been shown in [29, 33].

Proposition 4.23. Let G = ⟨h_{λ} : λ ∈ Λ⟩ ∈ G_{dis}. If {h_{λ} : λ ∈ Λ} is compact in Poly, then B_{min} is a proper non-empty subset of Λ under the above notation.

Proof. The result follows since by Proposition 4.1 both J_{min} and J_{max} have property (⋆λ). □

It is natural to investigate the relationship between H_{min}(G) and G. Specifically we ask, and answer, the following questions for a semigroup G ∈ G_{dis}:

(1) Must J(H_{min}(G)) = J_{min}(G)?  (2) Must J(H_{min}(G)) be connected?
(3) Must J_{min}(H_{min}(G)) = J_{min}(G)? (4) Must H_{min}(H_{min}(G)) = H_{min}(G)?

The answer to each of these questions is NO, as we see in this next example.

Example 4.24. We will construct a single 3-generator polynomial semigroup G ∈ G_{dis} which negatively answers questions (1)-(4). Furthermore, we will show that #J = ℵ_{0}.

Let h_{1}(z) = −z^{2} and f_{2}(z) = z^{2}/\sqrt{2} and note that J(h_{1}) = C(0, 1) and J(f_{2}) = C(0, \sqrt{2}). We set h_{2} = f_{2}^{m_{2}} where conditions on the large m_{2} ∈ ℕ will be specified later. Choose point P ∈ h_{2}^{−1}(J(h_{1})) such that P > 0. Note that P = \sqrt{2} − δ where δ is small for m_{2} large. Hence P^{2} = 2 − 2\sqrt{2}δ + δ^{2} and P^{4} = 4 − 8\sqrt{2}δ + O(δ^{2}).

Setting r = \frac{P^{4}+P^{2}}{2} = 3 − M_{1}δ + O(δ^{2}) and ϵ = \frac{P^{4}−P^{2}}{2} = 1 − M_{2}δ + O(δ^{2}) where M_{1}, M_{2} > 0, we see that for f_{3}(z) = \frac{(z−ϵ)^{2}}{r} + ϵ both h_{1}(P) = −P^{2} = ϵ − r and h_{1}^{2}(e^{iπ/4}P) = P^{4} = r + ϵ lie in C(ϵ, r) = J(f_{3}).

Suppose there exists w ∈ h_{1}^{−1}(J(f_{3})) ∩ J(f_{3}), i.e., h_{1}(w) ∈ J(f_{3}) and w ∈ J(f_{3}). Then |h_{1}(w)| = |w|^{2} ≥ | − P^{2}|^{2} = P^{4} since −P^{2} is the point in J(f_{3}) of smallest modulus. Since P^{4} is the point in J(f_{3}) of largest modulus, we see that h_{1}(w) could only be in J(f_{3}) if w = −P^{2}. But h_{1}(−P^{2}) = −P^{4} ∉ J(f_{3}), and so we have h_{1}^{−1}(J(f_{3})) ∩ J(f_{3}) = ∅ and thus h_{1}^{−1}(J(f_{3})) < J(f_{3}) (see Figure 1).

Since for m_{2} large we clearly have h_{2}^{−1}(J(f_{3})) < J(f_{3}), we are then free to choose A′ to be any closed annulus such that A′ ⊂ B(ϵ, r), int(A′) ≠ ∅, and A′ > h_{1}^{−1}(J(f_{3})) ∪ h_{2}^{−1}(J(f_{3})).

Choose m_{3} ∈ ℕ large enough so that h_{3} = f_{3}^{m_{3}} maps A′ into B(0, 1) (which can be done since the super attracting fixed point of f_{3} is ϵ ∈ B(0, 1)). This then implies that B(0, 1) is forward invariant under each map h_{1}, h_{2} and h_{3} and P^{∗}(G) ⊂ B(0, 1) = intK̂(G) ⊂ F (G) where G = ⟨h_{1}, h_{2}, h_{3}⟩.

Let A = J(h_{2}) ∪ h_{2}^{−1}(J(h_{1})) ∪ h_{1}^{−1}(J(h_{3})) ⊂ J(G). Because h_{1}^{−1}(J(h_{3})) is connected (Corollary 2.9) and contains both the point iP^{2}, which lies outside the circle J(h_{2}), and the point P ∈ h_{2}^{−1}(J(h_{1})), which lies inside the circle J(h_{2}), we see that the set A is connected.

Note that e^{iπ/4}P ∈ h_{2}^{−1}(J(h_{1})) ⊂ A and e^{iπ/4}P ∈ h_{1}^{−1}(h_{1}^{−1}(J(h_{3}))) ⊂ h_{1}^{−1}(A) since h_{1}^{2}(e^{iπ/4}P) = P^{4} ∈ J(h_{3}). Thus h_{1}^{−1}(A)∩A ≠ ∅ and since h_{1}^{−n}(A) is connected for each n ∈ ℕ by Lemma 2.8, we see that Lemma 2.16 implies A and J(h_{1}) are contained in a single J ∈ J. Since P^{∗}(G) ⊂ B(0, 1) ⊂ F (G), we see that J = J_{min}. Thus both J(h_{1}) and J(h_{2}) are contained in J_{min}(G). Note that ∂K̂(G) = J(h_{1}) ≠ J_{min}(G).

Both h_{1} and h_{2} map A′ into the unbounded component of \overline{ℂ} \ J(h_{3}) (since A′ is outside of both h_{1}^{−1}(J(h_{3})) and h_{2}^{−1}(J(h_{3}))), which is forward invariant under each map h_{1}, h_{2} and h_{3}. The map h_{3} maps A′ into B(0, 1), which is also forward invariant under each map h_{1}, h_{2} and h_{3}. Hence for any g ∈ G we have that g(A′) ∩ A′ = ∅ and so int(A′) ⊂ F (G). We conclude that J(h_{3}) is not contained in J_{min}.

Thus we have that H_{min}(G) = ⟨h_{1}, h_{2}⟩. One can easily show that J(H_{min}(G)) is disconnected (Cantor set of circles) and thus J(H_{min}(G)) ≠ J_{min}(G). Also H_{min}(H_{min}(G)) = ⟨h_{1}⟩ ≠ ⟨h_{1}, h_{2}⟩ = H_{min}(G) and J_{min}(G) ≠ J_{min}(H_{min}(G)).

h_{1}^{−1}(J(f_{3}))  J(f_{3}) h_{1}^{−2}(J(f_{3})) h_{2}^{−1}(J(h_{1}))  •P  •P^{4} •0 −P^{2}•  P e^{iπ/4}•

Figure 2. This figure shows the sets (from outside to inside) J(f_{3}), h_{1}^{−1}(J(f_{3})), h_{2}^{−1}(J(h_{1})), and h_{1}^{−2}(J(f_{3})). For this picture we chose m_{2} = 5 and P = 2^{31/64} ≈ 1.39898.

We now show #J_{G} = ℵ_{0}. Consider the set B = J_{min}(G) ∪ ⋃_{n=1}^{∞} h_{3}^{−n}(J_{min}(G)) ∪ J(h_{3}) ⊂ J(G), which clearly contains more than three points. We have J(h_{3}) ⊂ J_{max}(G) and h_{1}^{−1}(J(h_{3})) ⊂ J_{min}(G). By Theorem 1.14 or Corollary 2.9, we obtain h_{1}^{−1}(B) ⊂ h_{1}^{−1}(J(G)) ⊂ J_{min}(G) ⊂ B. Similarly, taking m_{2} so large, we may assume h_{2}^{−1}(J(h_{3})) ⊂ J_{min}(G), and it implies h_{2}^{−1}(B) ⊂ h_{2}^{−1}(J(G)) ⊂ J_{min}(G) ⊂ B. Since B is closed and backward invariant under each generator of G (and hence under every g ∈ G), we must have that B = J(G). Also, since h_{3}^{−1}(J_{min}(G)) is connected (by Corollary 2.9) and does not meet J_{min}(G), we see that J_{min}(G) < h_{3}^{−1}(J_{min}(G)). Repeated application of Lemma 2.8 shows us that J_{min}(G) < h_{3}^{−1}(J_{min}(G)) < h_{3}^{−2}(J_{min}(G)) < ⋯ < h_{3}^{−n}(J_{min}(G)) < . . .. From this we may conclude that J = {J_{min}(G), J(h_{3}), h_{3}^{−n}(J_{min}(G)) : n ∈ ℕ}, thus demonstrating that #J_{G} = ℵ_{0}.

Remark 4.25. Note that Example 4.24 does not settle (in the negative) the open question stated above since Claim 4.21 with J_{1} = J_{min} shows that each J ∈ J_{G} contains J(g) for some g ∈ G. One could also note that every J ∈ J_{G} other than J(h_{3}) is isolated in J_{G} and so from Lemma 4.17 each such J has property (⋆).

Question: Does there exists an example of some G ∈ G which can negatively answer questions (1)-(4) addressed by Example 4.24, but where #J_{G} is finite? The answer, as we see in the next example, is YES. We will also see that this example will settle two other questions that naturally arise when considering the two following results. In [29, 33] it was shown that, for each positive integer k, there exists a semigroup G ∈ G_{dis} with 2k generators such that J(G) has exactly k components. Furthermore, in [25] it was shown that any semigroup in G generated by exactly three elements will have a Julia set with either one or infinitely many components. Hence we have the following questions.

(5) What is the fewest number of generators that can produce a semigroup G ∈ G_{dis} with #J = 3?

(6) For fixed integer k > 3, what is the fewest number of generators that can produce a semigroup G ∈ G_{dis} with #J = k? The answer to both of these questions is four as stated in Theorem 1.23 whose proof is given now.

Proof of Theorem 1.23. Fix k ≥ 2 since the k = 1 case is trivial. Let maps h_{1}, f_{2}, and f_{3} and integer m_{2} ∈ ℕ be defined as in Example 4.24. Again, we set h_{2} = f_{2}^{m_{2}} and h_{3} = f_{3}^{m_{3}} where large m_{3} ∈ ℕ will be specified to fit the stipulations given below. Letting γ_{1} denote C(0, 1) and γ_{2} denote the boundary of the unbounded component of ℂ \ (h_{1}^{−1}(J(f_{3})) ∪ h_{2}^{−1}(J(f_{3}))), we set B = \overline{Ann(γ_{1}, γ_{2})}. Let A′ be any closed annulus such that A′ ⊂ B(ϵ, r), int(A′) ≠ ∅, and A′ > B. We choose m_{3} ∈ ℕ large enough so that h_{3}^{−1}(B) > A′ > B. Set G = ⟨h_{1}, h_{2}, h_{3}⟩ and note that for m_{3} large enough, γ_{1} ∪ γ_{2} ⊂ J_{min}(G) ⊂ B, as in Example 4.24.

Set ℓ = k − 2. Let C = C(ϵ_{0}, r_{0}) be the circle which is internally tangent to the circle J(h_{3}) at the point ϵ + r = P^{4} such that C meets h_{3}^{−(ℓ+1)}(B) and \overline{B(ϵ_{0}, r_{0})} ⊃ h_{3}^{−(ℓ+1)}(B). Hence C must necessarily meet h_{3}^{−(ℓ+1)}(γ_{2}) ⊂ h_{3}^{−(ℓ+1)}(J_{min}(G)) and C > h_{3}^{−ℓ}(B) > ⋯ > h_{3}^{−1}(B) > B. Note that as m_{3} → ∞, we have ϵ_{0} → ϵ and r_{0} → r. We may assume then that m_{3} has been chosen large enough so that ϵ_{0} ∈ B(0, 1).

Set f_{4}(z) = \frac{(z−ϵ_{0})^{2}}{r_{0}} + ϵ_{0} and observe that J(f_{4}) = C. Let R > 0 be large enough so that f_{j}(ℂ \ \overline{B(0, R)}) ⊂ ℂ \ \overline{B(0, R)} for j = 1, …, 4 (where f_{1} := h_{1}). Let A′′ be a closed annulus such that int(A′′) ≠ ∅ and h_{3}^{−ℓ}(B) < A′′ < h_{3}^{−(ℓ+1)}(B). Then h_{3}^{ℓ+1}(A′′) ⊂ B(0, 1). Let A_{0} := A′ ∪∪_{j=0}^{ℓ}h_{3}^{j}(A′′). We define h_{4} = f_{4}^{m_{4}} where m_{4} ∈ ℕ is large enough such that (i) h_{4}(A_{0}) ⊂ B(0, 1), (ii) h_{4}^{−1}(γ_{1}) meets h_{3}^{−(ℓ+1)}(J_{min}(G)) (this is possible since the connected set h_{3}^{−(ℓ+1)}(J_{min}(G)) meets, but is not contained in, C), (iii) h_{4}(∪_{j=1}^{2} ∪_{h∈⟨h_{1},h_{2},h_{3}⟩∪\{id\}} h ∘ h_{j}(A_{0})) ⊂ ℂ \ \overline{B(0, R)} (note that \overline{h_{1}(A_{0}) ∪ h_{2}(A_{0})} ⊂ \overline{ℂ} \ K(h_{3}), which is equal to the connected component of F (⟨h_{1}, h_{2}, h_{3}⟩) containing ∞), and (iv) h_{4}^{−1}(γ_{1}) > h_{3}^{−ℓ}(B).

Set G′ = ⟨h_{1}, h_{2}, h_{3}, h_{4}⟩. Since B(0, 1) is forward invariant under each map in G′, we conclude B(0, 1) ⊂ F (G′) and P^{∗}(G′) ⊂ B(0, 1). Thus G′ ∈ G. Also, int(A_{0}) ⊂ F (G′) since one can show that g(A_{0}) ⊂ A_{0} ∪ B(0, 1) ∪ ℂ̂ \ K(h_{3}) for all g ∈ G′. Hence G′ ∈ G_{dis}. By applying Lemma 2.16 and Lemma 2.14 (noting that J_{min}(G′) < int(A′) < J(h_{3})), we have that h_{3}^{−n}(J_{min}(G′)) < h_{3}^{−(n+1)}(J_{min}(G′)) for all n ≥ 0. Further, J_{max}(G′), which must contain J(h_{3}) and J(h_{4}) by Proposition 4.1, must also contain h_{3}^{−n}(J_{min}(G′)) for all n > ℓ. By examining the dynamics one can then show that J(G′) = J_{min}(G′) ∪ J_{max}(G′) ∪ h_{3}^{−1}(J_{min}(G′)) ∪ ⋯ ∪ h_{3}^{−ℓ}(J_{min}(G′)), since this set is closed and backward invariant under each generator of G′. Moreover, since h_{3}^{−ℓ}(γ_{1}) < A′′ < h_{3}^{−(ℓ+1)}(γ_{1}), h_{3}^{−ℓ}(γ_{1}) ⊂ h_{3}^{−ℓ}(J_{min}(G′)), h_{3}^{−(ℓ+1)}(γ_{1}) ⊂ J_{max}(G′) and int(A′′) ⊂ F (G′), we have h_{3}^{−ℓ}(J_{min}(G′)) < int(A′′) < J_{max}(G′). Thus we see that J(G′) has exactly k components. □

Remark 4.26. The addition of one generating function in the proof of Theorem 1.23 to the semigroup in Example 4.24 illustrates something of a general principle (which we decline to attempt to make precise) at work when dealing with the dynamics of semigroups in G_{dis}. Namely, if one adds a generator (or a whole family of generators) whose Julia set does not meet J_{min} of the new larger semigroup, then key properties of the dynamics can often be preserved. See for example Theorems 1.27 and 1.29.

However, as we see in this next lemma, adding “too many” new functions will necessarily destroy certain critical aspects of the dynamics. In particular, if we look to produce a new semigroup in G by adding “too many” generating polynomials of small degree (such that G_{k} defined in the lemma is not pre-compact) to a semigroup G ∈ G_{dis}, then the new semigroup will necessarily have a connected Julia set.

Lemma 4.27. Let G = ⟨h_{λ} : λ ∈ Λ⟩ ∈ G_{dis}. Then each G_{k} = {g ∈ G : deg(g) ≤ k} is pre-compact in Poly and, in particular, each {h_{λ} : λ ∈ Λ} ∩ G_{k} is pre-compact in Poly.

Remark 4.28. As stated earlier, a possibly generating set for G is G itself, which is necessarily not pre-compact (since it contains elements of arbitrarily high degree). Thus it is impossible to strengthen Lemma 4.27 to conclude that {h_{λ} : λ ∈ Λ} is pre-compact.

Proof. Note that J(G) is bounded in ℂ since Theorem 1.20 yields ∞ ∈ F (G). Choose R > 0 such that J(G) ⊂ \overline{B(0, R)}. Then Cap(J(g)) ≤ R for all g ∈ G, where Cap(E) denotes the logarithmic capacity of the set E (see [1] for definition and properties). Also, since G ∈ G_{dis} we have int(K̂(G)) ≠ ∅ (see Theorem 1.20 or [29]), and so there exists a ball of some radius r > 0 in K̂(G). Thus Cap(J(g)) ≥ r for all g ∈ G.

Let H_{n} = {g ∈ G : deg(g) = n}. In order to show that G_{k} = ∪_{n=1}^{k}H_{n} is pre-compact, it suffices to show that each H_{n} is pre-compact. We now fix g(z) = a_{n}z^{n} + ⋯ + a_{0} in H_{n} and proceed to show that |a_{n}| is uniformly bounded below by R^{1−n} and uniformly bounded above by r^{1−n}, and that the remaining coefficients a_{n−1}, …, a_{0} of g(z) are uniformly bounded (above) by positive constants which only depend on r, R and n. Recalling Remark 1.25, it follows then that H_{n} is pre-compact. Since |a_{n}|^{−1/(n−1)} = Cap(J(g)) (see [8], p. 35), we see that r^{1−n} ≥ |a_{n}| ≥ R^{1−n}. Express g′(z) = β(z − α_{1}) … (z − α_{n−1}) where β = na_{n} and the α_{j} are the critical points of g which, since G ∈ G and ∞ ∈ F (G), must lie in ℂ \ F_{∞}(G) ⊂ \overline{B(0, R)}, where F_{∞}(G) denotes the connected component of F (G) containing ∞.

One can multiply out the terms in the expansion of g′(z) and find an antiderivative to see that the a_{n−1}, …, a_{1} coefficients of g(z) are also bounded by constants which depend only on r, R and n. Now fix z_{0} ∈ K̂(G). Since g(K̂(G)) ⊂ K̂(G) ⊂ \overline{B(0, R)}, we have |g(z_{0})| = |a_{n}z_{0}^{n} + ⋯ + a_{0}| ≤ R. Thus, since |a_{n}|, …, |a_{1}| are bounded by constants depending only on r, R and n, the same is true for |a_{0}|. □

Remark 4.29. The proof of Lemma 4.27 also holds for any G ∈ G such that there exists both lower and upper bounds on Cap(J(g)) for all g ∈ G (e.g., when K̂(G) contains some non-degenerate continuum and ∞ ∈ F (G)).

5. Proof of Theorems 1.27 and 1.29

Example 5.1. Let f_{1}(z) = z^{2} + c where c > 0 is small (thus J(f_{1}) is a quasi-circle). Let z_{0} ∈ ℝ denote 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), the semigroup G = ⟨H, h_{1}⟩ is not hyperbolic even though J(h_{1}) ∩ J_{min}(G) = ∅.

By conjugating h_{2} by a suitable 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 1.27 does imply that G = ⟨H, h_{1}⟩ is semi-hyperbolic.

Remark 5.2. In contrast to the analogous behavior of Iterated Function Systems where contraction in each generating map leads to a semigroup (IFS) that is overall contracting, we see that in Example 5.1 each map of the semigroup G is hyperbolic, yet the entire semigroup G fails to be hyperbolic. To see this, note that each map h_{2}^{n} is hyperbolic and for each map g ∈ G \ {h_{2}^{n}} we have P^{∗}(g) ⊂ P^{∗}(G) ⊂ \overline{B(z_{0}, |c − z_{0}|)} and J(g) > \overline{B(z_{0}, |c − z_{0}|)} which implies g is hyperbolic.

We now state a lemma which we will use the proof of Theorem 1.27. Lemma 5.3. Let H_{1} be a polynomial semigroup in G and let Γ be a compact family in Poly. Let H_{2} = ⟨H_{1}, Γ⟩ be the semigroup generated by H_{1} and Γ. Suppose

(1) H_{2} ∈ G_{dis}, and

(2) J(γ) ∩ J_{min}(H_{2}) = ∅ for γ ∈ Γ. Then K̂(H_{1}) = K̂(H_{2}), which then implies J_{min}(H_{1}) ⊂ J_{min}(H_{2}) since ∂K̂(H_{1}) ⊂ J_{min}(H_{1}) and ∂K̂(H_{2}) ⊂ J_{min}(H_{2}).

Remark 5.4. We recall the facts given in [29, 33] that for any G ∈ G we have intK̂(G) = K̂(G) ∩ F (G). Moreover, for any G ∈ G_{dis}, we have intK̂(G) ≠ ∅ and g(K̂(G) ∪ J_{min}(G)) ⊂ intK̂(G) for any g ∈ G such that J(g) ∩ J_{min}(G) = ∅.

Proof. We begin by first showing that J_{min}(H_{1}) ⊂ J_{min}(H_{2}). Let C be the set of all connected components of \overline{⋃_{γ∈Γ} J(γ)}. By Lemma 4.8, M_{1} := min_{C∈C} C exists with respect to the surrounding order. Let J_{1} ∈ J_{H_{2}} be the element containing M_{1}. Let J_{2} ∈ J_{H_{2}} be the element containing J_{min}(H_{1}). Let J_{0} := min{J_{1}, J_{2}} ∈ J_{H_{2}}. Then for each g ∈ H_{1}∪Γ, either J_{0} < J(g) or J(g) ⊂ J_{0}. By Lemma 2.14 and Lemma 2.16, we obtain that for each g ∈ H_{1} ∪ Γ, either g^{−1}(J_{0}) > J_{0} or g^{−1}(J_{0}) ⊂ J_{0}. By Corollary 2.9 or Theorem 1.14, it follows that A := ⋃_{J∈J_{H_{2}} ,J≥J_{0}} J is closed, ♯A ≥ 3, and g^{−1}(A) ⊂ A for each g ∈ H_{2}. Therefore J(H_{2}) ⊂ A. Thus J(H_{2}) = A and hence J_{0} = J_{min}(H_{2}). From assumption (2), however, it must be the case that J_{0} = J_{2}. Therefore J_{min}(H_{1}) ⊂ J_{2} = J_{min}(H_{2}) as desired.

By Remark 5.4 and Theorem 1.20, it follows that for each γ ∈ Γ, we have γ(∂K̂(H_{1})) ⊂ γ(J_{min}(H_{1})) ⊂ γ(J_{min}(H_{2})) ⊂ intK̂(H_{2}). Let γ ∈ Γ and let U be any connected component of intK̂(H_{1}), which we note is simply connected by the maximum principle. Then ∂γ(U) ⊂ γ(∂U) ⊂ intK̂(H_{2}). Let V be the connected component of intK̂(H_{2}) containing the connected set γ(∂U). Then ∂γ(U) ⊂ V. By the maximum principle, \overline{ℂ}\\overline{V} is connected and unbounded. Hence γ(U) ⊂ V. From this argument, it follows that for each γ ∈ Γ, we have γ(K̂(H_{1})) ⊂ K̂(H_{2}) ⊂ K̂(H_{1}), where the last inclusion holds since H_{1} ⊂ H_{2}. Thus since K̂(H_{1}) is forward invariant under each γ ∈ Γ and under each map in H_{1}, it is also forward invariant under each map in H_{2}. We then conclude that K̂(H_{1}) ⊂ K̂(H_{2}), which together with the reverse inclusion already noted gives K̂(H_{1}) = K̂(H_{2}). □

Definition 5.5. 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 5.6. Let G be a rational semigroup. We define

UH(G) = \overline{ℂ} \ ∪_{N=1}^{∞}SH_{N}(G).

Remark 5.7. For a rational semigroup G we note that each SH_{N}(G) is open and thus UH(G) is closed.

Remark 5.8. 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 5.9. We note from Lemma 1.14 in [22] 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.

Proof of Theorem 1.27. Assume the conditions stated in the hypotheses. By the definition of semi-hyperbolic, our goal is to show J(G) ⊂ SH_{K}(G) for some K ∈ ℕ. We will show the equivalent statement that J(G) ∩ UH(G) = ∅. Since UH(G) ⊂ P (G) and P^{∗}(G) ∩ J(G) ⊂ J_{min}(G), we have only to show J_{min}(G) ⊂ ℂ \ UH(G).

By Theorem 1.20 or [29, 33] we know that ♯J_{min}(G) ≥ 3. Thus hypothesis (2) and Lemma 2.16 imply γ^{−1}(J_{min}(G))∩J_{min}(G) = ∅ for γ ∈ Γ, which in turn implies (by Lemma 2.14) γ^{−1}(J_{min}(G)) > J_{min}(G). Thus, for all γ ∈ Γ,

(I)  γ^{−1}(J(G)) ∩ A = ∅

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

Since Γ is compact in Poly, d = min_{γ∈Γ} dist(γ^{−1}(J(G)), A) > 0. By (I) there exists d_{1} > 0 such that for all γ ∈ Γ, for all z ∈ J(G), and all components U of γ^{−1}(B(z, d_{1})) we have

(II)  U ∩ B(A, d/2) = ∅.

Now by Lemma 5.3 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 complements, J_{min}(G) ⊂ ℂ \ UH(H).

Claim: There exists b ∈ UH(H) ∩ intK̂(H). Proof of claim: Lemma 5.3 and Remark 5.4 show 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 5.9). Thus by Sullivan’s No Wandering Domains Theorem the orbit {g_{0}^{n}(w)} must be drawn toward an attracting cycle in ℂ. By replacing, if necessary, g_{0} by an iterate we may assume that g_{0}^{n}(w) approaches a finite 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 Lemma 1.10 in [22], there exists 0 < δ_{2} < δ_{1} such that the component W of (h ∘ ϕ_{V,h})^{−1}(B(z, δ_{2})) containing 0 is such that diamW ≤ c where c > 0 is a small number independent of h, to be specified later.

Note that, in the above, the set V does depend on h ∈ H. Yet for each h ∈ H, the set ϕ_{V,h}(B(0, 1)) = V does not meet B(b, ϵ_{1}) and so the family {ϕ_{V,h}}_{h∈H} is normal on B(0, 1). Thus

(III)  diamϕ_{V,h}(W) < d_{1}/10

when c is sufficiently small.

Let g ∈ G. If g ∈ H, then (since δ_{2} < δ_{1}) we have 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 = hγg_{1} where g_{1} ∈ G∪{id}, h ∈ H ∪{id} and γ ∈ Γ. Let V_{0} be a component of γ^{−1}h^{−1}(B(z, δ_{2})). Thus we have deg(hγ : V_{0} → B(z, δ_{2})) ≤ NM where M = max_{γ∈Γ}{deg γ}. By (III) we have diamγ(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 hγ 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). □

Proof of Theorem 1.29. The proof follows the same line as the proof of Theorem 1.27. We note that the usual Koebe Distortion Theorem applies (without needing to invoke the distortion Lemma 1.10 in [22]), and on the domains of interest in the proof each γ is one-to-one by hypothesis (4) and each h ∈ H is one-to-one by hypothesis (3). We omit the details. □

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(Rich Stankewitz) Department of Mathematical Sciences, Ball State University, Muncie, IN 47306, USA

   E-mail address: rstankewitz@bsu.edu

(Hiroki Sumi) Department of Mathematics, Graduate School of Science, Osaka University, 1-1, Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

   E-mail address: sumi@math.sci.osaka-u.ac.jp