Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups

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 $g \in G$. 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 (\cite{Su01,Su9}). 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 $g \in G$ 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.


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 λn • · · · • h λ1 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 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 [11], who were interested in 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. 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 ∞ : C → [0, 1] of the probability of tending to ∞. It turns out that under certain conditions, the function T ∞ is continuous on 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). 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].
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 n c (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 ∈ C | G is normal in a neighborhood of z} and J(G) = 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 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) = 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 −1 λ (J(G)). This property is called the backward self-similarity. In particular, if G = h 1 , . . . , h m , then J(G) = m j=1 h −1 j (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]).
Remark 1.3. Theorem 1.2 can be used to easily show that F ( h λ : λ ∈ Λ ) is precisely the set of z ∈ 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 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 and callK(G) the smallest filled-in Julia set. Remark 1.5. We note that for all g ∈ G, we haveK(G) ⊂ K(g) and g(K(G)) ⊂ K(G). Definition 1.6. The postcritical set of a rational semigroup G is defined by 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 [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 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}.
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) ⊂K(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 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 C \ K 2 we say that K 1 is "outside" K 2 .
(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 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 such a G ∈ G dis is shown in Figure 1. 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 ∈ 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.   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 C \ J x is a John domain, and the bounded component of C\J x is not a John domain (see [31,33]). 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 m1 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) ∂K. 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, intK(G) = ∅. Furthermore, we have the following •K(G), and therefore P * (G), is contained in the polynomial hull of each J ∈ J .
Remark 1.21. We see that ∂K(G) ⊂ J(G) when G ∈ G, but, in general, we do not have ∂K(G) = J(G), unlike in iteration theory where ∂K(g) = J(g) for polynomials g of degree two or more. In fact, ∂K(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 ∂K 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 1.22 below. 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 : C → C | h is a polynomial}, endowed with the topology of uniform convergence on C with respect to the spherical metric.
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.  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.

Background and Tools
We first state some notation to be used later.

Notation: Given any set
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: 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 Proof. By Theorem 1.20 we have P * (G) ⊂ P H(J) for all J ∈ J . By Remark 1.11 . The other cases then follow from Lemma 2.5.
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 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.
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: 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.
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 β0 = r 0 , then clearly, by Lemma 2.11, C β0 = inf β∈B C β . Select a strictly decreasing sequence r βn → r 0 . By Lemma 2.11, we have that C β1 > C β2 > . . . . Let z βn ∈ C βn be arbitrary. Without loss of generality we may assume that z βn → a 0 ∈ C. By hypothesis there exists C α0 which contains a 0 . We will now show that C α0 = 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 βn ) < C β for large n (whenever r βn < r β ). Thus we must have C α0 < C β for all β ∈ B. Hence C α0 is a lower bound for {C β } β∈B . Suppose that C α1 > C α0 . It must then be the case that {a 0 } < C α1 and so it follows that {z βn } < C α1 for large n. Thus C βn < C α1 for large n, implying that C α1 is not a lower bound for {C β } β∈B . We conclude that C α0 = 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 ∪ 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 ∪ 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.
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).
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]).
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 2 1 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. Proof. The lemma follows from the fact that the connected set ∪ ∞ 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. Proof. Choose any z ∈ C and let α n ∈ A be such that dist(z, C αn ) → 0. By compactness in the topology generated by the Hausdorff metric on the space of non-empty compact subsets of C, we then may conclude (by passing to subsequence if necessary) that C αn → 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, ǫ) ⊃ B(K, ǫ) ⊃ C αn .
In particular, we apply Theorem 1.2 to obtain that if {g λ } λ∈Λ = G ∈ G and J ∈ J , then for every ǫ > 0 there exists g ∈ G such that J(g) ⊂ B(J, ǫ).

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 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.
The other result is proved similarly.
We will require the following result which was proved via fiberwise quasiconformal surgery by the second author. . Let G = α 1 , α 2 ∈ G be hyperbolic such that P * (G) is contained in a single component of int(K(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) = ∪ 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) ⊂ ∪ n∈N g −n (J(f )), and each g −n (J(f )) ∩ A = ∅, there exists n 0 ∈ N such that g −n0 (J(f )) > γ A and thus g −n0 (J(f )) > A. By Lemma 3.3 there exists k ∈ G such that A < g −n0 (J(f )) < J(k) < g −(n0+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 m1 and β 2 = g m2 generate a sub-semigroup H ′ of G where J(H ′ ) is disconnected and contained in 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 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 x1 = J x2 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 . 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 x1 = J x2 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.
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 =K. 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.

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 2.23 in [29]. 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.  Proof. Suppose J(g) ⊂ J min for g = h λ1 •· · ·•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 and Lemma 2.16 we have J min < h −1 λ (J min ). So it also follows from Corollary 2.10 that J(g) < h −1 λ (J(g)) for all λ ∈ Λ. Thus J(g) < h −1 λ1 (J(g)) and by Lemma 2.8 J(g) < h −1 λ2 (J(g)) < h −1 λ2 h −1 λ1 (J(g)). By repeated application of this argument we then get that J(g) < h −1 λ k (J(g)) < · · · < h −1 λ k . . . h −1 λ1 (J(g)) = g −1 (J(g)) = J(g), which is a contradiction. From this part (a) follows.
Part (b) follows in a similar manner.
Proof. Suppose int(J min ) = ∅. Since J(G) = ∪ 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).
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 theÃ n are separated from each other, each J n is isolated from the other J m , i.e., for each n there exists ǫ n > 0 such that the ǫ n -neighborhood B(J n , ǫ n ) does not meet any other J m ∈ J .
Let C = C(0, 2). Since C ⊂ J(G) we see that J(G) = G −1 (C). We now show that for each g ∈ G, the set g −1 (C) ⊂ J n for some n. Write g = g i1 • . . . g ij where each g i ℓ is a generator for G. Suppose that g ij = h n for some n. Then, by the backward invariance of A under each map in G, we have that , 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 ij = 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) = G −1 (C) ⊂ ∪ 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 ∪ 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.
Then both M ′ = min C∈C C and M ′′ = max C∈C C exist (with respect to the surrounding order ≤). Also, P H(C) ⊃K(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 ∈K(G) \ P H(C). Let γ be a curve in C \ P H(C) connecting z 0 to ∞ and set ǫ = dist(γ, P H(C)). By Corollary 2.18, there exists λ ∈ Λ such that J(g λ ) ⊂ B(C, ǫ). Hence, we see that γ is outside J(g λ ) implying that g n λ (z 0 ) → ∞ and thus contradicting the fact that z 0 ∈K(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 A, respectively.
We now prove (2)  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 ∂K(G) ⊂ J min . Let γ be a curve in U which connects w to some z 0 ∈K(G) and set ǫ = dist(γ, M ′′ ) > 0. By Corollary 2.18 there exists a generator g λ0 ∈ G such that J(g λ0 ) ⊂ B(M ′′ , ǫ). Thus γ ∩ J(g λ0 ) = ∅. SinceK(G) ⊂ P H(J(g λ0 )), we see that z 0 ∈ γ ∩ P H(J(g λ0 )) and so γ < J(g λ0 ). Hence {w} < J(g λ0 ) which implies (by Corollary 2.10) either J max < J(g λ0 ) or J max ∩ J(g λ0 ) = ∅. 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 A ⊂ U and so J max = M ′′ ⊂ A ⊂ 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.
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 − ǫ) 2 + ǫ, and f 3 (z) = z 2 /4. For ǫ > 0 small and m 1 , m 2 , m 3 all large we set g 1 = f m1 1 , g 2 = f m2 2 , and g 3 = f m3 3 and note that G = g 1 , g 2 , g 3 ∈ G dis . Thus 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.  Proof. Assume that ǫ > 0 is such that B(J, ǫ) does not meet any other set in J . Since J(G) = ∪ g∈G J(g) by Theorem 1.2, we see that any point in J must have, within a distance ǫ, 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 .
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 .
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. 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? The answer to each of these questions is NO, as we see in this next example.
Let h 1 (z) = −z 2 and f 2 (z) = z 2 / √ 2 and note that J(h 1 ) = C(0, 1) and J(f 2 ) = C(0, √ 2). We set h 2 = f m2 2 where conditions on the large m 2 ∈ N will be specified , and so we have Figure 1). Since for m 2 large we clearly have h −1 2 (J(f 3 )) < J(f 3 ), we are then free to choose A ′ to be any closed annulus such that A ′ ⊂ B(ǫ, r), int(A ′ ) = ∅, and A ′ > h −1 ). Choose m 3 ∈ N large enough so that h 3 = f m3 3 maps A ′ into B(0, 1) (which can be done since the super attracting fixed point of f 3 is ǫ ∈ B(0, 1)). This then implies that B(0, 1) is forward invariant under each map h 1 , h 2 and h 3 and ) 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 −1 2 (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 −1 2 (J(h 1 )) ⊂ A and e iπ/4 P ∈ h −1 . Thus h −1 1 (A)∩A = ∅ and since h −n 1 (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 ∂K(G) = J(h 1 ) = J min (G). Both ) and h −1 2 (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)). We , 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 Similarly, taking m 2 so large, we may assume 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 −1 3 (J min (G)) is connected (by Corollary 2.9) and does not meet J min (G), we see that J min (G) < h −1 3 (J min (G)). Repeated application of Lemma 2.8 shows us that J min (G) < h −1 3 (J min (G)) < h −2 3 (J min (G)) < · · · < h −n 3 (J min (G)) < . . . . From this we may conclude that J = {J min (G), J(h 3 ), h −n 3 (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.
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.
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) ⊂ 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(K(G)) = ∅ (see Theorem 1.20 or [29]), and so there exists a ball of some radius r > 0 inK(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 = ∪ k n=1 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) ⊂ 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 ∈K(G). Since g(K(G)) ⊂ K(G) ⊂ B(0, R), we have |g(z 0 )| = |a n z n 0 + · · · + 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., whenK(G) contains some non-degenerate continuum and ∞ ∈ F (G)).
By conjugating h 2 by a suitable rotation we may assume that {h k 2 (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 2 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 n 2 is hyperbolic and for each map g ∈ G \ {h n 2 } we have P * (g) ⊂ P * (G) ⊂ B(z 0 , |c − z 0 |) and J(g) > 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. (1) H 2 ∈ G dis , and Remark 5.4. We recall the facts given in [29,33] that for any G ∈ G we have intK(G) =K(G) ∩ F (G). Moreover, for any G ∈ G dis , we have intK(G) = ∅ and g(K(G) ∪ J min (G)) ⊂ intK(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 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 H2 be the element containing M 1 . Let J 2 ∈ J H2 be the element containing J min (H 1 ). Let J 0 := min{J 1 , J 2 } ∈ J H2 . 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∈JH 2 ,J≥J0 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.
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 ∈ 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 ).
Remark 5.7. For a rational semigroup G we note that each SH N (G) is open and thus U H(G) is closed.
Remark 5.8. For a rational semigroup G we see that U H(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 U H( g ), for any polynomial g with deg(g) ≥ 2. Hence we may conclude that such points are also in U H(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) ∩ U H(G) = ∅. Since U H(G) ⊂ P (G) and P * (G) ∩ J(G) ⊂ J min (G), we have only to show J min (G) ⊂ C \ U H(G).
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.
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 N M (G), thus giving z / ∈ U H(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.