Dynamics of postcritically bounded polynomial semigroups I: connected components of the Julia sets

We investigate the dynamics of semigroups generated by a family of polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. The Julia set of such a semigroup may not be connected in general. We show that for such a polynomial semigroup, if $A$ and $B$ are two connected components of the Julia set, then one of $A$ and $B$ surrounds the other. From this, it is shown that each connected component of the Fatou set is either simply or doubly connected. Moreover, we show that the Julia set of such a semigroup is uniformly perfect. An upper estimate of the cardinality of the set of all connected components of the Julia set of such a semigroup is given. By using this, we give a criterion for the Julia set to be connected. Moreover, we show that for any $n\in \Bbb{N} \cup \{\aleph_{0}\} ,$ there exists a finitely generated polynomial semigroup with bounded planar postcritical set such that the cardinality of the set of all connected components of the Julia set is equal to $n.$ Many new phenomena of polynomial semigroups that do not occur in the usual dynamics of polynomials are found and systematically investigated.


Introduction
The theory of complex dynamical systems, which has its origin in the important work of Fatou and Julia in the 1910s, has been investigated by many people and discussed in depth. In particular, since D. Sullivan showed the famous "no wandering domain theorem" using Teichmüller theory in the 1980s, this subject has attracted many researchers from a wide area. For a general reference on complex dynamical systems, see Milnor's textbook [16] or Beardon's textbook [3].
There are several areas in which we deal with generalized notions of classical iteration theory of rational functions. One of them is the theory of dynamics of rational semigroups (semigroups generated by a family of holomorphic maps on the Riemann sphereĈ), and another one is the theory of random dynamics of holomorphic maps on the Riemann sphere.
In this paper, we will discuss the dynamics of rational semigroups. A rational semigroup is a semigroup generated by a family of non-constant rational maps onĈ, whereĈ denotes the Riemann sphere, with the semigroup operation being functional composition ( [13]). A polynomial semigroup is a semigroup generated by a family of non-constant polynomial maps. Research on the dynamics of rational semigroups was initiated by A. Hinkkanen and G. J. Martin ( [13,14]), who were interested in the role of the dynamics of polynomial semigroups while studying various one-complex-dimensional moduli spaces for discrete groups, and by F. Ren's group( [48,12]), who studied such semigroups from the perspective of random dynamical systems. Moreover, the research on rational semigroups is related to that on "iterated function systems" in fractal geometry. In fact, the Julia set of a rational semigroup generated by a compact family has " backward self-similarity" (cf. . For other research on rational semigroups, see [19,20,21,47,22,24,44,43,45,46], and [27]- [40]. The research on the dynamics of rational semigroups is also directly related to that on the random dynamics of holomorphic maps. The first study in this direction was by Fornaess and Sibony ( [10]), and much research has followed. (See [4,6,7,5,11,33,34,37,38,39,40]. ) We remark that the complex dynamical systems can be used to describe some mathematical models. For example, the behavior of the population of a certain species can be described as the dynamical system of a polynomial f (z) = az(1 − z) such that f preserves the unit interval and the postcritical set in the plane is bounded (cf. [8]). It should also be remarked that according to the change of the natural environment, some species have several strategies to survive in the nature. From this point of view, it is very important to consider the random dynamics of such polynomials (see also Example 1.4). For the random dynamics of polynomials on the unit interval, see [26].
We shall give some definitions for the dynamics of rational semigroups: Definition 1.1 ( [13,12]). Let G be a rational semigroup. We set 2. Let G be a rational semigroup. We set P (G) := g∈G CV (g) (⊂Ĉ). This is called the postcritical set of G. Furthermore, for a polynomial semigroup G, we set P * (G) := P (G) \ {∞}. This is called the planar postcritical set (or finite postcritical set) of G. We say that a polynomial semigroup G is postcritically bounded if P * (G) is bounded in C.
Remark 1.3. Let G be a rational semigroup generated by a family Λ of rational maps. Then, we have that P (G) = g∈G∪{Id} g( h∈Λ CV (h)), where Id denotes the identity map onĈ, and that g(P (G)) ⊂ P (G) for each g ∈ G. From this formula, one can figure out how the set P (G) (resp. P * (G)) spreads inĈ (resp. C). In fact, in Section 2.6, using the above formula, we present a way to construct examples of postcritically bounded polynomial semigroups (with some additional properties). Moreover, from the above formula, one may, in the finitely generated case, use a computer to see if a polynomial semigroup G is postcritically bounded much in the same way as one verifies the boundedness of the critical orbit for the maps f c (z) = z 2 + c. Remark 1.5. It is well-known that for a polynomial g with deg(g) ≥ 2, P * ( g ) is bounded in C if and only if J(g) is connected ([16, Theorem 9.5]).
As mentioned in Remark 1.5, the planar postcritical set is one piece of important information regarding the dynamics of polynomials. Concerning the theory of iteration of quadratic polynomials, we have been investigating the famous "Mandelbrot set".
When investigating the dynamics of polynomial semigroups, it is natural for us to discuss the relationship between the planar postcritical set and the figure of the Julia set. The first question in this regard is: some additional properties (disconnectedness of Julia set, semi-hyperbolicity, hyperbolicity, etc.) (cf. Proposition 2.40, Theorem 2.43, Theorem 2.45). For example, by Proposition 2.40, there exists a 2-generator postcritically bounded polynomial semigroup G = h 1 , h 2 with disconnected Julia set such that h 1 has a Siegel disk.
As we see in Example 1.4 and Section 2.6, it is not difficult to construct many examples, it is not difficult to verify the hypothesis "postcritically bounded", and the class of postcritically bounded polynomial semigroups is very wide.
Throughout the paper, we will see many new phenomena in polynomial semigroups that do not occur in the usual dynamics of polynomials. Moreover, these new phenomena are systematically investigated.
In Section 2, we present the main results of this paper. We give some tools in Section 3. The proofs of the main results are given in Section 4.
There are many applications of the results of postcritically bounded polynomial semigroups in many directions. In the sequel [36], by using the results in this paper, we investigate the fiberwise (sequencewise) and random dynamics of polynomials and the Julia sets. We present a sufficient condition for a fiberwise Julia set to be of measure zero, a sufficient condition for a fiberwise Julia set to be a Jordan curve, a sufficient condition for a fiberwise Julia set to be a quasicircle, and a sufficient condition for a fiberwise Julia set to be a Jordan curve which is not a quasicircle. Moreover, using uniform fiberwise quasiconformal surgery on a fiber bundle, we show that for a G ∈ G dis , there exist families of uncountably many mutually disjoint quasicircles with uniform dilatation which are parameterized by the Cantor set, densely inside J(G). In the sequel [37], we classify hyperbolic or semi-hyperbolic postcritically bounded compactly generated polynomial semigroups, in terms of the random complex dynamics. It is shown that in one of the classes, for almost every sequence γ, the Julia set J γ of γ is a Jordan curve but not a quasicircle, the unbounded component ofĈ \ J γ is a John domain, and the bounded component of C \ J γ is not a John domain. Moreover, in [37,36], we find many examples with this phenomenon. Note that this phenomenon does not hold in the usual iteration dynamics of a single polynomial map g with deg(g) ≥ 2. In the sequel [38,42], we investigate the Markov process onĈ associated with the random dynamics of polynomials and we consider the probability T ∞ (z) of tending to ∞ ∈Ĉ starting with the initial value z ∈Ĉ. Applying many results of this paper, it will be shown in [42] that if the associated polynomial semigroup G is postcritically bounded and the Julia set is disconnected, then the function T ∞ defined onĈ has many interesting properties which are similar to those of the Cantor function. In fact, under certain conditions, T ∞ is continuous onĈ and varies precisely on the Julia set, of which Hausdorff dimension is strictly less than two. (For example, if we consider the random dynamics generated by two polynomials h 1 := g 2 1 , h 2 := g 2 2 , where g 1 (z) := z 2 − 1, g 2 (z) := z 2 /4, then T ∞ is continuous onĈ and T ∞ varies precisely on the Julia set ( Figure 1) of the semigroup generated by h 1 , h 2 . See [38,33].) Such a kind of "singular functions on the complex plane" appear very naturally in random dynamics of polynomials, and the results of this paper (for example, the results on the space of all connected components of a Julia set) are the keys to investigating that. (The above results have been announced in [33,34,39].) Moreover, as illustrated before, it is very important for us to recall that the complex dynamics can be applied to describe some mathematical models. For example, the behavior of the population of a certain species can be described as the dynamical systems of a polynomial h such that h preserves the unit interval and the postcritical set in the plane is bounded. When one considers such a model, it is very natural to consider the random dynamics of polynomial with bounded postcritical set in the plane (see Example 1.4).
In the sequel [24], we give some further results on postcritically bounded polynomial semigroups, by using many results in this paper and [36,37]. Moreover, in the sequel [35], we define a new kind of cohomology theory, in order to investigate the action of finitely generated semigroups (iterated function systems), and we apply it to the study of the dynamics of postcritically bounded finitely generated polynomial semigroups G. In particular, by using this new cohomology theory, we can describe the space J G of connected components of Julia sets of G, we can give some estimates on the cardinality of J G , and we can give a sufficient condition for the cardinality of the space of connected components of the Fatou set of G to be infinity. In [38,40,41], we investigate the random complex dynamics and the dynamics of transition operator, by developing the theory of random complex dynamics and that of dynamics of rational semigroups, simultaneously. It is shown that regarding the random dynamics of complex polynomials, generically the chaos of the averaged system disappears due to the cooperation of the generators, even though each map itself in the system has a chaotic part. We call this phenomenon "cooperation principle". Moreover, we see that under certain conditions, in the limit state, complex analogues of singular functions (continuous functions onĈ which vary only on the Julia set of associated rational semigroup G) naturally appear. The above function T ∞ is a typical example of this complex analogue of singular function.
Acknowledgement: The author thanks R. Stankewitz for many valuable comments.

Main results
In this section we present the statements of the main results. Throughout this paper, we deal with semigroups G that might not be generated by a compact family of polynomials. The proofs are given in Section 4.

Space of connected components of a Julia set, surrounding order
We present some results concerning the connected components of the Julia set of a postcritically bounded polynomial semigroup. The proofs are given in Section 4.1.
Theorem 2.1. Let G be a rational semigroup generated by a family {h λ } λ∈Λ . Suppose that there exists a connected component A of J(G) such that ♯A > 1 and λ∈Λ J(h λ ) ⊂ A. Moreover, suppose that for any λ ∈ Λ such that h λ is a Möbius transformation of finite order, we have h −1 λ (A) ⊂ A. Then, J(G) is connected. Definition 2.2. We set Rat : = {h :Ĉ →Ĉ | h is a non-constant rational map} endowed with the topology induced by uniform convergence onĈ with respect to the spherical distance. We set Poly := {h :Ĉ →Ĉ | h is a non-constant polynomial} endowed with the relative topology from Rat. Moreover, we set Poly deg≥2 := {g ∈ Poly | deg(g) ≥ 2} endowed with the relative topology from Rat.  • 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}.
Notation: For a polynomial semigroup G, we denote by J = J G the set of all connected components J of J(G) such that J ⊂ C. Moreover, we denote byĴ =Ĵ G the set of all connected components of J(G). Remark 2.5. If a polynomial semigroup G is generated by a compact set in Poly deg≥2 , then ∞ ∈ F (G) and thus J =Ĵ . Definition 2.6. For any 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 . Furthermore, "K 1 < K 2 " indicates K 1 ≤ K 2 and K 1 = K 2 . Note that "≤" is a partial order in the space of all non-empty compact connected sets in C. This "≤" is called the surrounding order. Theorem 2.7. Let G ∈ G (possibly generated by a non-compact family). Then we have all of the following.
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 For the figures of the Julia sets of semigroups G ∈ G dis , see figure 1 and figure 2.

Upper estimates of ♯(Ĵ )
Next, we present some results on the spaceĴ and some results on upper estimates of ♯(Ĵ ). The proofs are given in Section 4.2 and Section 4.3.
1. For a polynomial g, we denote by a(g) ∈ C the coefficient of the highest degree term of g.

2.
We set RA := {ax + b ∈ R[x] | a, b ∈ R, a = 0} endowed with the topology such that, a n x + b n → ax + b if and only if a n → a and b n → b. The space RA is a semigroup with the semigroup operation being functional composition. Any subsemigroup of RA will be called a real affine semigroup. We define a map Ψ : Poly → RA as follows: For a polynomial g ∈ Poly, we set Ψ(g)(x) := deg(g)x + log |a(g)|.
3. We setR := R ∪ {±∞} endowed with the topology such that {(r, +∞]} r∈R makes a fundamental neighborhood system of +∞, and such that {[−∞, r)} r∈R makes a fundamental neighborhood system of −∞. For a real affine semigroup H, we set where the closure is taken in the spaceR. Moreover, we denote by M H the set of all connected components of M (H). 4. We denote by η : RA → Poly the natural embedding defined by η(x → ax+b) = (z → az +b), where x ∈ R and z ∈ C.
Remark 2.11. The above "≤ r " is a partial order in the space of non-empty connected subsets of R. Moreover, for each real affine semigroup H, (M H , ≤ r ) is totally ordered.
The following theorem gives us some upper estimates of ♯(Ĵ G ).
The following three theorems give us sufficient conditions for the Julia set of a G ∈ G to be connected.  Theorem 2.16. Let G be a polynomial semigroup in G generated by a (possibly non-compact) family {h λ } λ∈Λ of polynomials. Let a λ be the coefficient of the highest degree term of the polynomial h λ . Suppose that for any λ, ξ ∈ Λ, we have (deg(h ξ ) − 1) log |a λ | = (deg(h λ ) − 1) log |a ξ |. Then, J(G) is connected.
Remark 2.17. In [35], a new cohomology theory for (backward) self-similar systems (iterated function systems) was introduced by the author of this paper. By using this new cohomology theory, for a postcritically bounded finitely generated polynomial semigroup G, we can describe the space of connected components of G and we can give some estimates on ♯(J G ) and ♯(M Ψ(G) ).

Properties of J
In this section, we present some results on J . The proofs are given in Section 4.3. Notation: For a set A ⊂Ĉ, we denote by int(A) the set of all interior points of A.
Notation: For a polynomial semigroup G with ∞ ∈ F (G), we denote by F ∞ (G) the connected component of F (G) containing ∞. Moreover, for a polynomial g with deg(g) ≥ 2, we set F ∞ (g) := F ∞ ( g ).
The following theorem is the key to obtaining further results of postcritically bounded polynomial semigroups in this paper, and those of related random dynamics of polynomials in the sequel [36,42]. We remark that Theorem 2.20-5 generalizes [47, Theorem 2].
Theorem 2.20. Let G ∈ G dis (possibly generated by a non-compact family). Then, under the above notation, we have the following.
1. We have that ∞ ∈ F (G) (thus J =Ĵ ) and the connected component containing ∞ is simply connected. Furthermore, the element J max = J max (G) ∈ J containing ∂F ∞ (G) is the unique element of J satisfying that J ≤ J max for each J ∈ J .
2. There exists a unique element J min = J min (G) ∈ J such that J min ≤ J for each element J ∈ J . Furthermore, let D be the unbounded component of C \ J min . Then, P * (G) ⊂K(G) ⊂ C \ D and ∂K(G) ⊂ J min .
3. If G is generated by a family {h λ } λ∈Λ , then there exist two elements λ 1 and λ 2 of Λ satisfying: (a) there exist two elements J 1 and J 2 of J with J 1 = J 2 such that J(h λi ) ⊂ J i for each i = 1, 2; and (d) h λ1 has an attracting fixed point z 1 in C, int(K(h λ1 )) consists of only one immediate attracting basin for z 1 , and K(h λ2 ) ⊂ int(K(h λ1 )). Furthermore, z 1 ∈ int(K(h λ2 )).

4.
For each g ∈ G with J(g) ∩ J min = ∅, we have that g has an attracting fixed point z g in C, int(K(g)) consists of only one immediate attracting basin for z g , and J min ⊂ int(K(g)). Note that it is not necessarily true that z g = z f when g, f ∈ G are such that J(g) ∩ J min = ∅ and J(f ) ∩ J min = ∅ (see Proposition 2.26).

5.
We have that int(K(G)) = ∅. Moreover, (a) C \ J min is disconnected, ♯J ≥ 2 for each J ∈Ĵ , and (b) for each g ∈ G with J(g)∩J min = ∅, we have that J min < g * (J min ), g −1 (J(G))∩J min = ∅, g(K(G) ∪ J min ) ⊂ int(K(G)), and the unique attracting fixed point z g of g in C belongs to int(K(G)).

Let
A be the set of all doubly connected components of F (G). Then, A∈A A ⊂ C and (A, ≤) is totally ordered.
We present a result on uniform perfectness of the Julia sets of semigroups in G.
Definition 2.21. A compact set K inĈ is said to be uniformly perfect if ♯K ≥ 2 and there exists a constant C > 0 such that each annulus A that separates K satisfies that mod A < C, where mod A denotes the modulus of A (See the definition in [15]).
1. Let G be a polynomial semigroup in G. Then, J(G) is uniformly perfect. Moreover, if z 0 ∈ J(G) is a superattracting fixed point of an element of G, then z 0 ∈ int(J(G)).
3. Suppose that G ∈ G dis . Let z 1 ∈ J(G) ∩ C be a superattracting fixed point of g ∈ G. Then z 1 ∈ int(J min ) and J(g) ⊂ J min .
We remark that in [14], it was shown that there exists a rational semigroup G such that J(G) is not uniformly perfect.
We now present results on the Julia sets of subsemigroups of an element of G dis .
Then, we have the following.
3. Let Q = {g ∈ G | ∃J ∈ J with J 1 ≤ J ≤ J 2 , J(g) ⊂ J} and let H be the subsemigroup of G generated by Q.
Proposition 2.24. Let G be a polynomial semigroup generated by a compact subset Γ of Poly deg≥2 . Suppose that G ∈ G dis . Then, there exists an element h 1 ∈ Γ with J(h 1 ) ⊂ J max and there exists an element h 2 ∈ Γ with J(h 2 ) ⊂ J min .

Finitely generated polynomial semigroups
In this section, we present some results on various finitely generated polynomial semigroups G ∈ G dis such that 2 ≤ ♯(Ĵ G ) ≤ ℵ 0 . The proofs are given in Section 4.4. It is well-known that for a rational map g with deg(g) ≥ 2, if J(g) is disconnected, then J(g) has uncountably many connected components (See [16]). Moreover, if G is a non-elementary Kleinian group with disconnected Julia set (limit set), then J(G) has uncountably many connected components. However, for general rational semigroups, we have the following examples.
As mentioned before, these results illustrate new phenomena which can hold in the rational semigroups, but cannot hold in the dynamics of a single rational map or Kleinian groups.
For the figure of the Julia set of a 3-generator polynomial semigroup G ∈ G dis such that ♯Ĵ G = ℵ 0 , see figure 2. Remark 2.30. In [35], a new cohomology theory for (backward) self-similar systems (iterated function systems) was introduced by the author of this paper. By using it, for a finitely generated G ∈ G, we can describe the space J G of connected components of J(G), and we can give some estimates on ♯(J G ). Moreover, by using this new cohomology, a sufficient condition for the cardinality of the set of all connected components of the Fatou set of a postcritically bounded finitely generated polynomial semigroup G to be infinity was given.

Hyperbolicity and semi-hyperbolicity
In this section, we present some results on hyperbolicity and semi-hyperbolicity.
Definition 2.31. Let G be a polynomial semigroup generated by a subset Γ of Poly deg≥2 . Suppose G ∈ G dis . Then we set Γ min := {h ∈ Γ | J(h) ⊂ J min }, where J min denotes the unique minimal element in (J , ≤) in Theorem 2.20-2. Furthermore, if Γ min = ∅, let G min,Γ be the subsemigroup of G that is generated by Γ min . Remark 2.32. Let G be a polynomial semigroup generated by a compact subset Γ of Poly deg≥2 . Suppose G ∈ G dis . Then, by Proposition 2.24, we have Γ min = ∅ and Γ \ Γ min = ∅. Moreover, Γ min is a compact subset of Γ. For, if {h n } n∈N ⊂ Γ min and h n → h ∞ in Γ, then for a repelling periodic The following Proposition 2.33 means that for a polynomial semigroup G ∈ G dis generated by a compact subset Γ of Poly deg≥2 , we rarely have the situation that "Γ \ Γ min is not compact." Proposition 2.33. Let G be a polynomial semigroup generated by a non-empty compact subset Γ of Poly deg≥2 . Suppose that G ∈ G dis and that Γ \ Γ min is not compact. Then, both of the following statements 1 and 2 hold.
is an immediate parabolic basin of a parabolic fixed point of h.
2. We say that G is semi-hyperbolic if there exists a number δ > 0 and a number N ∈ N such that for each y ∈ J(G) and each g ∈ G, we have deg(g : V → B(y, δ)) ≤ N for each connected component V of g −1 (B(y, δ)), where B(y, δ) denotes the ball of radius δ with center y with respect to the spherical distance, and deg(g : · → ·) denotes the degree of a finite branched covering.
Remark 2.35. There are many nice properties of hyperbolic or semi-hyperbolic rational semigroups. For example, for a finitely generated semi-hyperbolic rational semigroup G , there exists an attractor in the Fatou set ( [27,30]), and the Hausdorff dimension dim H (J(G)) of the Julia set is less than or equal to the critical exponent s(G) of the Poincaré series of G ( [30]). If we assume further the "open set condition", then dim H (J(G)) = s(G) ( [32,45]). Moreover, if G ∈ G is generated by a compact set Γ and if G is semi-hyperbolic, then for each sequence γ ∈ Γ N , the basin of infinity for γ is a John domain and the Julia set of γ is locally connected ( [30]). In [37], by using the above result, we classify hyperbolic or semi-hyperbolic postcritically bounded compactly generated polynomial semigroups, in terms of the random complex dynamics. It is shown that in one of the classes, for almost every sequence γ, the Julia set J γ of γ is a Jordan curve but not a quasicircle, the unbounded component ofĈ \ J γ is a John domain, and the bounded component of C \ J γ is not a John domain. Moreover, in [37,36], we find many examples with this phenomenon. Note that this phenomenon does not hold in the usual iteration dynamics of a single polynomial map g with deg(g) ≥ 2.
We now present some results on semi-hyperbolic or hyperbolic polynomial semigroups in G dis . These results are used to construct examples of semi-hyperbolic or hyperbolic polynomial semigroups G ∈ G dis (see the proof of Proposition 2.40). Therefore these are important in terms of the sequel [36,37].
Theorem 2.36. Let G be a polynomial semigroup generated by a non-empty compact subset Γ of Poly deg≥2 . Suppose that G ∈ G dis . If G min,Γ is semi-hyperbolic, then G is semi-hyperbolic. Theorem 2.37. Let G be a polynomial semigroup generated by a non-empty compact subset Γ of Poly deg≥2 . Suppose that G ∈ G dis . If G min,Γ is hyperbolic and ( h∈Γ\Γmin CV * (h)) ∩ J min (G) = ∅, then G is hyperbolic.
Remark 2.38. In [24], it will be shown that in Theorem 2.37, the condition ( h∈Γ\Γmin CV * (h))∩ J min (G) = ∅ is necessary. For the figures of the Julia sets of hyperbolic polynomial semigroups G ∈ G dis , see figure 1 and figure 2.
Proposition 2.39. Let G be a polynomial semigroup generated by a non-empty compact subset Γ of Poly deg≥2 . Suppose that G ∈ G dis and that Γ \ Γ min is not compact. Suppose that statement 2a in Theorem 2.33 holds. Then, both of the following statements hold.
1. We have that G min,Γ is hyperbolic and G is semi-hyperbolic.

Construction of examples
In this section, we present a way to construct examples of semigroups G in G dis (with some additional properties). These examples are important in terms of the sequel [36,37].
Moreover, in addition to the assumption above, if G is semi-hyperbolic (resp. hyperbolic), then the above H Γ,V ′ is semi-hyperbolic (resp. hyperbolic).
Remark 2.41. By Proposition 2.40, there exists a 2-generator polynomial semigroup G = h 1 , h 2 in G dis such that h 1 has a Siegel disk.
We set Y d := {h ∈ Poly | deg(h) = d} endowed with the relative topology from Poly.
. Then, both of the following statements hold.
Theorem 2.45. Under the above notation, all of the following statements hold.

Tools
To show the main results, we need some tools in this section.
The following Lemma 3.1 and Theorem 3.2 will be used in the proofs of the main results. 1. For each h ∈ G, we have h(F (G)) ⊂ F (G) and h −1 (J(G)) ⊂ J(G). Note that we do not have that the equality holds in general.

If
). More generally, if G is generated by a compact subset Γ of Rat, then J(G) = h∈Γ h −1 (J(G)). (We call this property of the Julia set of a compactly generated rational semigroup "backward self-similarity." ) 6. If ♯(J(G)) ≥ 3 , then J(G) is the smallest closed backward invariant set containing at least three points. Here we say that a set A is backward invariant under G if for each where m(g, z) denotes the multiplier of g at z ( [3]). In particular, J(G) = g∈G J(g).
. By the Riemann-Hurwitz formula, it follows that h −1 1 (F ∞ (h 2 )) is connected and simply connected.
Lemma 3.6. Let Γ be a non-empty compact subset of Poly deg≥2 and let G be a polynomial semigroup generated by Γ. Let R > 0, ǫ > 0, and

Proofs of the main results
In this section, we demonstrate the main results.

Proofs of results in 2.1
In this section, we demonstrate the results in 2.1.
Proof of Theorem 2.1: First, we show the following: Then h λ is either identity or an elliptic Möbius transformation. By hypothesis and Lemma 3.1-1, we obtain h −1 λ (A) ⊂ A. Hence, we have shown the claim. Combining the above claim with ♯A ≥ 3, by Lemma 3.1-6 we obtain J(G) ⊂ A. Hence J(G) = A and J(G) is connected.
We need the following lemmas to prove the main results. d(z, J ′ ) → 0 as j → ∞. Since J(g nj ) is compact and connected for each j, we may assume, passing to a subsequence, that there exists a non-empty compact connected subset K ofĈ such that J(g nj ) → K as j → ∞, with respect to the Hausdorff metric. Then K ∩ J = ∅ and K ∩ J ′ = ∅. Since K ⊂ J(G) and K is connected, it contradicts J ′ = J.
Lemma 4.2. Let G ∈ G. Then given J ∈ J and ǫ > 0, there exists an element g ∈ G such that J(g) ⊂ B(J, ǫ).
Proof. We take a point z ∈ J. Then, by Theorem 3.2, there exists a sequence {g n } n∈N in G such that d(z, J(g n )) → 0 as n → ∞. By Lemma 4.1, we conclude that there exists an n ∈ N such that J(g n ) ⊂ B(J, ǫ).
by Lemma 3.1-6. Then J(G) = A and it causes a contradiction, since J(G) is disconnected.
We now demonstrate Theorem 2.7. Proof of Theorem 2.7: First, we show statement 1. Suppose the statement is false. Then, there exist elements However, this implies a contradiction since P * (G) is bounded in C. Hence we have shown statement 1.
Next, we show statement 2. Let F 1 be a connected component of F (G). Suppose that there exist three connected components J 1 , J 2 and J 3 of J(G) such that they are mutually disjoint and such that ∂F 1 ∩ J i = ∅ for each i = 1, 2, 3. Then, by statement 1, we may assume that we have either (1): J i ∈ J for each i = 1, 2, 3 and J 1 < J 2 < J 3 , or (2): J 1 , J 2 ∈ J , J 1 < J 2 , and ∞ ∈ J 3 . Each of these cases implies that J 1 is included in a bounded component of C \ J 2 and J 3 is included in the unbounded component ofĈ \ J 2 . However, it causes a contradiction, since ∂F 1 ∩ J i = ∅ for each i = 1, 2, 3. Hence, we have shown that we have either Suppose that we have Case I. Let J 1 be the connected component of J(G) such that ∂F 1 ⊂ J 1 . Let D 1 be the connected component ofĈ\J 1 containing F 1 . Since ∂F 1 ⊂ J 1 , we have ∂F 1 ∩D 1 = ∅. Hence, we have F 1 = D 1 . Therefore, F 1 is simply connected.
Suppose that we have Case II. Let J 1 and J 2 be the two connected components of J(G) such that J 1 = J 2 and Hence, we have F 1 = D. Therefore, F 1 is doubly connected. Thus, we have shown statement 2.
We now show statement 3. Let g ∈ G be an element and J a connected component of J(G). Suppose that g −1 (J) is disconnected. Then, by Lemma 3.7, there exist at most finitely many connected components C 1 , . . . , C r of g −1 (J) with r ≥ 2. Then there exists a positive number ǫ such that denoting by B j the connected component of g −1 (B(J, ǫ)) containing C j for each j = 1, . . . , r, {B j } are mutually disjoint. By Lemma 3.7, we see that, for each connected component B of g −1 (B(J, ǫ)), g(B) = B(J, ǫ) and B ∩ C j = ∅ for some j. Hence we get that g −1 (B(J, ǫ)) = r j=1 B j (disjoint union) and g(B j ) = B(J, ǫ) for each j. By Lemma 4.2, there exists an element h ∈ G such that J(h) ⊂ B(J, ǫ). Then it follows that g −1 (J(h)) ∩ B j = ∅ for each j = 1, . . . , r. Moreover, we have g −1 (J(h)) ⊂ g −1 (B(J, ǫ)) = r j=1 B j . On the other hand, by Lemma 3.4, we have that g −1 (J(h)) is connected. This is a contradiction. Hence, we have shown that, for each g ∈ G and each connected component J of J(G), g −1 (J) is connected. By Lemma 4.3, we get that if J ∈ J , then g * (J) ∈ J . Let J 1 and J 2 be two elements of J such that J 1 ≤ J 2 . Let U i be the unbounded component of C \ J i , for each i = 1, 2. Then U 2 ⊂ U 1 . Let g ∈ G be an element. Then g −1 (U 2 ) ⊂ g −1 (U 1 ). Since g −1 (U i ) is the unbounded connected component of C \ g −1 (J i ) for each i = 1, 2, it follows that g −1 (J 1 ) ≤ g −1 (J 2 ). Hence g * (J 1 ) ≤ g * (J 2 ), otherwise g * (J 2 ) < g * (J 1 ), and it contradicts g −1 (J 1 ) ≤ g −1 (J 2 ).

Proofs of results in 2.2
In this section, we prove the results in Section 2.2, except Theorem 2.12-2 and Theorem 2.12-3, which will be proved in Section 4.3.
To demonstrate Theorem 2.12, we need the following lemmas.
Lemma 4.4. Let G be a polynomial semigroup in G dis . Let J 1 , J 2 ∈Ĵ be two elements with J 1 = J 2 . Then, we have the following.
1. If J 1 , J 2 ∈ J and J 1 < J 2 , then there exists a doubly connected component A of F (G) such that J 1 < A < J 2 .
2. If ∞ ∈ J 2 , then there exists a doubly connected component A of F (G) such that J 1 < A.
Proof. First, we show statement 1. Suppose that J 1 , J 2 ∈ J and J 1 < J 2 . We set B = J∈J ,J1≤J≤J2 J. Then, B is a closed disconnected set. Hence, there exists a multiply connected component A ′ of C \ B. Since A ′ is multiply connected, we have that A ′ is included in the unbounded component of Hence, A must be equal to A ′ . Since A ′ is multiply connected, Theorem 2.7-2 implies that A = A ′ is doubly connected. Let J be the connected component J(G) such that J < A and J ∩ ∂A = ∅. Then, since A ′ = A is included in the unbounded component ofĈ \ J 1 , we have that J does not meet any bounded component of C \ J 1 . Hence, we obtain J 1 ≤ J, which implies that J 1 ≤ J < A. Therefore, A is a doubly connected component of F (G) such that J 1 < A < J 2 . Hence, we have shown statement 1. Next, we show statement 2. Suppose that ∞ ∈ J 2 . We set B = ( J∈J ,J1≤J J) ∪ J 2 . Then, B is a disconnected closed set. Hence, there exists a multiply connected component A ′ ofĈ \ B. By the same method as that of proof of statement 1, we see that A ′ is equal to a doubly connected component A of F (G) such that J 1 < A. Hence, we have shown statement 2.
We need the notion of Green's functions, in order to demonstrate Theorem 2.12. 3. there exists a Borel subset A of ∂D such that the logarithmic capacity of (∂D) \ A is zero and such that for each ζ ∈ A, we have ϕ(D, z) → 0 as z → ζ. 3. It is well-known that for any g ∈ Poly deg≥2 , (See [25, p147].) Note that the point − 1 deg(g)−1 log |a(g)| ∈ R is the unique fixed point of Ψ(g) in R.
In order to demonstrate Theorem 2.12-1, we will prove the following lemma. (Theorem 2.12-2 and Theorem 2.12-3 will be proved in Section 4.3.) To show this claim, let R > 0 be a number such that J(G) ⊂ D(0, R). Then, for any g ∈ G, we have K(g) < ∂D(0, R). By Lemma 4.8, we get that there exists a constant C > 0 such that for each g ∈ G, −1 deg(g)−1 log |a(g)| ≤ C. Hence, it follows that M (Ψ(G)) ⊂ [−∞, C]. Therefore, we have shown Claim 1.
We now prove the statement of the lemma in the case G ∈ G con . If ∞ ∈ F (G), then claim 1 implies that M (Ψ(G)) ⊂R \ {+∞} and the statement of the lemma holds. We now suppose ∞ ∈ J(G). We put L g := max z∈J(g) |z| for each g ∈ G. Moreover, for each non-empty compact subset E of C, we denote by Cap (E) the logarithmic capacity of E. We remark that Cap(E) = exp(lim z→∞ (log |z| − ϕ(D E , z))), where D E denotes the connected component ofĈ \ E containing ∞. We may assume that 0 ∈ P * (G). Then, by [1], we have Cap(J(g)) ≥ Cap ([0, L g ]) ≥ L g /4 for each g ∈ G. Combining this with ∞ ∈ J(G), Theorem 3.2, and Remark 4.7-3, we obtain +∞ ∈ M Ψ(G) and definingΨ(J(G)) to be the connected component of M Ψ(G) containing +∞, the statement of the lemma holds.
We now prove the statement of the lemma in the case G ∈ G dis . Let {J λ } λ∈Λ be the setĴ G of all connected components of J(G). By Lemma 4.2, for each λ ∈ Λ and each n ∈ N, there exists an element g λ,n ∈ G such that We have that the fixed point of Ψ(g λ,n ) in R is equal to −1 deg(g λ,n )−1 log |a(g λ,n )|. We may assume that −1 deg(g λ,n )−1 log |a(g λ,n )| → α λ as n → ∞, where α λ is an element ofR. For each λ ∈ Λ, let B λ ∈ M Ψ(G) be the element with α λ ∈ B λ . LetΨ(J λ ) = B λ for each λ ∈ Λ. We will show the following claim. Claim 2: If λ, ξ are two elements in Λ with λ = ξ, then To show this claim, let λ and ξ be two elements in Λ with λ = ξ. We have the following two cases: Case 1: J λ , J ξ ∈ J G and J λ < J ξ .
We now demonstrate Theorem 2.15. Proof of Theorem 2.15: Let C be a set of polynomials of degree two such that C generates G. Suppose that J(G) is disconnected. Then, by Theorem 2.1, there exist two elements h 1 , h 2 ∈ C such that the semigroup H = h 1 , h 2 satisfies that J(H) is disconnected. For each j = 1, 2, let a j be the coefficient of the highest degree term of polynomial h j . Let α := min j=1,2 { −1 deg(hj)−1 log |a j |} and β := max j=1,2 { −1 deg(hj )−1 log |a j |}. Then we have that α = min j=1,2 {− log |a j |} and β = max j=1, 2 . Hence, by Theorem 2.14, it must be true that J(H) is connected. However, this is a contradiction. Therefore, J(G) must be connected.
We now demonstrate Theorem 2.16. Proof of Theorem 2.16: For each λ ∈ Λ, let b λ be the fixed point of Ψ(h λ ) in R. It is easy to see that b λ = −1 deg(h λ )−1 log |a λ |, for each λ ∈ Λ. From the assumption, it follows that there exists a point b ∈ R such that for each λ ∈ Λ, b λ = b. This implies that for any element g ∈ G, the fixed point b(g) ∈ R of Ψ(g) in R is equal to b. Hence, we obtain M (Ψ(G)) = {b}. Therefore, M (Ψ(G)) is connected. From Theorem 2.12-1, it follows that J(G) is connected.

Proofs of results in 2.3
In this section, we prove the results in 2.3, Theorem 2.12-2 and Theorem 2.12-3.
In order to demonstrate Theorem 2.20, Theorem 2.12-2, and Theorem 2.12-3, we need the following lemma. Proof. Suppose that G ∈ G dis and ∞ ∈ J(G). We will deduce a contradiction. By Lemma 4.3, the element J ∈Ĵ G with ∞ ∈ J satisfies that J = {∞}. Hence, by Lemma 4.2, for each n ∈ N, there exists an element g n ∈ G such that J(g n ) ⊂ B(∞, 1 n ). Let R > 0 be any number which is sufficiently large so that P * (G) ⊂ B(0, R). Since we have that P * (G) ⊂ K(g) for each g ∈ G, it must hold that there exists a number n 0 = n 0 (R) ∈ N such that for each n ≥ n 0 , B(0, R) < J(g n ). From Lemma 4.8, it follows that lim z→∞ (log |z| − ϕ(F ∞ (g n ), z)) → +∞ as n → ∞. Hence, we see that Furthermore, by Theorem 2.12-1, we must have that M (Ψ(G)) is disconnected. We now consider the polynomial semigroup H = {z → |a(g)|z deg(g) | g ∈ G} ∈ G. By Theorem 3.2, we have J(H) = h∈H J(h). Since the Julia set of polynomial |a(g)|z deg(g) is equal to where the closure is taken inĈ. Moreover, J(Θ(G)) = J(H). Combining it with (10), (11), and Corollary 2.13, we see that Furthermore, we have that Combining (13), (14), and (15), we see that By Lemma 4.3 and (12), we get that the connected component J of J(H) containing ∞ satisfies that Combined with Lemma 4.2, we see that for each n ∈ N, there exists an element h n ∈ H such that Combining (11), (13), (17), and (18), we obtain the following claim.
We now demonstrate Proposition 2.19. Proof of Proposition 2.19: Let U be a connected component of F (G) with U ∩K(G) = ∅. Let g ∈ G be an element. Then we haveK(G) ∩ F (G) ⊂ int(K(g)). Since h(F (G)) ⊂ F (G) and h(K(G) ∩ F (G)) ⊂K(G) ∩ F (G) for each h ∈ G, it follows that h(U ) ⊂ int(K(g)) for each h ∈ G. Hence U ⊂ int(K(G)). From this, it is easy to seeK(G) ∩ F (G) = int(K(G)). By the maximum principle, we see that U is simply connected.
We now demonstrate Theorem 2.20. Proof of Theorem 2.20: First, we show statement 1. By Lemma 4.10, we have that ∞ ∈ F (G). Let J ∈ J be an element such that ∂F ∞ (G) ∩ J = ∅. Let D be the unbounded component ofĈ \ J. Then F ∞ (G) ⊂ D and D is simply connected. We show F ∞ (G) = D. Otherwise, there exists an element J 1 ∈ J such that J 1 = J and J 1 ⊂ D. By Theorem 2.7-1, we have either J 1 < J or J < J 1 . Hence, it follows that J < J 1 and we have that J is included in a bounded component Next, let J max be the element of J with ∂F ∞ (G) ⊂ J max , and suppose that there exists an element J ∈ J such that J max < J. Then J max is included in a bounded component of C \ J. On the other hand, F ∞ (G) is included in the unbounded component ofĈ \ J. Since ∂F ∞ (G) ⊂ J max , we have a contradiction. Hence, we have shown that J ≤ J max for each J ∈ J .
Therefore, we have shown statement 1. However, this is a contradiction, since g l n (x) → ∞ as l → ∞, and x ∈K(G). Hence, we have shown statement 2.
Next, we show statement 3. By Theorem 2.1, there exist λ 1 , λ 2 ∈ Λ and connected components J 1 , J 2 of J(G) such that J 1 = J 2 and J(h λi ) ⊂ J i for each i = 1, 2. By Lemma 4.3, we have J i ∈ J for each i = 1, 2. Then J(h λ1 ) ∩ J(h λ2 ) = ∅. Since P * (G) is bounded in C, we may assume J(h λ2 ) < J(h λ1 ). Then we have K(h λ2 ) ⊂ int(K(h λ1 )) and J 2 < J 1 . By statement 2, J 1 = J min . Hence J(h λ1 ) ∩ J min = ∅. Since P * (G) is bounded in C, we have that K(h λ2 ) is connected. Let U be the connected component of int(K(h λ1 )) containing K(h λ2 ). Since P * (G) ⊂ K(h λ2 ), it follows that there exists an attracting fixed point z 1 of h λ1 in K(h λ2 ) and U is the immediate attracting basin for z 1 with respect to the dynamics of h λ1 . Furthermore, by Lemma 3.4, h −1 λ1 (J(h λ2 )) is connected. Therefore, h −1 λ1 (U ) = U. Hence, int(K(h λ1 )) = U. Suppose that there exists an n ∈ N such that h −n λ1 (J(h λ2 )) ∩ J(h λ2 ) = ∅. Then, by Lemma 3.4, A := s≥0 h −ns λ1 (J(h λ2 )) is connected and its closure A contains J(h λ1 ). Hence J(h λ1 ) and J(h λ2 ) are included in the same connected component of J(G). This is a contradiction. Therefore, for each , we obtain z 1 ∈ int(K(h λ2 )). Hence, we have proved statement 3.
We now prove statement 4. Let g ∈ G be an element with J(g) ∩ J min = ∅. We show the following: Claim 2: J min < J(g).
To show the claim, suppose that J min is included in the unbounded component U of C \ J(g). Since ∅ = ∂K(G) ⊂ J min , it follows thatK(G) ∩ U = ∅. However, this is a contradiction. Hence, we have shown Claim 2.
Combining Claim 2, Theorem 3.2 and Lemma 4.1, we get that there exists an element h 1 ∈ G such that J(h 1 ) < J(g). From an argument which we have used in the proof of statement 3, it follows that g has an attracting fixed point z g in C and int(K(g)) consists of only one immediate attracting basin for z g . Hence, we have shown statement 4.
By statement 4, g has a unique attracting fixed point z g in C. Then, z g ∈ P * (G) ⊂K(G). Hence, z g = g(z g ) ∈ g(K(G)) ⊂ int(K(G)). Hence, we have shown statement 5.
We now show statement 6. Since F ∞ (G) is simply connected (statement 1), we have A∈A A ⊂ C. Suppose that there exist two distinct elements A 1 and A 2 in A such that A 1 is included in the unbounded component of C \ A 2 , and such that A 2 is included in the unbounded component of C\A 1 . For each i = 1, 2, let J i ∈ J be the element that intersects the bounded component of C\A i . Then, J 1 = J 2 . Since (J , ≤) is totally ordered (Theorem 2.7-1), we may assume that J 1 < J 2 . Then, it implies that A 1 < J 2 < A 2 , which is a contradiction. Hence, (A, ≤) is totally ordered. Therefore, we have proved statement 6.
Thus, we have proved Theorem 2.20.
We now demonstrate Theorem 2.22. Proof of Theorem 2.22: First, we show Theorem 2.22-1. If G ∈ G con , then J(G) is uniformly perfect.
We now suppose that G ∈ G dis . Let A be an annulus separating J(G). Then A separates J min and J max . Let D be the unbounded component of C \ J min and let U be the connected component of C \ J max containing J min . Then it follows that A ⊂ U ∩ D. Since ♯J min > 1 and ∞ ∈ F (G) (Theorem 2.20), we get that the doubly connected domain U ∩ D satisfies mod (U ∩ D) < ∞. Hence, we obtain mod A ≤ mod (U ∩ D) < ∞. Therefore, J(G) is uniformly perfect.
If a point z 0 ∈ J(G) is a superattracting fixed point of an element g ∈ G, then, combining uniform perfectness of J(G) and [14,Theorem 4.1], it follows that z 0 ∈ int(J(G)). Thus, we have shown Theorem 2.22-1.
Hence, we have shown Theorem 2.22.
We now demonstrate Theorem 2.12-2. Proof of Theorem 2.12-2: Suppose G ∈ G dis . Then, by Lemma 4.10, we obtain ∞ ∈ F (G). Hence, there exists a number R > 0 such that for each g ∈ G, J(g) < ∂B(0, R). From Lemma 4.8, it follows that there exists a constant C 1 > 0 such that for each g ∈ G, This implies that there exists a constant C 2 ∈ R such that Moreover, by Theorem 2.20-5, we have that int(K(G)) = ∅. Let B be a closed disc in int(K(G)).
We now demonstrate Proposition 2.23. Proof of Proposition 2.23: First, we show statement 1. Let g ∈ Q 1 . We show the following: Claim 1: For any element J 3 ∈ J with J 1 ≤ J 3 , we have J 1 ≤ g * (J 3 ). To show this claim, let J ∈ J be an element with J(g) ⊂ J. We consider the following two cases; Case 1: J ≤ J 3 , and Case 2: Suppose that we have Case 1. Then, J 1 ≤ J = g * (J) ≤ g * (J 3 ). Hence, the statement of Claim 1 is true.
Next, we show statement 2. Let g ∈ Q 2 . Then, by the same method as that of the proof of Claim 1, we obtain the following. Claim 2: For any element J 4 ∈ J with J 4 ≤ J 2 , we have g * (J 4 ) ≤ J 2 . Now, let K 2 := J(G) ∩ (C \ A 2 ). Then, by Claim 2, we obtain g −1 (K 2 ) ⊂ K 2 , for each g ∈ Q 2 . From Lemma 3.1-6, it follows that J(H 2 ) ⊂ K 2 . Hence, we have shown statement 2.
Next, we show statement 3. By statements 1 and 2, we obtain J(H) Hence, we have proved Proposition 2.23.

Proofs of results in 2.4
In this section, we prove the results in 2.4.
To prove Theorem 2.27, we need the following notation.
3. L has infinitely many connected components. 4. Let x := (1, 1, 1, . . .) ∈ Σ m and let x ′ ∈ Σ m be an element with x = x ′ . Then, for any y ∈ L x and y ′ ∈ L x ′ , there exists no connected component A of L such that y ∈ A and y ′ ∈ A.
Proof. We show statement 1 by induction on k. We have Hence, by statement 2, we conclude that L has infinitely many connected components. We now show statement 4. Let k 0 := min{l ∈ N | x ′ l = 1}. Then, by (25) and statement 1, we get that there exist compact sets B 1 and B 2 in L such that Hence, statement 4 holds.
We now demonstrate Theorem 2.27. Proof of Theorem 2.27: By Theorem 2.20-1 or Remark 2.5, we haveĴ = J . Let J 1 ∈Ĵ be the element containing J(h m ). By Theorem 2.1, we must have J 0 = J 1 . Then, by Theorem 2.7-1, we have the following two possibilities.
Thus, we have proved Proposition 2.28.
We now show Proposition 2.29. Proof of Proposition 2.29: In fact, we show the following claim: Claim: There exists a polynomial semigroup G = h 1 , h 2 , h 3 in G such that all of the following hold.
Definition 4.15. Let G be a rational semigroup and N a positive integer. We denote by SH N (G) the set of points z ∈Ĉ satisfying that there exists a positive number δ such that for each g ∈ G, deg(g : V → B(z, δ)) ≤ N , for each connected component V of g −1 (B(z, δ)). Moreover, we set U H(G) :=Ĉ \ N ∈N SH N (G). Lemma 4.16. Let G be a polynomial semigroup generated by a compact subset Γ of Poly deg≥2 . Suppose that G ∈ G dis and that Γ \ Γ min is not compact. Moreover, suppose that (a) in Proposition 2.33-2 holds. Then, there exists an open neighborhood U of Γ min in Γ and an open set U in int(K(G)) with U ⊂ int(K(G)) such that: 2. h∈U CV * (h) ⊂ U , and 3. denoting by H the polynomial semigroup generated by U, we have that P * (H) ⊂ int(K(G)) ⊂ F (H) and that H is hyperbolic.
Proof. Let h 0 ∈ Γ min be an element. Let E := {ψ(z) = az +b | a, b ∈ C, |a| = 1, ψ(J(h 0 )) = J(h 0 )}. Then, by [2], E is compact in Poly. Moreover, by [2], we have the following two claims: Claim 1: If J(h 0 ) is a round circle with the center b 0 and radius r, is not a round circle, then ♯E < ∞. Let z 0 be the unique attracting fixed point of h 0 in C. Let g ∈ G min,Γ . By [2], for each n ∈ N, there exists an ψ n ∈ E such that h n 0 g = ψ n gh n 0 . Hence, for each n ∈ N, h n 0 g(z 0 ) = ψ n gh n 0 (z 0 ) = ψ n g(z 0 ). Combining it with Claim 1 and Claim 2, it follows that there exists an n ∈ N such that h n 0 (g(z 0 )) = z 0 . For this n, g(z 0 ) = ψ −1 n (h n 0 (g(z 0 ))) = ψ −1 n (z 0 ) ∈ ψ∈E ψ(z 0 ). Combining it with Claim 1 and Claim 2 again, we see that the set C := g∈Gmin,Γ {g(z 0 )} is a compact subset of int(K(G)). Let d H be the hyperbolic distance on int(K(G)). Let R > 0 be a large number such that setting U := {z ∈ int(K(G)) | min a∈C d H (z, a) < R}, we have h∈Γmin CV * (h) ⊂ U. Then, for each h ∈ Γ min , h(U ) ⊂ U. Therefore, there exists an open neighborhood U of Γ min in Γ such that h∈U h(U ) ⊂ U , and such that h∈U CV * (h) ⊂ U. Let H be the polynomial semigroup generated by U. From the above argument, we obtain P * (H) = g∈H CV * (g) ⊂ g∈H∪{Id} g h∈U CV * (h) ⊂ g∈H∪{Id} g(U ) ⊂ U ⊂ int(K(G)) ⊂ F (H). Hence, H is hyperbolic. Thus, we have proved Lemma 4.16.
We now demonstrate Theorem 2.36. Proof of Theorem 2.36: Suppose that G min,Γ is semi-hyperbolic. We will consider the following two cases: Case 1: Γ \ Γ min is compact. Case 2: Γ \ Γ min is not compact.
We now suppose that we have Case 2. Then, by Proposition 2.33, we have that for each h ∈ Γ min , K(h) =K(G) and int(K(h)) is non-empty and connected. Moreover, for each h ∈ Γ min , int(K(h)) is an immediate basin of an attracting fixed point z h ∈ C. Let U be the open neighborhood of Γ min in Γ as in Lemma 4.16. Denoting by H the polynomial semigroup generated by U, we have P * (H) ⊂ int(K(G)). Therefore, there exists a number δ > 0 such that D(J(G), δ) ⊂ C \ P (H).
Moreover, combining Theorem 2.20-5b and that Γ\U is compact, we see that there exists a number ǫ > 0 such that for each connected component V of h −1 (D(z, cδ)). Let z ∈ J min (G) and g ∈ G. We will show that z ∈ C \ U H(G). Suppose that g ∈ H. Then, (62) implies that for each connected component V of g −1 (D(z, cδ)), deg(g : V → D(z, cδ)) = 1.
Thus, we have proved Theorem 2.36.
We now demonstrate Theorem 2.37. Proof of Theorem 2.37: We use the same argument as that in the proof of Theorem 2.36, but we modify it as follows:  (1)) (z → ∞) uniformly on Γ, it follows that if c is small enough, then for any a ∈ C with 0 < |a| < c and for any h ∈ Γ, . This implies that for each h ∈ Γ, Moreover, if c is small enough, then for any a ∈ C with 0 < |a| < c and any h ∈ Γ, Let a ∈ C with 0 < |a| < c. By which implies that int(K(G)) ∪ (Ĉ \ D a ) ⊂ F (H Γ,V ), where H Γ,V denotes the polynomial semigroup generated by the family Γ ∪ V. By (71), we obtain that for any non-empty subset V ′ of V , where H Γ,V ′ denotes the polynomial semigroup generated by the family Γ ∪ V ′ . If the compact neighborhood V of g a is so small, then g∈V CV * (g) ⊂ int(K(G)).
We now show that for any non-empty subset V ′ of V , J(H Γ,V ′ ) is disconnected and (Γ∪V ′ ) min ⊂ Γ. Let Then, for any h ∈ Γ, h(U ) ⊂Ĉ \ D a .
Moreover, for any g ∈ V , g(U ) ⊂ int(K(G)). Combining it with (73), (76), and Lemma 3.1-2, it follows that U ⊂ F (H Γ,V ). If the neighborhood V of g a is so small, then there exists an annulus A in U such that for any g ∈ V , A separates J(g) and h∈Γ h −1 (J(g)). Hence, it follows that for any non-empty subset V ′ of V , the polynomial semigroup H Γ,V ′ generated by the family Γ ∪ V ′ satisfies that J(H Γ,V ′ ) is disconnected and (Γ ∪ V ′ ) min ⊂ Γ. We now suppose that in addition to the assumption, G is semi-hyperbolic. Let V ′ be any non-empty subset of V. Since (Γ ∪ V ′ ) min ⊂ Γ, Theorem 2.36 implies that the above H Γ,V ′ is semi-hyperbolic.
We now suppose that in addition to the assumption, G is hyperbolic. Let V ′ be any non-empty subset of V. By (74) and (75), we have g∈Γ∪V ′ CV * (g) ⊂ int(K(H Γ,V ′ )).
If h 1 is semi-hyperbolic, then using the same method as that of Case 1 in the proof of Theorem 2.36, we obtain that G is semi-hyperbolic.
We now suppose that h 1 is hyperbolic. By (78), we have m j=2 CV * (h j ) ⊂ int(K(G)). Combining it with the same method as that in the proof of Theorem 2.37, we obtain that G is hyperbolic. Hence, we have proved statement 1.