DYNAMICS OF REGULARLY RAMIFIED RATIONAL MAPS: I. JULIA SETS OF MAPS IN ONE-PARAMETER FAMILIES

. In [6], regularly ramiﬁed rational maps are constructed and Julia sets of these maps in some one-parameter families are explored through computer-generated pictures. It is observed that they have classiﬁcations similar to the Julia sets of maps in the families f cn ( z ) = z n + cz n , where n ≥ 2 and c is a complex number. A new type of Julia set is also presented, which has not appeared in the literature. We call such a Julia set an exploded McMullen necklace . We prove in this paper: if a map f in the one-parameter families given in [6] has a superattracting ﬁxed point of order greater than 2, then its Julia set J ( f ) is either connected, a Cantor set, or a McMullen necklace (either exploded or not); if such a map f has a superattracting ﬁxed point of order equal to 2, then J ( f ) is either connected or a Cantor set.


1.
Introduction. Let C be the Riemann sphere. A rational map f : C → C is said to be regularly ramified if there exists a group G of conformal automorphisms of C such that two points z 1 and z 2 have the same image under f if and only if there is an element g of G with g(z 1 ) = z 2 ( [8]). In fact, there are only five types of such maps, which correspond to the only five types of finite Kleinian groups acting on C ( [5]). Let G be such a group. Then the quotient spaceĈ/G ofĈ under G is conformally equivalent toĈ and hence the projection map fromĈ toĈ/G =Ĉ is a regularly ramified rational map, denoted by R G .
Up to conjugacy by a Möbius transformation, a finite Kleinian group G is equal to one of the following five groups: 1. A cyclic group G 1 of order ν generated by z → e 2πi ν z, where ν is any positive integer; 2. A dihedral group G 2 generated by the symmetries of a regular polygon of ν sides, where ν is a positive integer ≥ 2; 3. The tetrahedral group G 3 generated by the symmetries of a regular tetrahedron; 4. The octahedral group G 4 generated by the symmetries of a regular octahedron or its dual, a cube; 5. The icosahedral group G 5 generated by the symmetries of a regular icosahedron or its dual, a regular dodecahedron. Elements of each group G j are rotations of finite order. Fixed points of nonidentity elements of G j are critical points of the corresponding projection map R Gj . Points on the same orbit under G j are mapped by R Gj to the same point. Since each group G j has only two or three distinct orbits of fixed points, the projection map R Gj has two or three critical values. In fact, R G1 has two critical values and all the other four projection maps R Gj , j = 2, 3, 4, 5, have three critical values. Up to conjugacy by a Möbius transformation, a regularly ramified rational map f is written as f = A • R Gj for some Möbius transformation A and some 1 ≤ j ≤ 5. Clearly, A • R Gj • g = A • R Gj for any g ∈ G j , hence we call f = A • R Gj a regularly ramified rational map with G j invariance. It also follows that for every point q ∈ C, all pre-images of q under f have equal indices (meaning that f has same local degrees at all pre-images of q). Furthermore, the modular space of A • R G1 in terms of dynamics is a two-dimensional space and the space of A • R Gj , 2 ≤ j ≤ 5, is a three-dimensional space. The former has been studied by Milnor in [9], but the latter has not been investigated much yet. Milnor showed in [9] that A • G 1 cannot have Herman rings in its Fatou set and its Julia set is either connected or a Cantor set. Some one-parameter and two-parameter families of A•G 2 have been studied by Devaney and his collaborators in several papers, among which [3] is most related to this paper. Some two-parameter families of A • G 2 are investigated in [14] and [15]. We are interested in studying classifications of the Julia sets of regularly ramified rational maps A • R Gj with 2 ≤ j ≤ 5, especially when 3 ≤ j ≤ 5.
Julia sets of regularly ramified maps A • R Gj with 2 ≤ j ≤ 5 are first explored in [6] through computer-generated pictures. Each of those maps A • R Gj satisfies the following two assumptions: 1. There exists a critical point fixed by the map, denoted by p; 2. There exists one critical value contained in (A • R Gj ) −1 ({p}) \ {p}, denoted by q. Without loss of generality, one can arrange p at the origin and q at infinity. Such a map A • R Gj is called a normalized projection map in [6], which is now called in this paper a normalized regularly ramified rational map and is briefly denoted by A • R Gj .
The goal of this paper is to classify Julia sets of normalized regularly ramified rational maps A • R Gj in some one-parameter families, where 2 ≤ j ≤ 5. We establish two classification results; that is, if the order of the fixed critical point p is more than 2, then the classification is a tetrachotomy; if the order of the fixed point p is 2, then the classification is a trichotomy.
The paper is organized as follows: Section 2 is a warm-up section. One-parameter families of maps of form A • R G2 or A • R G2 • φ are explored in this section, where A and φ are two Möbius transformations. They are different from the family F ν c (z) = z ν + c z ν . But the strategy and techniques to draw classification results on the Julia sets of the maps in these one-parameter families are quite similar to those applied to the family F ν c , and in fact one of these families can be embedded into a two-parameter family studied in [14]. So we mostly give statements of results without recapitulating details from the works of [3] and [14] to prove them. Section 3 is the main work of this paper. We give proofs to classification results of the Julia sets of the maps in two one-parameter families of maps of form A • R G4 or A•R G4 •φ. In Subsection 3.1, we give statements of three main results: nonexistence of Herman rings, a tetrachotomy of the Julia sets of the maps in one family, and a trichotomy of the Julia sets of the maps in another family. In Subsection 3.2, we prove the result on nonexistence of Herman rings for the maps in either of the two families. In Subsection 3.3, we prove the tetrachotomy result. In Subsection 3.4, we prove the trichotomy result. Section 4 is the last section. The first half considers two one-parameter families of maps of form A • R G5 or A • R G5 • φ explored in [6] and states, without proofs, three main results: nonexistence of Herman rings, a tetrachotomy of the Julia sets of the maps in one family, and a trichotomy of the Julia sets of the maps in another family. The second half constructs two one-parameter families of maps of form A • R G3 or A • R G3 • φ satisfying the two assumptions given in this section. Then we finish the section by stating the corresponding results for the maps in these two families.
2. The dihedral case. Let ν be a positive integer ≥ 2. The symmetric group of a regular polygon of ν sides, called a dihedral group and denoted by G 2 , can be generated by rotations z → e 2πi ν z and z → 1 z . The orbit of fixed points of group elements of order ν is given by one orbit of fixed points of order 2 is given by and another orbit of fixed points of order 2 is given by A normalized projection map A • R G2 is given in [6] by The critical points of f (2,ν) (z) are comprised of the points in Julia sets are explored in [6] for rational maps A • R G2 in the following oneparameter families where ν is an integer greater than 1 and λ is a nonzero complex number. From computer-generated pictures, the Julia sets of rational maps in this family have a similar classification as the rational maps in the following family of singularly perturbed monomials where ν is an integer ≥ 2. The family F ν c has been extensively studied by Professor Robert L. Devaney and his collaborators. The Julia sets of the maps F ν c have a major difference in classification when ν > 2 and ν = 2 ([2]). That is, when ν > 2, there is a so-called McMullen domain containing the origin in the parameter plane such that for each nonzero parameter c in the domain, the Julia set of the map F ν c is a Cantor set of disjoint Jordan curves surrounding the origin, which we briefly call a McMullen necklace ( [7]); when ν = 2, such a McMullen domain doesn't exist. Whether ν > 2 or ν = 2, it is proved that the Julia set of F ν c is connected if the nonzero critical values don't escape to infinity ( [4]). When the nonzero critical values do escape to infinity, the main result of [3] covers the statement that if ν > 2, then the Julia set of F ν c is a Cantor set, a McMullen necklace or a Sierpinski curve; if ν = 2, then the Julia set of F ν c is a Cantor set or a Sierpinski curve, where by a Sierpinski curve we mean a planar set homeomorphic to the well-known Sierpinski carpet fractal. Our computer-generated pictures of Julia sets indicate that the family f λ (2,ν) (z) behaves very similarly to F ν c (z) when ν > 2 and when ν = 2. It is given in [6] by the Möbius transformation z → ν−1 √ λ z . Clearly,f λ (2,ν) and F ν c (z) are two distinct one-parameter families of singularly perturbed monomials. Therefore, one cannot apply the results of the family F ν c (z) to draw same results for f λ (2,ν) . On the other REGULARLY RAMIFIED RATIONAL MAPS 3193 hand, f λ (2,ν) is a one-parameter family embedded in the following two-parameter family where c = 0 and ν > 2. Classification of the Julia sets of the maps in this twoparameter family is given in [14]. Strategies, frameworks and techniques in [3] and [14] can be applied quite directly to classify the Julia sets of the maps in the familyf λ (2,ν) and hence in the family f λ (2,ν) . Without rewriting those frameworks and details, we give the statements of results for the family f λ (2,ν) . Using the method of [13] (or a simplified one given in [4]) to show the nonexistence of Herman rings for maps in the family F ν c , we first obtain the following theorem. Theorem 2.1. For each ν ≥ 2 and each nonzero parameter λ, f λ (2,ν) has no Herman rings.
Let A(0) be the global attracting basin of f λ (2,ν) at the origin and let B(0) be the immediate attracting basin at the origin.
We found that if ν > 2, there is a tetrachotomy classification of Julia sets and four types of Julia sets are illustrated in Figure 2 for ν = 4; if ν = 2, there is a trichotomy classification and three types of Julia sets are illustrated in Figure 3.
In the next part of this section, by proving the following proposition we show that for each ν ≥ 2, there exist actual parameter values of λ such that J(f λ (2,ν) ) is a Cantor set; for each ν > 2, there exist actual values of λ such that J(f λ (2,ν) ) is a McMullen necklace; if ν = 2, there is no value of λ such that J(f λ (2,ν) ) is a McMullen necklace.  To prove (1) of Proposition 2.4, we conjugate f λ (2,ν) to g λ (2,ν) (z) = 1 λ (z ν −1) 2 z ν by using the map z → 1 z . For brevity of notation, we prove a corresponding result for the family f µ n (z) = µ (z n −1) 2 z n instead of g λ (2,ν) (z), where µ = 1 λ and n = ν. Clearly, f µ n is a rational map invariant under pre-composition by the elements of the dihedral group; that is, f µ n (z) = f µ n (ω n z) for any n th root ω n of unity and f µ n (z) = f µ n (1/z). It is easy to check that f µ n (z) has 2n + 2 critical points {0, ∞, e πik/n , k = 1, · · · , 2n} and three critical values 0, ∞ and −4µ, it has a superattracting fixed point at ∞ and 0 is mapped to ∞. Thus, v = −4µ is the only critical value depending on µ. In details, the critical points {e 2πik/n , k = 1, · · · , n} of multiplicity 2 are mapped to 0; the critical point 0 of multiplicity n is mapped to ∞; ∞ is a superattracting fixed point f µ n of order n; the critical points {e πi(2k+1)/2n , k = 1, · · · , n} of multiplicity 2 are mapped to v = −4µ.
Let B(∞) be the immediate attracting basin of f µ n at ∞. Now in order to prove (1) of Proposition 2.4, it is equivalent to show v = −4µ ∈ B(∞) if µ is large enough.
It follows Hence, Corollary 2.6. Assume that n ≥ 2. Then for any |µ| > and z ∈ D. By the previous lemma, f µ n maps D into the interior of D. Using the Schwartz Lemma, we conclude that the iterates of z under f µ n converges to the unique fixed point ∞. Hence, z ∈ B(∞). Then D ⊂ B(∞) and v = −4µ ∈ B(∞).
Next, we apply Riemann-Hurwitz formula to prove (3) of Proposition 2.4; that is, we show if ν = 2, then −4a cannot fall into the trap door T even if it exists. Our proof applies the same idea used in ( [3]) to classify the Julia sets of the rational maps in the family (2.1). Let us first recall Riemann-Hurwitz formula. Then there exists an integer d ≥ 1 such that f is a branched covering map from U onto V with degree d and where χ(·) denotes the Euler characteristic and δ f (U ) denotes the total number of the critical points of f in U (counted with multiplicity).
Suppose that T exists and −4a ∈ T . Applying Riemann-Hurwitz formula to the mapf a (2,ν) : B(∞) → B(∞), we can see χ(B(∞)) = 1. This means B(∞) is simply connected. Furthermore, we know T is simply connected. The two critical values 0 and −4a contained in T have their corresponding critical points on two different orbits of a corresponding dihedral group. Using the fact that the dihedral group is generated by two elements fixing two critical points on different orbits respectively, we can show that (f a (2,ν) ) −1 (T ) is connected (this method is also used to prove the first half of Lemma 3.16). Furthermore, by applying Riemann-Hurwitz formula . Then A(0) and A(∞) are two disjoint annuli contained in the annulus A and they share one boundary with A respectively. Therefore,

M odulus(A(0)) + M odulus(A(∞)) < M odulus(A).
On the other hand,f a (2,ν) : This is a contradiction to the previous strict inequality. Therefore, if T is not empty, then −4a / ∈ T . Let G 2 be the symmetry group of a regular polygon with ν sides. Even if ν > 2, one can construct normalized regularly ramified rational maps A • R G2 with a superattracting fixed point p of order 2. For example, let ν be an even positive integer (denoted by 2ν with ν > 1) and take p = 1 and q = −1. After conjugation by the map φ(z) = z−1 z+1 , one of such maps can be expressed as for which 0 is a superattracting fixed point of order 2, ∞ is mapped to 0, and the two critical points of order 4 are mapped to ∞. Therefore, one can consider the following one-parameter family , where λ is a complex parameter. In Figure 4, three types of Julia sets are given for the family Let v λ be the non-zero critical value of h λ (2,2ν) for each λ = 0, where ν > 1. Similarly, we can prove that h λ (2,2ν) has no Herman rings and there is also a trichotomy theorem for the Julia sets of the maps in this family. Theorem 2.8. For each ν > 1 and λ = 0, h λ (2,2ν) has no Herman rings. Theorem 2.9. Let ν > 1. The following trichotomy holds: ) is connected. As pointed out in the introduction, this section is a warm-up section. So we claim the main results for the previous three one-parameter families of the maps of form A • R G2 or A • R G2 • φ without providing proofs. Detailed proofs of similar results for the maps of form A • R G4 or A • R G4 • φ are given in the next sections. The proofs of the main results in this section can be constructed in a straightforward way by using the strategies and techniques presented in the next section.

Statements of results.
In this section, we prove the classification results on the Julia sets of normalized regularly ramified rational maps A • R G4 in some oneparameter families, that are observed in [6] through computer-generated pictures. We continue to use the notation of p and q given in the introduction. By setting the fixed attracting point p at 0 and the point q at ∞, these maps A • R G4 form oneparameter families, which can be divided into two groups according to the order of p. If the order of p is bigger than 2, then the classification of Julia sets is a tetrachotomy; if the order of p is equal to 2, then the classification is a trichotomy.
For example, if the order of 0 is 4 and the order of the critical points mapped to ∞ is 2, then this family is given by where λ is a complex parameter.
Recall that f λ is invariant under pre-composition by any element of the symmetry group G 4 of a regular octahedron. Given a fixed point x of some (non-identity) element of G 4 , the stabilizer G 4 (x) is the collection of all elements of G 4 fixing x.
In this case, all elements of G 4 (x) share the same rotation axis. Note that the Julia set J(f λ ) is symmetric with respect to the axis of any element of G 4 (x); if x belongs to a Fatou component Ω of f λ then Ω is symmetric with respect to G 4 (x). Now we consider that the octahedron is embedded on the Riemann sphereĈ with six vertices at 0, ∞, e iπk/2 , k = 0, 1, 2, 3, and with edges and faces onĈ. Then the 6 vertices are the critical points of f λ of order 4 which are mapped to 0; the 12 middle points of the edges e iπ(2k+1)/4 , (1 ± √ 2)e iπk/2 , k = 0, 1, 2, 3, are the critical points of order 2 which are mapped to ∞; the 8 centers of the faces (1± e iπk/2 , k = 0, 1, 2, 3, are the critical points of order 3 which are mapped to the free critical value v λ . If we denote by Crit(f λ ) the set of critical points of f λ , then : k = 0, 1, 2, 3}, If z ∈ C(k), then the local degree of f λ is k, k = 2, 3, 4, and where v λ = −λ/108. The critical values of f λ are 0, ∞ and v λ = −λ/108, and 0 is a super-attracting fixed point. Define the attracting basin of f λ at 0 as and B(∞) be the connected Fatou components of f λ containing 0 and ∞ respectively. In fact, B(0) is called the immediate attracting basin of f λ at 0. It is possible that In this section, we first prove the following two main theorems for f λ .    Remark 3.3. Patterns of connectivity numbers of Fatou components of rational maps are investigated in [1] and [11]. We have not found the pattern of connectivity numbers of the Fatou components of f λ = f λ (2,3,4) in the literature. In the second half of this section, we consider the case when the order of the fixed critical point p is 2. One example of such a one-parameter family was given in [6], for which p is set at ∞. Precisely, it is defined there as follows. Let is a one-parameter family of normalized regularly ramified rational maps of form A • R G4 • φ, for which 0 and ∞ are critical points of order 2 and ∞ is fixed. We briefly denote by h λ = h λ (2,3,4) , and let B(∞) denote the immediate attracting basin of h λ at ∞ and T the connected and h λ has the same three critical values as f λ . Now we state the classification theorem for the family h λ . The main difference of this case is that the free critical value v λ can never enter the trap door T if it is not empty. Thus the Julia set J(f λ ) cannot be an exploded McMullen necklace. In the last part of this section, we prove the following two theorems for h λ .
Theorem 3.4. Each rational map h λ has no Herman rings.
Theorem 3.5. For the maps in the family h λ with λ = 0, the following trichotomy holds: Before we present the proof, let us first have an agreement on notation concerning the finite Kleinian group related to f λ and h λ . Let G (2,3,4) be the group of the symmetries of the regular octahedron with vertices at the points in C(4) and Through the rest of this section, we will use G (2,3,4) to denote either G (2,3,4) or G ′ (2,3,4) ; that is, if the involved map is f λ , then G (2,3,4) is G (2,3,4) ; but if the involved map is h λ , then G (2,3,4) is G ′ (2,3,4) . Similarly, for n = 2, 3 or 4, we use C(n) to denote either C(n) or φ −1 (C(n)); that is, if the involved map is f λ , then C(n) is C(n); but if the involved map is h λ , then C(n) is φ −1 (C(n)).
Proof. Suppose that f has a cycle of Herman rings {R 0 , R 1 , · · ·, R p−1 }. Then f p is conjugate to the irrational rotation z → λz on R 0 , where λ = exp(2πiα) and α is an irrational number. For each 0 ≤ k ≤ p − 1, f : R k → R k+1 is a conformal mapping. Take a Jordan curve γ in R 0 invariant under f p which separates two connected components of ∂R 0 . Let U be the bounded connected component of C \ γ. Then U ∩ J = ∅. Since γ m ≡ f m (γ) ⊂ R k , where k = m mod p, and R k is disjoint from B(0) and B(∞) for all k, we have {f m (γ)} ∞ m=0 is uniformly bounded. Let U k denote the bounded component ofĈ \ γ k for k = 0, 1, · · · , p − 1 and U p = U 0 .
We first show that U k cannot contain any pole of f . Note that C(2) is the set of all poles of f . So we show U k ∩ C(2) = ∅ for each k = 0, 1, · · · , p − 1. On the contrary, if this is not true, then some U k contains at least an element y ∈ C (2). Note that f injectively maps the boundary curve γ k of U k onto the curve γ k+1 , and the winding number of γ k+1 with respect to the origin is 0, 1 or −1. Let N (f, U k ) denote the number of zeros of f in U k and P (f, U k ) denote the number of poles of f in U k . By the argument principle, we obtain Clearly, C(4) is the set of all zeros of f . Set k j = ♯(U k ∩ C(j)) for j = 2, 4. We obtain that N (f, U k ) = 4k 4 , P (f, U k ) = 2k 2 and 4k 4 − 2k 2 = 1, 0 or −1.
is conformal, all zeros and poles in U k belong to V k . By the assumption k 2 = 0, we know k 4 = 0 and U k contains at least one point x ∈ C(4) not equal to ∞, that is, k 4 ≥ 1. Next, we show k 4 has to be equal to 6 by eliminating other choices of k 4 , for which we use the following observation.
Observation. Let g ∈ G (2,3,4) . If γ k separates the fixed points of g, then g(γ k ) = γ k and hence g(U k ) = U k . Now let g be an element of order 4 with one fixed point at x. If k 4 = 1, then the other fixed point of g is not in U k , and hence g(U k ) = U k by the above observation. Then the orbit of y under g belongs to U k . It follows k 2 ≥ 4 and k 4 ≥ 2. Let x 1 and x 2 denote two points in C(4) ∩ U k . There are two cases to consider depending on whether or not x 1 and x 2 are fixed points of the same element g of G (2,3,4) . Case 1. Assume x 1 and x 2 are fixed points of two different elements of G (2,3,4) . Let g be an element of G (2,3,4) of order 4 with a fixed point at x 1 . Using the observation, we conclude the orbit of x 2 under g is contained in U k . Hence, k 4 ≥ 5. If k 4 = 5, then k 2 = 10 and there exits g ∈ G (2,3,4) such that only one fixed point of g belongs to U k . By the observation, the orbits of points of U k ∩ C(2) are contained in U k and hence k 2 is divisible by 4. Thus, k 2 = 12 and hence k 4 = 6.
Case 2. Assume x 1 and x 2 are fixed points of the same element g of G (2,3,4) of order 4. Note that U k is a simply connected domain and both g 2 (U k ) and U k contain x 1 and x 2 . Hence, g 2 (U k ) ∪ U k is a connected domain and it is also invariant under g 2 . Then this union is either a connected domain with connectivity number at least 2 or it is the whole Riemann sphere. In the former situation, the boundaries of g 2 (U k ) and U k intersect and the intersecting points are critical points of f . So this situation cannot happen since γ k ⊂ R k and f is univalent on R k . Therefore, the union g 2 (U k ) ∪ U k is the Riemann sphere. Since U k and g 2 (U k ) contain the same number of the critical points in C(4), U k has to contain at least two more points of C(4) besides x 1 and x 2 , which are denoted by x 3 and x 4 . Furthermore, we show it is impossible to have the situation in which k 4 = 4 and x 3 and x 4 are the fixed points of the same element g ′ of G (2,3,4) of order 4. Otherwise, the complement W k of U k contains exactly two points of U k ∩ C(4) that are the fixed points of the same element g ′′ of G (2,3,4) of order 4, denoted by x 5 and x 6 . Using the above argument, we conclude that W k contains two more elements of U k ∩ C(4) besides x 5 and x 6 . This is impossible. Hence, either k 4 ≥ 5 or k 4 = 4 with x 3 and x 4 being fixed points of two different elements of G (2,3,4) of order 4. If k 4 ≥ 5, then we can show k 4 = 6 by using the same argument at the end of the previous paragraph. If the latter situation happens, we can conclude k 4 = 6 by applying the observation to the elements of order 4 with one fixed point at x 3 and x 4 respectively.
In summary, we have shown k 4 = 6 and then k 2 = 12, which is impossible since ∞ / ∈ U k . This means U k contains no pole of f for all k = 0, 1, · · · , p − 1. Then f : U k → U k+1 is holomorphic for k = 0, 1, · · · , p − 1. Since f maps γ k onto γ k+1 injectively, f : U k → U k+1 is in fact conformal by using the argument principle. Thus, {f n } ∞ n=1 is a normal family on U 0 , which contradicts U 0 ∩ J = ∅. Therefore, we conclude that f has no Herman rings in its Fatou set.
3.3. Proof of Theorem 3.2. This is a long subsection. We first prepare some background and lemmas before giving a proof to Theorem 3.2.
Let B(0) be the immediate attracting basin of f λ at the origin. We use M to denote the connected component of C\B(0) containing ∞ if exists. In the following, we first mention a symmetric property of M . This set is studied in Proposition 3.18, which contains some main ingredients to prove Theorem 3.2.
Let U be a subset of C and λ ∈ C. We denote by λU := {λz : z ∈ U } and by ∂U the boundary of U , and denote by ♯(U ) the cardinality of U . Recall that f λ is invariant under precomposition by any element g of G (2,3,4) . In particular, for any ω ∈ {i k : k = 1, 2, 3, 4}, the map z → ωz belongs to G (2,3,4) ; the map h : z → 1 z also belongs to G (2,3,4) . Therefore, the following lemma holds. (3) Let U be a Fatou component of f λ . Then either iU = U (for which we say that U has a 4-order rotation symmetry) or U , iU , i 2 U and i 3 U are pairwise disjoint.
The following lemma is given in [15].

Lemma 3.7 ([15, Lemma 2.9]). If a rational function f has no Herman rings and each Fatou component contains at most one critical value, then the Julia set of f is connected.
Obviously, if a simply connected domain U ⊂ C * = C \ {0} does not contain any critical value, then f −1 λ (U ) consists of exactly 24 connected components and each of them is simply connected. For the case that U contains the critical value v λ , we have the following lemma.
Hence V is a proper sub-domain of C, which and χ(V ) < 2. It follows that χ(V ) = 1 and k 3 = 1. This means V is simply connected and contains only one point of C (3). Therefore, f −1 λ (U ) consists of 8 simply connected components.  Proof. Suppose B(0) = B(∞); that is, ∞ ∈ B(0). Since 0 is a super-attracting fixed point, we can choose a small simply connected neighborhood Ω 0 of 0 such that ∞ ∈ Ω 0 , f λ (Ω 0 ) ⊂ Ω 0 and ∂Ω 0 is a Jordan curve. For m ≥ 0, let Ω m be the connected component of f −m λ (Ω 0 ) containing Ω 0 . Then we have Ω 0 ⊂ Ω 1 ⊂ Ω 2 ⊂ · · · ⊂ Ω m ⊂ · · · and B(0) = m≥0 Ω m . Since ∞ ∈ B(0), there must exist m 0 ≥ 1 such that ∞ ∈ Ω m0 \Ω m0−1 . By Lemma 2.7, it follows that Ω m0−1 is simply connected. Now, we consider the branching covering map f λ : Ω m0 → Ω m0−1 . Set k 4 = ♯(Ω m0 ∩ C(4)) ≥ 2. Then by Lemma 2.7, we obtain Then χ(Ω m0 ) = k 4 ≥ 2, which is impossible since χ(Ω m0 ) is at most equal to 1. Therefore, ∞ / ∈ B(0). It follows that each Ω m is simply connected and hence B(0) = m≥0 Ω m is simply connected. Using Lemma 2.7 again, we obtain that B(∞) is simply connected too. Now we start to prove Theorem 3.2. The proof is quite long since there are four cases to consider. Before doing that, let us recall one more known result.  In the second part of this subsection, we prove that if v λ ∈ A(0) \ (B(0) ∪ B(∞)), then the Julia set is a Sierpinski curve. Recall that by a Sierpinski curve we mean a planar set homeomorphic to the well-known Sierpinski carpet fractal. In [12], Whyburn given a topological characterization of the set. Theorem 3.13 ([12, Theorem 3]). Any non-empty planar set that is compact, connected, locally connected, nowhere dense, and has the property that any two complementary domains are bounded by disjoint simple closed curves is homeomorphic to the Sierpinski carpet.
The previous characterization of a Sierpinski curve is used in [3] to prove Julia sets to be such curves for maps in the family f λ (z) = z n + λ/z m . We use it in this paper too. In the following, we first recall a result on local connectivity of Julia sets. By a hyperbolic rational map we mean a rational map with every critical point approaching an attracting periodic cycle under iteration.

Lemma 3.14 ([8]). (1) If the Julia set of a hyperbolic rational map is connected, then it is locally connected.
(2) If U is a simply connected Fatou component of a hyperbolic rational map, then the boundary ∂U is locally connected.  For a given angle θ ∈ [0, 1), by definition, the internal ray in B(0) with angle θ is γ(θ) = {z ∈ B(0) : φ(z) = re 2πiθ , r < 1}. Obviously, f λ (γ(θ)) = γ(4θ). If γ(θ) → z ∈ ∂B(0) as r → 1, we say that the internal ray γ(θ) lands at z, and z is a landing point of γ(θ). Since f λ is hyperbolic, ∂B(0) is locally connected by Lemma 3.10 and Lemma 3.14.  Since v λ ∈ B(∞), there is a point z 1 of C(3) and a point z 2 of C(2) contained in U . Let g 1 and g 2 be non-identity elements of G 4 with fixed point at z 1 and z 2 respectively. Then G 4 is generated by g 1 and g 2 . For any other point z ′ 1 on the orbit of z 1 under the group G 4 , there is a finite product g k1 g k2 · · · g kj of g 1 and g 2 such that g k1 g k2 · · · g kj (z 1 ) = z ′ 1 . Clearly, U is invariant under g 1 and g 2 ; that is, g 1 (U ) = U and g 2 (U ) = U . This implies that g k1 g k2 · · · g kj (U ) = U and hence z ′ 1 ∈ U . Thus, C(3) ⊂ U . Similarly, we can show C(2) ⊂ U . Now it is clear that f λ : U → B(∞) is a holomorphic map of degree 24. Therefore, f −1 λ (B(∞)) has only one component, which means it is connected. Applying Riemann-Hurwitz formula, we obtain that ∂f −1 λ (B(∞)) has 6 connected components, which means its connectivity number is 6.
A rational map f is said to be subhyperbolic if every critical point in the Julia set is preperiodic and every critical point in the Fatou set is attracted to an attracting cycle. Morosawa established in [10] a sufficient condition for boundary of a Fatou component to be a Jordan curve.    0)). Without loss of generality, we may assume ∂V ⊂ ∂B(0). Then by the order-4 rotation symmetry of ∂B(0), we obtain ∂(i k V ) ⊂ ∂B(0) for each k = 0, 1, 2, 3. Other components of f −1 λ (U ) are expressed by h(i k V ), k = 0, 1, 2, 3, where h : z → 1 z . Note that h(∂B(0)) = ∂B(∞) and hence ∂h(i k V ) ⊂ ∂B(∞) for each k = 0, 1, 2, 3. Now we take a point z ∈ ∂U and consider the number of the preimages of z. Clearly, f λ : V → U is a branched covering with degree 3 and it defines analytically on a neighborhood ofV . We also know ∂U is locally connected since it is a connected subset of a connected and locally connected set ∂B(0) without any interior point. Using the fact that there is no critical value on ∂U , we can conclude that z also has exactly three preimages on ∂V . Similarly, z has three preimages on the boundary of each component of f −1 λ (U ). Therefore, z has 24 preimages on ∪ 3 k=0 (∂i k V ∪ ∂h(i k V )) ⊂ ∂B(0) ∪ ∂B(∞). Now we consider z as a point on ∂B(0) and use f λ : B(i k ) → B(0) for each k = 0, 1, 2, 3. Applying a similar argument, we know z has 4 preimages on ∂B(i k ) for each k = 0, 1, 2, 3. Since the boundaries of the components of f −1 λ (B(0)) are pairwise disjoint, it follows that z has 16 more distinct preimages on ∪ 3 k=0 ∂B(i k ). All together, z has at least 40 distinct preimages. This is impossible since the degree of f λ is 24. Thus, we conclude that v λ ∈ M .
(2) We have proved in (1) that v λ ∈ M . There are two cases to consider based on whether or not v λ ∈ B(∞).
Then it is contained in another component of C \ B(0), denoted by U . It follows that ∂U ⊂ ∂B(0) and ∂h(U ) ⊂ ∂B(∞), On the other hand, we also know h(f −1 Now we take a point z ∈ ∂B(v λ ) and a point z * ∈ ∂M . By Lemma 3.10 and Lemma 3.14, ∂B(0) is locally connected. Then ∂M is also locally connected. Thus, there is a curve γ in M connecting z and z * such that γ ∩ f −1 λ (B(0)) = {z * }. For each j = 0, 1, 2, 3, f λ : i j D → B(v λ ) is a branched covering map of degree 3 and hence there are 3 preimages of z on ∂i j D. We consider the preimages of γ intersecting i j D. They are three curves emanating from the critical point in i j D and landing at three points on the boundary of V j . The landing points are distinct (otherwise one of them becomes a critical point on ∂V j ). Thus, z * has 3 distinct preimages on ∂V j for each j = 0, 1, 2, 3. Hence, z * have 24 distinct preimages on the boundaries of V j 's and h(V j )'s, which are contained in ∂B(0) ∪ ∂B(∞). On the other hand, by considering z * as a point on the boundary of B(0), we know that it has 4 distinct preimages on ∂B(i k ) for each k = 1, 2, 3, 4. All together, we have found 40 distinct preimages for z * . This is impossible since the degree of f λ is 24. Therefore, the assumption that there is a component D of  In the next part of this subsection, we consider the third case of the tetrachotomy given in Theorem 3.2, that is when B(∞) = B(0) and v λ ∈ B(∞). In this case, B(∞) is called the trap door for f λ , which is often denoted by T in the papers by Devaney and his collaborators.
Let γ ⊂ C be a Jordan curve and denote by D(γ) the bounded component of C \ γ. Given two Jordan curves γ 1 and γ 2 with γ 1 ⊂ D(γ 2 ), denote by A(γ 1 , γ 2 ) the closed annulus bounded by γ 1 , γ 2 . We use D(0) and D(∞) to denote the connected components of C \ f  (2) Under the assumption, it is clear that B(i k ) is simply connected and then each component of f −j λ (B(i k )) is simply connected, where k = 1, 2, 3, 4 and j is a positive integer.
The classification of the Julia sets of the maps in the family h λ is similar to the one of the maps in the family f λ with one exception: absence of (exploded) McMullen necklace. The main reason is the following proposition. 4. The tetrahedral and icosahedral cases. In this last section, we consider oneparameter families of normalized regularly ramified rational maps of form A • R Gj or A • R Gj • φ for j = 3 or j = 5, where A and φ are Möbius transformations. We first consider such maps related to G 5 since they have been explored in [6] through computer generated pictures. Some one-parameter families of rational maps related to G 3 are also explored in [6], but those maps don't satisfy the two assumptions for the regularly ramified rational maps investigated in this paper. Therefore, we need to construct the ones of form A • R G3 or A • R G3 • φ satisfying the two assumptions, which we carry out in the second half of this section. The following one-parameter family of normalized regularly ramified rational maps A • R G5 is given in [6]: f λ (2,3,5) (z) = λz 5 (z 10 + 1) 2 j=0,1 where λ is a complex parameter. All points of C(5) are critical points of f λ (2,3,5) of order 5 and are mapped to 0; all points of C(2) are critical points of order 2 and are mapped to ∞; all points of C(3) are critical points of order 3 and are mapped to the same non-zero value, denoted by v λ . Clearly, 0 is fixed by f λ (2,3,5) and ∞ is mapped to 0. Similar to f λ (2,3,4) , we define B(0) and B(∞) for f λ (2,3,5) . Julia sets of the rational maps in this family are explored through computer-generated pictures in [6]. Using the same strategies, but modified details, to prove Theorems 3.1 and 3.2, we obtain corresponding results for the maps in the family f λ (2,3,5) . Four types of Julia set for the maps in the family f λ (2,3,5) are illustrated in Figure 8.  Each rational map f λ (2,3,5) has no Herman rings.
Theorem 4.2. For the maps in the family f λ (2,3,5) with λ = 0, the following tetrachotomy holds: each connected component of (f λ (2,3,5) ) −m (B(w)) is simply connected; (iii) for each m ≥ 1, each connected component of (f λ (2,3,5) ) −m (B(∞)) is multiply connected with connectivity number 2(5 m ) + 2; (iv) every simply connected Fatou component B of f λ (2,3,5) is surrounded by a Cantor set of Jordan curve components of J(f λ (2,3,5) ). 4. If v λ / ∈ A(0), then the Julia set J(f λ (2,3,5) ) is connected. Now we consider another type of one-parameter family of normalized regularly ramified rational maps of form A • R G5 • φ similar to the one considered for the octahedral case in the previous section. These rational maps fix one endpoint p of a rotation axis of an element of G 5 of order 2, map the antipodal point q of p to p, and map all fixed points of rotation axes of elements of G 5 of order 5 to q. Through conjugation by a Möbius transformation, we may assume that p is arranged at ∞ and q at 0. There is a short cut to construct such rational maps, which goes as follows. Let φ(z) = iz + 1 z + i , and define h λ (2,3,5) (z) = λf 1 (2,3,5) (φ(z)). Then h λ (2,3,5) is a family of normalized regularly ramified rational maps of form A • R G5 • φ, where A is a Möbius transformation. All points in φ −1 (C(2)), including 0 and ∞, are critical points of h λ (2,3,5) of order 2 and are mapped to ∞; all points in φ −1 (C(5)) are critical points of order 5 and are mapped to 0; all points in φ −1 (C(3)) are critical points of order 3 and are mapped to the same value, denoted by v λ again. Julia sets of the maps in this family are explored in [6] and a trichotomy is observed there. Similar to h λ (2,3,4) , we define A(∞) and B(∞) for h λ (2,3,5) . Using the same strategies, but modified details, to prove Theorems 3.4 and 3.5 for h λ (2,3,4) , we draw similar conclusions for h λ (2,3,5) . Three types of Julia set for the maps in h λ (2,3,5) are illustrated in Figure 9. Theorem 4.4. For the maps in the family h λ (2,3,5) with λ = 0, the following trichotomy holds: 1.

2.
If v λ ∈ A(∞) \ B(∞), then the Julia set J(h λ (2,3,5) ) is a Sierpinski curve. 3. If v λ / ∈ A(∞), then the Julia set J(h λ (2,3,5) ) is connected. 4.2. The tetrahedral case. As the last part of this paper, we consider oneparameter families of normalized regularly ramified rational maps of form A • R G3 or A • R G3 • φ. In [6], a regular tetrahedron is imbedded on a Riemann Sphere with 3 vertices at points in the set and middle points of 6 edges at points in the set