Escape quartered theorem and the connectivity of the Julia sets of a family of rational maps

In this paper, we investigate the dynamics of the following family of rational maps \begin{document}$ \begin{equation*} f_{\lambda}(z) = \frac{z^{2n} - \lambda^{3n+1}}{z^n(z^{2n} - \lambda^{n - 1})} \end{equation*} $\end{document} with one parameter \begin{document}$ \lambda \in \mathbb{C}^* - \{\lambda: \lambda^{2n + 2} = 1\} $\end{document} , where \begin{document}$ n\geq 2 $\end{document} . This family of rational maps is viewed as a singular perturbation of the bi-critical map \begin{document}$ P_{-n}(z) = z^{-n} $\end{document} if \begin{document}$ \lambda \neq 0 $\end{document} is small. It is proved that the Julia set \begin{document}$ J(f_\lambda) $\end{document} is either a quasicircle, a Cantor set of circles, a Sierpinski carpet or a degenerate Sierpinski carpet provided the free critical orbits of \begin{document}$ f_\lambda $\end{document} are attracted by the super-attracting cycle \begin{document}$ 0\leftrightarrow\infty $\end{document} . Furthermore, we prove that there exists suitable \begin{document}$ \lambda $\end{document} such that \begin{document}$ J(f_\lambda) $\end{document} is a Cantor set of circles but the dynamics of \begin{document}$ f_{\lambda} $\end{document} on \begin{document}$ J(f_{\lambda}) $\end{document} is not topologically conjugate to that of any known rational maps with only one or two free critical orbits (including McMullen maps and the generalized McMullen maps). The connectivity of \begin{document}$ J(f_{\lambda}) $\end{document} is also proved if the free critical orbits are not attracted by the cycle \begin{document}$ 0\leftrightarrow\infty $\end{document} . Finally we give an estimate of the Hausdorff dimension of the Julia set of \begin{document}$ f_\lambda $\end{document} in some special cases.


1.
Introduction. Let f be a rational map with degree d ≥ 2 on the Riemann sphere C = C ∪ {∞}, where C is the complex plane. Let f •k be the k-th iteration of f , where k ∈ N. The Julia set of f , denoted by J(f ), is defined as the set of points at which the family of iterates {f •k : k ∈ N} fails to be a normal family in the sense of Montel. The complement C \ J(f ) of the Julia set is the Fatou set of f and is denoted by F (f ). It is easy to see that F 1.1. Background and motivation. The topology of the Julia sets of rational maps, such as the connectivity and local connectivity, is an interesting and important research subject in complex dynamics. The dynamics of the unicritical polynomial P n (z) = z n (n ≥ 2) is simple. Naturally, one wishes to study the dynamics of the rational maps nearby P n by a small perturbation on P n . McMullen is the first one who investigated the small singular perturbation on P n (z) (see [17]) and he considered the following family of rational maps F λ (z) = z n + λ/z m , where n ≥ 2, m ≥ 1 and λ ∈ C * = C\{0}. This family is referred as McMullen maps later (see [5], [21] for example), which exhibits very rich dynamical behaviors and can be viewed as a singular perturbation of P n (z) acting on C. In 2005, Devaney, Look and Uminsky obtained an Escape Trichotomy Theorem for F λ (z) according to the free critical orbits of F λ (z) [5]. They proved that the Julia set J(F λ ) is either a Cantor set, a Cantor set of circles or a Sierpiński carpet if the free critical orbits are attracted by the infinity.
A Sierpiński carpet is a planar set which is homeomorphic to the standard Sierpiński carpet fractal. A subset S of the Riemann sphere C is a Sierpiński carpet if and only if it has empty interior and can be written as S = C\ k∈N U k , where U k ⊆ C, k ∈ N, are pairwise disjoint Jordan disks with ∂U k ∩ ∂U j = ∅ for k = j, and diam(U k ) → 0 as k → ∞. Furthermore, we call a compact set S in C degenerate Sierpiński carpet if it satisfies all the conditions of the Sierpiński carpet except it allows that the intersection of the boundaries of complementary domains can be non-empty. In addition, a subset of the Riemann sphere C is called a Cantor set of circles (or Cantor circles in short) if it consists of uncountably many closed Jordan curves which is homeomorphic to C × S 1 , where C is the Cantor middle third set and S 1 is the unit circle.
For Cantor circle Julia sets, the first example is due to McMullen ( [17]). He proved that the Julia set of the rational map F λ (z) = z 2 + λ/z 3 is a Cantor set of circles if λ = 0 is small enough. As a variation of F λ (z), the generalized McMullen maps F λ,η (z) = z n + λ/z m + η are also investigated by many people. There exists some additional dynamical phenomenon for the family F λ,η (z). We refer the reader to [6], [13], [30], [12] and the references therein. It is known that the Cantor circle Julia sets and Sierpiński carpet Julia sets can appear in McMullen family and the generalized McMullen family (see [5,30]). For the study of singular perturbation of unicritical polynomials, one may also refer to [27], [31], [32], [15] and [16].
It is not difficult to show that there are some generalized McMullen maps F λ,η with their Julia sets being Cantor circles and the dynamics of F λ,η in a neighborhood of the Julia set is quasiconformally conjugate to that of some McMullen maps (the corresponding Julia sets are also Cantor circles). However, by quasiconformal surgery, Haïssinsky and Pilgrim [14] proved that there are some other types of rational maps whose Julia sets are Cantor circles but the dynamics on the Julia sets are not topologically conjugate to any McMullen map on their corresponding Julia sets. Later, the specific expressions of these rational maps were given in [23]. Recently Fu and Yang [9] studied the following one-dimensional family of rational maps where n ≥ 2 and λ ∈ C * − {λ : λ 2n−2 = 1}. They proved that the Julia sets of some maps in this family are Cantor circles and each of these maps has essentially only one free critical orbit. Moreover, its dynamics on the Julia set are not topologically conjugate to any McMullen map on their corresponding Julia sets. We are interested in the problem to find a one-dimensional family of such rational maps, such that the Julia sets of some maps in this family are Cantor circles but its dynamics is neither topologically conjugate to any McMullen map nor the rational map h λ on their corresponding Julia sets but for each map, there exists essentially only one free critical orbit. For this, we consider the following family of rational maps where n ≥ 2 and λ ∈ Λ := C * − {λ : λ 2n+2 = 1}. The rational map f λ degenerates to the bi-critical rational map P −n (z) = z −n if λ = 0 or λ 2n+2 = 1. Thus the map f λ with λ ∈ Λ can be viewed as a perturbation of the simple map P −n if λ = 0 lies in a small punctured neighborhood of the origin. This perturbation is essentially different from that of McMullen maps and the family h λ since f λ keeps the dynamics of P −n near the orbit 0 ↔ ∞ of f λ with period two.
1.2. Statement of the main results. A straightforward computation shows that f λ has a super-attracting periodic orbit 0 ↔ ∞. We denote by D 0 and D ∞ , respectively, the immediate attracting basins of 0 and ∞. One can observe easily that D 0 is always different from D ∞ since a Fatou component cannot contain two different periodic points. The map f λ has 6n − 2 critical points (counted with multiplicity) since the degree of f λ is 3n. Note that the local degrees of 0 and ∞ are both n, and there are 4n critical points other than 0 and ∞, which are called the free critical points (for the definition of free critical point, see Section 2). The forward orbits of 0 and ∞ are trivial since they lie in the super-attracting orbit 0 ↔ ∞. We will show in Section 2 that the remaining 4n critical points behave symmetrically. So we just have essentially one free critical orbit for each f λ . The dynamics of f λ are determined by the forward orbit of any free critical point c λ . Under the assumption that the free critical orbits are attracted by the cycle 0 ↔ ∞, we obtain the following main result.
Theorem 1.1. Suppose that the free critical point c λ is attracted by 0 or ∞. Then there are following four cases.
Otherwise, the free critical orbits are not attracted by the cycle 0 ↔ ∞ and J(f λ ) is connected.
The four cases in Theorem 1.1 happen indeed, see Figure 1 for examples (see also Section 5). We call Theorem 1.1 the Escape Quartered Theorem. As one comparison between McMullen maps and our family, the Julia set of a McMullen map cannot be a quasicircle nor a degenerate Sierpiński carpet if the free critical orbits are attracted by the cycle 0 ↔ ∞. As another comparison between h λ and f λ , the Fatou components of f λ containing the origin and the infinity are of period two, which is different from the family h λ .
For the connectivity of the Julia sets under the assumption that the free critical orbits are bounded, the case of McMullen maps has been studied in [29] and [7], and the case of h λ is considered in [10].
As a consequence of Theorem 1.1, we obtain the following result.
Theorem 1.2. Suppose that one of the free critical orbits of f λ is attracted by the cycle 0 ↔ ∞. Then (a) The boundary components of all Fatou components of f λ are quasicircles; For the Julia set of f λ being a Cantor set of circles, one can prove that there exists a punctured region of λ = 0 in which the Julia set of f λ is a Cantor set of circles (see Section 5). This punctured region is called the McMullen domain (see Figure 2). As it was mentioned above, our primary motivation of studying the family (2) is to obtain a one-dimensional family of rational maps with Cantor circle Julia sets which are neither topologically conjugate to any McMullen map nor to the above family h λ on their corresponding Julia sets. The following result shows that f λ is the desired family.
1.3. Organization of the paper. This paper is organized as follows: In Section 2 we give some preliminary results on the rational map f λ , including the analysis of two kinds of symmetries and the dynamical properties of f λ . We also give the proof of Theorem 1.1 (a). In Section 4 we deal with the case that all the free critical orbits are not attracted by the cycle 0 ↔ ∞ and give the proof of the connectivity of the Julia set of f λ , which completes the proof of Theorem 1.1.
In Section 5 we study the dynamical properties of f λ when λ = 0 is real or small. Under these settings we give specific examples such that the Julia set of f λ is a quasicircle or a Cantor set of circles.

Preliminaries and basic settings.
2.1. The symmetric property. In this paper, we will need some preliminary properties of f λ , including the symmetric distribution of critical points and the symmetric dynamical behaviors. In the following, we assume that n ≥ 2 is an integer if not special specified. Recall that D ∞ (respectively, D 0 ) is the immediate super-attracting basin of ∞ (respectively, 0) and f λ has a super-attracting periodic orbit 0 ↔ ∞. Let U ⊂ C be a set and a ∈ C \ {0}, we denote aU = {az : z ∈ C}. .

Proof. (a) A direct calculation shows that
By induction it is easy to see that (c) The proof of this part is straightforward and we omit the details.
From Lemma 2.1, one knows that all the orbits of points with the form ω j z (j = 0, 1, 2, · · · , 2n−1) behave "symmetrically" under f λ . For example, on the one hand, f •k λ (z) tends to origin (or infinity) as k tends to infinity if and only if f •k λ (ω j z) tends to origin (or infinity) as k tends to infinity. On the other hand, f •k λ (z) tends to origin (or infinity) as k tends to infinity if and only if f •k λ (λ 2 /z) tends to infinity (or origin) as k tends to infinity. We thus obtain the result as follows.
is also (here U is allowed to be equal to τ (U )). In particular, τ where ω satisfies ω 2n = 1 and ω j0 = 1. Then ω j z 0 ∈ U for each integer j. In particular, U has 2n-fold symmetry and winds arround the origin.
Remark. The proof of Corollary 2.2 is straightforward and one can refer to [5, Lemma 1.3] for a similar proof. Suppose that z 0 lies in a Fatou component U , while ω j z 0 does not lie in this component for some j. Corollary 2.2 implies that there are 2n such Fatou components which surround the origin (or infinity) with 2n-fold symmetry.

The quasicircle case. A direct computation implies that
Obviously 0 is a critical point of f λ with multiplicity n − 1. Lemma 2.1 implies that ∞ is also a critical point of f λ with multiplicity n − 1. If c λ is one of the critical points other than 0 and ∞, then the resting 4n critical points (including c λ ) of f λ have the following form: where ω 2n = 1. These 4n critical points are called the free critical points. It follows from Lemma 2.1 again that the properties of the Julia set of f λ depend only on one of the free critical orbits. Proof. Suppose that the free critical point c λ ∈ D 0 . Then Lemma 2.1 shows that λ 2 /c λ ∈ D ∞ . In viewed of Lemma 2.1 again, one gets that ω j c λ ∈ D 0 and ω j λ In fact, let d be the degree of the restriction of f λ on D ∞ . From the assumption that f λ : D ∞ → D 0 is proper and the local degree of f λ at ∞ is n, it follows that d ≥ n. If 0 has preimages other than ∞ in D ∞ , one must deduce from Lemma 2.1 that d ≥ 3n. It is easily obtained that d = 3n since the degree of f λ is 3n. Hence d = n or d = 3n.
In the following we shall show that d = n is impossible. Assume by contradiction that d = n. By the argument above it follows from c λ ∈ D 0 that the 2n − 1 free critical points {ω j c λ : 1 ≤ j ≤ 2n − 1} also lie in D 0 . From the definition of f λ (see (2)), one can conclude that f λ (c λ ) has at least 2n preimages {ω j c λ : 0 ≤ j ≤ 2n − 1} contained in D 0 (counted with multiplicity). Therefore d ≥ 2n, reaching a contradiction with the assumption d = n. It then follows that d = 3n and D ∞ is the unique component of f −1 λ (D 0 ). By the similar argument as before, we can prove that D 0 is also the unique component of f −1 λ (D ∞ ). Note that all the critical points are attracted by 0 and ∞. We claim that there are no Fatou components other than D 0 and D ∞ for f λ . Indeed, firstly if there were either (super-)attracting periodic basins or parabolic periodic basins which are different from D 0 and D ∞ , then those periodic basins should contain at least one critical point, which is impossible. Finally, if there were either Herman rings or Siegel disks, then J(f λ ) would contain at least one critical value. Clearly this is also impossible since all the critical points lie in D 0 and D ∞ . Thus f λ has only two Fatou components D 0 , D ∞ and f λ is hyperbolic. According to [3, p.102], J(f λ ) is a quasicircle. This ends the proof of Lemma 2.3 and hence Theorem 1.1(a).
2.3. Some dynamical properties in the non-quasicircle case. In the following we investigate the case that the free critical points fail to lie in D 0 and D ∞ .
We then invoke Corollary 2.2 (2n-fold symmetry of Fatou component) to obtain that A ∞ has only one or 2n components. Suppose that A ∞ has 2n components, then 0 has 5n preimages (counted with multiplicity) since each component of A ∞ is mapped to D 0 by degree 2 and there are 2n such components in this case. This is impossible since the degree of f λ is 3n by the definition. Thus A ∞ consists of one component and it is connected. In view of the Riemann-Hurwitz formula for f λ : A ∞ → D 0 , one can deduce that χ(A ∞ )+2n = 2nχ(D 0 ). Together with Lemma 2.1, this implies that χ(A ∞ ) = 0 and A ∞ is an annulus surrounding the origin with 2n-fold symmetry. The completely similar arguments can also be applied to A 0 . Hence D ∞ is a simply connected domain and A 0 is an annulus surrounding the origin with 2n-fold symmetry.
Recall that the Sierpiński carpet is a planar set which is compact, connected, locally connected, nowhere dense, and has the property that any two complementary domains are bounded by disjoint simple closed curves ( [28]). As a kind of Julia sets, the Sierpiński carpets attract much interest recently. One may refer to [4], [2], [34], [11], [25] and the references therein.
In this subsection, we will consider a sufficient and necessary condition when the Julia set of f λ is a Sierpiński carpet by studying the "escaping times" of the free critical points. By Lemmas 2.3 and 2.4 (see also Theorem 1.1), when J(f λ ) is a quasicircle or a Cantor set of circles, then the free critical points must belong to . Therefore, we should exclude these cases.
In the following we will see that the first preimages of D 0 and D ∞ may not be annuli. We write respectively. In the rest of this subsection, we also assume that one of the free critical points c λ belongs to U k 0 for some k ≥ 2. From the dynamical symmetry stated in Corollary 2.2, it can be deduced that λ 2 /c λ ∈ U k ∞ and f λ is n-to-1 on D 0 and D ∞ . Note that there are no critical points in U 1 0 . The first preimage U 1 0 of D 0 has 2n-symmetric components and it cannot be an annulus. If it is proved that D 0 is a Jordan disk and all components of its preimages are simply connected, then it is easy to show that the Julia set is a Sierpiński carpet.
Lemma 2.5. If one of the free critical points c λ lies in U k 0 (or U k ∞ ) for some k ≥ 2, then all Fatou componets of f λ are simply connected. Furthermore J(f λ ) is compact, connected, locally connected and nowhere dense.
Proof. By the similar argument of Lemma 2.4, one can obtain that the immediate super-attracting basins D 0 and D ∞ are simply connected. Just as the stated above, the preimage Fatou components, all of which are simply connected because f λ maps each one of them onto D 0 conformally. Since c λ ∈ U k 0 for k ≥ 2, each component of U j 0 is simply connected for 1 ≤ j ≤ k−1. Moreover, the components in U j 0 consists of at least 2n such components which are located symmetrically around the origin for each j.
Let V be a simply connected component in U k−1 0 . If V contains no critical orbits, then all the components of the preimages of V are simply connected. If V contains some free critical value and one component P of f −1 λ (V ) is not simply connected, then P contains at least two free critical points. But, in view of the symmetry of Fatou components, there are 2n − 1 distinct Fatou components ω l P , where ω 2n = 1 and 1 ≤ l ≤ 2n − 1, such that there are at least two free critical points in each component of ω l P . This implies that f λ has at least 4n free critical points, which is impossible. Hence, if there is a free critical value being to V , then all components of f −1 λ (V ) are also simply connected. This means that all components in U k 0 are simply connected.
Note that U k 0 contains no critical values. Each component of the preimages of D 0 is simply connected. By the similar argument, each component of the preimages of D ∞ is also simply connected. From the symmetry in Corollary 2.2, it follows that each Fatou component of f λ is simply connected. Then the Julia set J(f λ ) = Since J(f λ ) is not equal to the whole Riemann sphere and has no interior points, it is nowhere dense. By [18,Theorem 19.2], the Julia set is locally connected since f λ is hyperbolic and its Julia set is connected.
3.1. The Cantor circle case. Since the quasicircle case has been considered in the last section, we now first consider the Cantor circle case.
Proof of Theorem 1.1 (b). By Lemma 2.4, one can deduce that A ∞ is contained in the component of C − A 0 containing 0. We claim that the closures of the following sets: D 0 , A ∞ , A 0 and D ∞ , are mutually disjoint. To see this, since f λ (D 0 ) = D ∞ , f λ (D ∞ ) = D 0 and D 0 ∩ D ∞ = ∅, it follows easily that A 0 ∩ A ∞ = ∅. One can see that A 0 and A ∞ are two annuli separating 0 from ∞.
Then it is clear that V is the closed set between D 0 and D ∞ . Suppose that V 1 , V 2 , V 3 are the closed sets between D ∞ and A 0 , A 0 and A ∞ , A ∞ and D 0 , respectively (see the bottom picture of Figure 3).
Taking a Jordan curve γ in A 0 ⊂ V such that it is smooth and surrounds the origin. Again we claim that the preimage of γ in V 3 is a smooth Jordan curve surrounding the origin. Indeed, since none of critical values belongs to V , the preimage of γ in V 3 has finitely many smooth Jordan curves. If one of them, denoted by γ 3 , does not wind around the origin, then int(γ 3 ) is a simply connected domain in V 3 , where int(Γ) (resp. ext(Γ)) denotes the bounded (resp. unbounded) component of C − Γ for a Jordan closed curve Γ ⊂ C. Then it follows that the image int(γ) = int(f λ (γ 3 )) is also a simply connected domain in V . Thus one gets a contradiction because γ ⊂ V surrounds the origin. This implies that the preimages of γ in V 3 are all smooth Jordan curves separating the origin from the infinity. Arranging f −1 λ (γ) such that it has two components in V 3 , then the annular region between these two Jordan curves will contain either zeros or poles, which is impossible. Then f −1 λ (γ) ∩ V 3 is a smooth Jordan curve surrounding the origin. We denote it by η for simplicity. Similarly, one can find a Jordan curve ξ in D ∞ such that ξ = f λ (γ) winds around the origin.
Note that the Jordan disk int(η) is compactly contained in the Jordan disk int(ξ). Then f λ maps int(η) onto ext(γ) properly with degree n and maps ext(γ) onto int(ξ) as covering with degree n. Hence the map f •2 λ : int(η) → int(ξ) is a polynomiallike mapping with degree n 2 . By Douady and Hubbard's Straightening Theorem [8, p.296], f •2 λ : int(η) → int(ξ) is quasiconformally equivalent to the uncritical polynomial P n 2 (z) = z n 2 . It is well known that the Julia set of P n 2 is the unit circle. Hence the boundary ∂D 0 is a quasicircle. Similarly, the boundary ∂D ∞ is also a quasicircle.
Since the super-attracting cycle 0 ↔ ∞ attracts all the critical points of f λ , it means that all the preimages of ∂D 0 and ∂D ∞ are quasicircles. It is evident that V 1 ∪ V 2 ∪ V 3 contains all the preimages of ∂D 0 and ∂D ∞ . Then all of them surround the origin by a similar argument as above.
From the previous argument, we know that each f λ : V i → V is a covering map in n-to-1 fashion (1 ≤ i ≤ 3) (see also the bottom picture of Figure 3).
. Then each component of J(f λ ) has the form J i1i2···i k = k≥1 g i k • · · · • g i2 • g i1 (V ), where i k ∈ {1, 2, 3} for each k ∈ N. By virtue of the construction of J(f λ ), we know that each component J i1i2···i k is a compact set separating 0 from ∞.  Figure 3. The above and below pictures illustrate the mapping relations of h λ (see (1)) and f λ respectively when D 0 contains one of the free critical values but contains no free critical points. One can observe clearly that f λ and h λ are not topologically conjugate on their corresponding Julia sets.
Let L = J ∪ J , where J = J 1,3,1,3,··· and J = J 3,1,3,1,··· ) are the boundaries ∂D ∞ and ∂D 0 respectively. Since V i is contained in V , the identity map id: V i → V is not homotopic to a constant map. We consider the following two cases. Case 2. The Julia component J i1i2···i k ,··· is eventually iterated onto J or J under the map f λ . Then J i1i2···i k ,··· is a quasicircle by the above argument. In either case, J i1i2···i k ,··· is always a Jordan curve.
On the one hand, we have already shown that all the Julia components of f λ are Jordan curves (actually quasicircles). On the other hand, there is an isomorphism between the dynamics on the Julia components J i1i2···i k ,··· and the one-sided shift on the space of 3 symbols Σ 3 = {1, 2, 3} N . Moreover, the Julia set J(f λ ) is homeomorphic to Σ 3 × S 1 , where S 1 := {z : |z| = 1}. This means that J(f λ ) is a Cantor set of circles. This finishes the proof of Theorem 1.1 (b).

3.2.
The carpet case. In this subsection, we consider the case that the free critical points enter into D 0 ∪ D ∞ by at least two iterates. In this case we prove that the Julia set of f λ is either a Sierpiński carpet or a degenerate Sierpiński carpet.
Proof of Theorem 1.1 (c). The proof will be divided into three main steps.
Step 1. All Fatou components of f λ are Jordan disks. To see this, it is sufficient to show that the boundary ∂D 0 is a Jordan curve by Lemma 2.5 and Corollary 2.2. Note that ∂D 0 is connected and locally connected. There are at most countably many Jordan disks in the complement C \ D 0 . Let us assume that W ∞ is the component of C \ D 0 such that it contains the infinity. Then ∂W ∞ is a Jordan curve. In what follows we shall prove that f −1 λ (W ∞ ) ⊂ D 0 . Clearly we have ∞ ∈ D ∞ ⊂ W ∞ . Therefore, we only need to show that f −1 λ (∞) ⊂ D 0 . Applying Lemma 2.1 and Corollary 2.2, the 2n poles f −1 λ (∞) \ {0} belong either to the same Fatou component U 0 surrounding the origin or are contained in 2n distinct Fatou components of f λ . For the first case, we assume that f −1 λ (∞) ⊂ U 0 . It then follows that the Fatou component U 0 must separate D 0 from W ∞ . This is a contradiction since ∂W ∞ ⊂ ∂D 0 . For the second case, there are 2n distinct components of C \ (D 0 ∪ W ∞ ). We denote them by V 0 , V 1 , · · · , V 2n−1 . Then each of these 2n distinct components contains at least one Fatou component.
It is easy to see that each point of ∂W ∞ ⊂ ∂D ∞ has at least 2n preimages on ∂D 0 . This is a contradiction since the degree of f λ : ∂D ∞ → ∂D 0 is n.
Let z be a point in ∂W ∞ . Note that f −1 λ (W ∞ ) ⊂ D 0 and z ∈ ∂W ∞ is not any exceptional point, we get that It means that ∂D 0 = ∂W ∞ is a Jordan curve. Indeed, by applying a standard argument and noting that f λ is hyperbolic, it is easy to obtain that ∂D 0 is a quasicircle (for a proof, see [25,Lemma 3.3]).
According to the Böttcher theorem, there exist a unique conformal map φ : D ∞ → C \ D tangent at ∞ and another unique conformal map ϕ : D 0 → D tangent at 0 such that φ • f λ (z) = (ϕ(z)) −n , where D = {z : |z| < 1}. Given an angle θ ∈ [0, 1), the external ray in D ∞ with angle θ is denoted by γ(θ) = {z ∈ D ∞ : φ(z) = re 2πiθ , r > 1}. The external ray γ(θ) is called landing at z if γ(θ) → z ∈ ∂D ∞ as r → 1 and z is said to be a landing point of γ(θ). Suppose that D is a component of is called an internal ray in D. When z ∈ ∂D ∞ is a landing point of γ(θ), f −k λ (z) ∈ ∂D is also a landing point of f −k λ (γ(θ)). Note that f λ (γ(θ)) = γ(nθ) and ∂D ∞ is locally connected. From Carathéodory's theorem we know that all external rays land and each point of ∂D ∞ is a landing point of an external ray.
Step 2. The Julia set of f λ is a Sierpiński carpet if ∂D 0 ∩ ∂D ∞ = ∅. Up to now, we have proved that all Fatou components of f λ are Jordan disks. The following argument will be divided into three cases. Case 1. Suppose that P and Q are two components of U k 0 (or U k ∞ ) satisfying There are two internal rays l 1 , l 2 in P and Q, respectively, such that both of them land at a point z ∈ P ∩ Q and γ = f ). This implies that z is a critical point of f , which is a contradiction since all critical points are attracted by the cycle 0 ↔ ∞.
Case 2. Suppose that P and Q are two components of U i ∞ and U j ∞ (or U i 0 and , by a similar argument as Case 1, we also have a contradiction. Case 3. Suppose that P and Q are two components of U i 0 and U j ∞ respectively, where (i, j) = (0, 0). Since ∂D 0 ∩ ∂D ∞ = ∅, it follows that P ∩ Q = ∅.
The above arguments imply that the closures of each two different Fatou components of f λ are disjoint. By Lemma 2.5, the Julia set J(f λ ) is a Sierpiński carpet. This finishes the proof of Theorem 1.1 (c1). (b) If the free critical points are attracted by the cycle 0 ↔ ∞, then f λ is hyperbolic. According to [26, p. 745], the Julia set of f λ has Hausdorff dimension strictly less than two 1 . Since f λ has two Fatou components D 0 and D ∞ which are Jordan domains, we have dim H J(f λ ) ≥ 1.
Since f •2 λ has two fixed super-attracting basins D 0 , D ∞ , and f •2 λ is neither conjugate to a Blaschke product nor a quotient of a Blaschke product, it means that dim H (∂D 0 ) > 1 by [20]. Note that ∂D 0 ⊂ J(f •2 λ ) = J(f λ ). We have 1 < dim H J(f λ ) < 2. This ends the proof of Theorem 1.2. Proof of Theorem 1.3. Suppose that J(f λ ) is a Cantor set of circles. Then it follows that D 0 and D ∞ must be both simply connected and all other Fatou components (other than D 0 and D ∞ ) are annuli which separate the origin from the infinity. Lemma 2.4 yields that there are two of the annular Fatou components containing both 2n free critical points. Using the Riemann-Hurwitz's formula, one can get that the first preimages of D 0 and D ∞ contain all of these free critical points. Furthermore, none of the free critical points belongs to D 0 and D ∞ since J(f λ ) is a Cantor set of circles (not a quasicircle).
From the proof of Theorem 1.1 (b), we have the bottom picture of Figure 3. The conformal moduli of annuli satisfy mod( which holds if and only if n ≥ 4. The proof of Theorem 1.3 is complete. 3.5. This family is different from known ones. In this subsection we show that the dynamics of f λ is neither conjugate to h λ nor to any McMullen maps. Moreover, we study the Hausdorff dimension of f λ when J(f λ ) is a Cantor set of circles.
Proof of Theorem 1.4. (a) On the one hand, from the proof of Theorem 1.1 (b) we know that the dynamics on the set of Julia components of f λ is isomorphic to the one-sided shift on three symbols Σ 3 := {1, 2, 3} N . But it is known that the dynamics on the set of Julia components of McMullen map F (z) = z n + λ/z m is conjugate to the one-sided shift on only two symbols Σ 2 := {1, 2} N . It follows that f λ cannot be topologically conjugate to f on their corresponding Julia sets (see Figure 4 for an example). On the other hand, from the proof of Theorem 1.1 (b), we know that f λ has a super-attracting cycle D 0 ↔ D ∞ with period 2. However h λ has only two fixed super-attracting domains D 0 and D ∞ . Therefore f λ cannot be topologically conjugate to h λ on their corresponding Julia sets (see also Figure 3).
(b) If J(f λ ) is a Cantor set of circles, there are four closed annuli V 1 , V 2 , V 3 and V = C \ (D 0 ∪ D ∞ ), which are introduced in the proof of Theorem 1.1 (b). Their boundaries are Jordan curves and we have (see also Figure 3) • V 1 , V 2 and V 3 are closed annuli between D ∞ and A 0 , A 0 and A ∞ , A ∞ and D 0 respectively; and • f λ : V i → V is a coverings map with degree n for i = 1, 2, 3.
According to [ On the other hand, we have dim H J(f λ ) < 2 since f λ is hyperbolic. This finishes the proof of Theorem 1.4.

4.
The connectivity of the Julia sets. In this section, we consider the case that the orbits of all free critical points are not attracted by the super-attracting cycle 0 ↔ ∞. In this case, we will prove the connectivity of the Julia set of f λ (the last statement of Theorem 1.1) and the idea is similar to [10]. Proof. Based on the argument in the last section, we know that both D 0 and D ∞ are simply connected and all the iterated preimages f −l λ (D 0 ) and f −l λ (D ∞ ) are also simply connected for all l ∈ N. Assume that there exists a periodic attracting basin or parabolic basin U with period p associating a periodic point z 0 , which is different from D 0 and D ∞ . Then the period of z 0 is p but z 0 ∈ {0, ∞}. Without loss of generality, we assume that U contains at least a critical point c λ (see [18, § §8-10]). Obviously f •p λ (U ) = U . Note that all the components of p-periodic attracting or parabolic basin have the same connectivity. It means that the connectivity of is the same as that of U . We thus only need to study the connectivity of U . Let k be the number of points in U ∩ C λ , where C λ is the finite set consisting of the free critical points of f λ . Specifically, we have C λ = {ω j c λ , ω j λ 2 /c λ : 0 ≤ j ≤ 2n − 1}.
We claim that k satisfies 1 ≤ k ≤ 2. Otherwise, 2 < k ≤ 4n. If 2 < k < 2n+2, by Lemma 2.1 and Corollary 2.2, then ω j U = U for each j ∈ N and there are at least such 2n free critical points ω j c λ (0 ≤ j ≤ 2n − 1) contained in U by the assumption that c λ ∈ U . On the one hand, taking a Jordan curve γ ⊂ U such that it surrounds the origin and ω j γ = γ for each j ∈ N. Again Lemma 2.1 implies that f •np λ (γ) must surround the origin for every n ∈ N. On the other hand, from the assumption that U is different from D 0 and D ∞ , it follows that for each γ ⊂ U , f np λ (γ) → z 0 as n → ∞. Note that z 0 is different from 0 and ∞, we conclude that f •np λ (γ) cannot wind around the origin for any n ∈ N. We thus obtain a contradiction. When 2n + 2 ≤ k ≤ 4n, there are 4n free critical points ω j c λ , ω j λ 2 /c λ (0 ≤ j ≤ 2n − 1) in U . One can obtain a contradiction by applying a similar argument as above. This ends the proof of the claim. In the following, we divide the argument into the following two cases.
Case 1. If k = 1, then U must be simply connected. To see this, we consider two subcases. When U is a periodic attracting basin, let z 0 ∈ U be the periodic attracting point with period p and define z j = f •j λ (z 0 ) ∈ f •j λ (U ) for 1 ≤ j < p. Let B 0 be a small disk centered at z 0 such that f •p λ (B 0 ) ⊂ B 0 and ∂B 0 does not contain any points in the critical orbits of f λ . For l ≥ 0 and 0 ≤ j < p, one uses B lp+j to denote the connected component of f Since each B lp+j contains at most one critical point and ∂B 0 does not contain any points in the critical orbits of f λ , it follows from the Riemann-Hurwitz formula that B lp+j is simply connected. Noting U = l≥0 B lp and the formula (4), we know that U is simply connected. Suppose that U is a periodic parabolic basin. Let z 0 ∈ ∂U be the parabolic periodic point such that By virtue of the dynamics of f λ on the parabolic basins, we may take a small disk and ∂B 0 does not contain any points in the critical orbits of f λ . By the argument similar to that of the attracting case, one can deduce that U is also simply connected. Case 2. If k = 2, then U contains two distinct free critical points, say c 1 and c 2 . If c 2 = ω j0 c 1 , Lemma 2.1 and Corollary 2.2 imply that the 2n free critical points ω j c 1 (0 ≤ j ≤ 2n − 1) belong to U , which is a contradiction. We thus obtain that c 2 = ω j0 τ (c 1 ) for some j 0 ∈ N.
In the following we claim that deg(f λ | U ) = 3. On the one hand, if deg(f λ | U ) > 3, then for each z ∈ f λ (U ), Lemma 2.1 implies that f λ has more than 3n preimages (counting multiplicity), which is impossible since deg(f λ ) = 3n. On the other hand, recall that z 0 ∈ U is a periodic point with period p, we conclude that there are at least 3 preimages of f λ (z 0 ) which lie in U (By a similar argument, there are at least 3 preimages of f λ (z 0 ) ∈ f λ (∂U ) in ∂U for the parabolic case). Indeed, it is easy to see that ω j U (0 ≤ j ≤ 2n − 1) contains 4n free critical points. Lemma 2.1 and Corollary 2.2 imply that ω j τ (U ) (0 ≤ j ≤ 2n − 1) are 2n different Fatou components containing such 4n free critical points. Therefore U = ω j0 τ (U ) and hence z 0 = ω j0 τ (z 0 ) for some j 0 .
Suppose that z 1 ∈ U satisfies f λ (z 1 ) = f λ (z 0 ). Then For simplicity, we set z 2 = ω j0 τ (z 1 ). If z 1 = z 2 , then z 1 = ±z 0 . Lemma 2.1 (2n-fold symmetry of Fatou components) rules out the case that z 1 = −z 0 . In fact, assume that z 1 = −z 0 , it is clear that U surrounds the origin and satisfies ω i U = U for every i ∈ N, which is impossible because U is different from D 0 . Then z 2 = z 1 = z 0 , which shows that the free critical point c λ = z 0 and then its local degree is at least 3, thus deg f λ | z0 = 3. If z 1 = z 0 , it then follows from the above argument that z 1 = z 2 . Therefore z 0 , z 1 and z 2 are three distinct points. The condition that U = ω j0 τ (U ) means z 2 ∈ U , which implies deg(f λ | U ) = 3. In either cases, deg(f λ | U ) = 3 and the claim is proved. By a similar argument as above, we take B 0 such that it satisfies B 0 = ω j0 τ (B 0 ). This implies that the two critical points c 1 and c 2 = ω j0 c 1 of f λ are contained in B 1 at the same time, where B 1 denotes the preimage of B 0 under f λ which is contained in the p-periodic attracting or parabolic cycle. By using the Riemann-Hurwitz formula, we can use a standard process to prove that the periodic attracting or parabolic basin U is simply connected. The proof is complete.
For a Jordan curve γ ⊂ C, recall that set ext(γ) is defined as the component of C \ γ containing ∞ and int(γ) the other. Let A ⊂ C be an annulus. Recall that the core curve, denoted by γ A , of A is defined as ψ −1 ( √ r), where ψ : A → A r := {z ∈ C : 0 < r < |z| < 1} is a conformal isomorphism. Denoted by E A and I A the exterior and interior boundaries of A, respectively. Since γ A is a smooth Jordan curve, clearly it can separates the Riemann sphere into two disjoint disks A ext := ext(γ A ) and A int := int(γ A ).
For any z ∈ C, define the forward orbit of z under the iteration of f by O f (z) := {f •n (z) : z ∈ N}. We call that two forward orbits O f (z 1 ) and O f (z 2 ) disjoint if the intersection of them are empty. To show that J(f λ ) is connected, we need to rule out the case that f λ has Herman rings. To do so, we need the following criterion which has been proved in [33,Corollary 2.2].
Proof. Let us assume by contradiction that f λ has a cycle of Herman rings {U 0 , U 1 , . . ., U p = U 0 }. Then f •p λ is conjugate to the irrational rotation z → µz on U 0 , where µ = e 2πiα and α is an irrational number. For each 0 ≤ j ≤ p − 1, f λ : U j → U j+1 is conformal. According to Lemma 2.1, f λ has symmetric properties and J(f λ ) is 2n-symmetric. We now consider a new rational map obtained by semiconjugacy. Specifically, we take ϕ(z) = z 2n and define .
Then ϕ • f λ = g λ • ϕ. This means that the dynamics of f λ is similar as that of g λ . In particular, g λ has a cycle of Herman rings if and only if f λ has. Lemma 2.1 and Corollary 2.2 imply that the component U 0 is bounded and does not wind around the origin since f λ is injective in U 0 as before. By the similar argument, one can show that U j is bounded and does not surround the origin for each 1 ≤ j ≤ p − 1. According to the semiconjugacy between f λ and g λ under ϕ, each of the corresponding Herman ring of g λ is also bounded and does not surround the origin. Again by Lemma 2.1, we then find 2n Fatou components ω i U 0 (1 ≤ i ≤ 2n) such that ω i U ∩ ω j U = ∅ for any i = j (mod 2n). It follows that the restriction of ϕ on U 0 is injective and ϕ(U 0 ) is a periodic Herman ring of g λ . Suppose that the period of ϕ(U 0 ) is p (Note that p is allowed to be equal to p but p is a divisor of p ).
The following aim is try to obtain a contradiction by Proposition 4.2. Set V 0 := ϕ(U 0 ) and V j = g •j λ (V 0 ), where 0 < j ≤ p−1. In particular, g λ (V p−1 ) = V 0 and then {V 0 , V 1 , . . . , V p = V 0 } is a p-periodic Herman rings of g λ . Denote by τ 0 (z) = λ 4n /z. One obtains that Note that deg(g λ ) = 3n and g λ (z) has 6n − 2 critical points (counting with multiplicity). The local degrees of 0 and ∞ are both n and the local degrees of λ 3n+1 and λ n−1 are both 2n. Hence this leaves 2 critical points. According to (5), the two critical points can be written by c λ and τ 0 ( c λ ) = λ 4n / c λ , and the local degree of them are both 2. It is easy to see that V j is bounded and does not surround the origin for any 0 ≤ j ≤ p − 1.
We now study the iteration g •p λ . It is clear that the free critical points of g •p λ are Then there exist at most 2p disjoint critical orbits of g •p λ and such critical orbits have the following form The collection of core curves {γ 0 , γ 1 , · · · , γ p−1 } of the p-periodic Herman rings of g λ separates C into p + 1 connected components, say W 0 , W 1 · · · , W p . In the following we consider two cases. Case 1. Suppose that τ 0 (V 0 ) = V 0 . It follows from (5) that τ 0 (V j ) = V j for all 0 ≤ j ≤ p − 1. For simplicity, the exterior and interior boundary components of V j are denoted by E j and I j respectively, where 0 ≤ j ≤ p − 1. We claim that τ 0 (E j ) = E j for each j ∈ N. To see this, recall that V j is bounded and does not wind around the origin, we conclude that the exterior component E i must contain the points in V j with the largest and smallest modulus. Since the point of V j with the largest modulus is mapped to the smallest one under τ 0 , then τ 0 (E j ) = I j and hence τ 0 (E j ) = E j for each j ∈ N. That means the claim holds. Then O g •p λ (c j ) and O g •p λ (τ 0 (c j )) are always contained in the same component W j . By Proposition 4.2, there are at least p + 1 disjoint infinite critical orbits However the 2p critical orbits of g •p λ can only belong to p of p + 1 components of the collection {W 0 , W 1 , · · · , W p }, which is impossible.
Case 2. Suppose that τ 0 (V 0 ) = V 0 . This implies from (5) that τ 0 (V j ) = V j for each 0 ≤ j ≤ p − 1. Therefore g •p λ must have 2p disjoint fixed Herman rings. On the other hand, there are only at most 2p disjoint critical orbits. By Proposition 4.2, this is a contradiction.
This means that g λ has no Herman rings and hence f λ has neither. Now we can finish the proof of Theorem 1.1.
Proof of Theorem 1.1 (The connectivity of the Julia sets). By Lemma 4.1, f λ has no infinitely connected Fatou components. According to Lemma 4.3, f λ has no Herman rings. This means that all periodic Fatou component of f λ are simply connected. Note that the iterated preimages of such periodic Fatou components do not contain any critical points. It means that each component of such iterated preimages is also simply connected. According to [1, p. 173], we conclude that J(f λ ) is connected. This finishes the last statement of Theorem 1.1 and hence completes the proof of Theorem 1.1.
Next we only need to show that none of the free critical points are contained in D 0 and D ∞ . To see this, by way of contradiction we assume that there is a free critical point, say c 1 , is contained in D 0 . By Lemma 2.1, one has c i ∈ D 0 and c i ∈ D ∞ for every 1 ≤ i ≤ 2n. From the proof of Theorem 1.1 (a) we know that J(f λ ) is quasicircle. Hence int(J(f λ )) is mapped into D ∞ under f λ , which is impossible because f λ (T a ) ⊂ D 0 and T a ⊂ int(J(f λ )).
Note that there exists a free critical point c i such that f λ (c i ) ∈ D ∞ . However c i ∈ D 0 when |λ| = 0 is small enough. From Theorem 1.1 (b), it follows that J(f λ ) is a Cantor set of circles.