On a Family of Rational Perturbations of the Doubling Map

The goal of this paper is to investigate the parameter plane of a rational family of perturbations of the doubling map given by the Blaschke products $B_a(z)=z^3\frac{z-a}{1-\bar{a}z}$. First we study the basic properties of these maps such as the connectivity of the Julia set as a function of the parameter $a$. We use techniques of quasiconformal surgery to explore the relation between certain members of the family and the degree 4 polynomials $\left(\overline{\overline{z}^2+c}\right)^2+c$. In parameter space, we classify the different hyperbolic components according to the critical orbits and we show how to parametrize those of disjoint type.


Introduction
Given a rational map f :Ĉ !Ĉ, whereĈ ¼ C < f1} denotes the Riemann sphere, we consider the dynamical system given by the iterates of f. The Riemann sphere splits into two totally f-invariant subsets: the Fatou set F ð f Þ, which is defined to be the set of points z [Ĉ where the family ff n ; n [ N} is normal in some neighbourhood of z, and its complement, the Julia set J ð f Þ. The dynamics of the points in F ð f Þ are stable whereas the dynamics in J ð f Þ present chaotic behaviour. The Fatou set F ð f Þ is open and therefore J ð f Þ is closed. Moreover, if the degree of the rational map f is greater than or equal to 2, then the Julia set J ð f Þ is not empty and is the closure of the set of repelling periodic points of f .
The connected components of F ð f Þ, called Fatou components, are mapped under f among themselves. Sullivan [33] proved that any Fatou component of a rational map is either periodic or preperiodic. By means of the Classification Theorem (see, e.g. [20]), any periodic Fatou component of a rational map is either the basin of attraction of an attracting or parabolic cycle or is a simply connected rotation domain (a Siegel disc) or is a doubly connected rotation domain (a Herman ring). Moreover, any such component is somehow related to a critical point, i.e. a point z [Ĉ such that f 0 ðzÞ ¼ 0. Indeed, the basin of attraction of an attracting or parabolic cycle contains, at least, a critical point whereas Siegel discs and Herman rings have critical orbits accumulating on their boundaries. For a background on the dynamics of rational maps we refer to [3,20].
The aim of this paper is to study the dynamics of the degree 4 Blaschke products given by where a; z [ C. This Blaschke family restricted to S 1 is the rational analogue of the double standard family A a;b ðuÞ ¼ 2u þ a þ ðb=pÞ sinð2puÞ ðmod 1Þ of periodic perturbations of the doubling map on S 1 . Indeed, when jaj tends to 1, the products B a tend to e 4pit z 2 uniformly on compact sets of the punctured plane C * ¼ Cnf0}, where t [ R=Z denotes the argument of a. Double standard maps extend to entire transcendental selfmaps of C * . Although there is no explicit simple expression for the restriction of B a to S 1 , the global dynamics are simpler than in the transcendental case. The double standard family has been studied in several papers such as [7,10,22,23].
For all values of a [ C, the points z ¼ 0 and z ¼ 1 are superattracting fixed points of local degree 3. We denote by Að0Þ and Að1Þ their basins of attraction and by A * ð0Þ and A * ð1Þ their immediate basins of attraction, i.e. the connected components of the basins containing the superattracting fixed points. If jaj # 1, Að0Þ ¼ A * ð0Þ ¼ D and Að1Þ ¼ A * ð1Þ ¼Ĉn D and hence J ðB a Þ ¼ S 1 (see Lemma 3.3). If jaj . 1, there are two free critical points c^. If jaj ¼ 2, the two free critical points collide into a single one. If jaj . 2 the two free critical points are symmetric with respect to S 1 (see Section 3) and so are their orbits. Consequently, if jaj . 2 it is enough to control one of the free critical orbits in order to understand the possible dynamics of B a . On the other hand, if 1 , jaj , 2 the Blaschke product B a has two different free critical points contained in S 1 which may lead to different dynamics.
The connectivity of the Julia set is a focus of attention when studying a family of rational functions (see, e.g. [9,26,27,32]). It is known that, given a polynomial P, its Julia set J ðPÞ is connected if and only if it has no free critical point captured by the basin of attraction of infinity (see [20]). However, such a classification does not exist for general rational maps which, unlike polynomials, may have Herman rings and even Cantor sets of Jordan curves [18]. It turns out that the family B a shares some of the features of polynomials in this respect such as the non-existence of Herman rings (see Proposition 3.5). We also prove the following criterion.
Theorem A. Given a Blaschke product B a as in (1), the following statements hold: Consequently, if jaj $ 2, then J ðB a Þ is connected if and only if c þ Ó A * ð1Þ. Next, we focus on the basins of attraction of attracting or parabolic cycles not contained in S 1 , other than 0 or 1. These may only exist when jaj . 2. We distinguish two cases depending on the location of the cycles with respect to D.
On the one hand, if B a , jaj . 2, has no attracting or parabolic cycle in S 1 , we can relate the Blaschke product with a cubic polynomial preserving the dynamics of all orbits contained in Cn D. Indeed, for such parameters, B a j S 1 is quasisymmetrically conjugate to the doubling map u ! 2u ðmod 1Þ and a quasiconformal surgery, consisting in gluing a superattracting cycle in D, can be performed obtaining cubic polynomials of the form M b ðzÞ ¼ bz 2 ðz 2 1Þ with b [ C (see [25] and Section 4.2). These polynomials have been the object of study of several papers (see, e.g. [21,29]). This surgery establishes a conformal conjugacy between M b and B a on the set of points which never enter D under iteration of B a and the points which are not attracted to z ¼ 0 under iteration of M b . In particular, if B a has an attracting or parabolic cycle contained in Cn D, this surgery conjugates B a with M b conformally in its basin of attraction.
On the other hand, if B a has an attracting or parabolic cycle with points both inside and outside D the situation is different. Although the previous surgery construction is still possible, a lot of information is lost since, under the new map, the critical point is always captured by the basin of z ¼ 0. Parameters for which the orbit of c þ [ Cn D enters the unit disc at least once are called swapping parameters and connected components of the set of swapping parameters are called swapping regions. Inside these regions, the nonholomorphic dependence of B a on the parameter a gives rise to what appear to be small 'copies' of the Tricorn, the connectedness locus of the antiholomorphic family (i.e. holomorphic on the variable z) of polynomials p c ðzÞ ¼ z 2 þ c (see [6] and Figure 9(a)). Milnor [19] showed that a similar situation takes place for real cubic polynomials introducing the concept of antipolynomial-like mapping. We distinguish between two types of attracting cycles for swapping parameters. We say that a parameter is bitransitive if it has a cycle whose basin of attraction contains the two free critical points. We say that a parameter is disjoint if there are two different cycles other than 0 or 1. The very special dynamics taking place for swapping parameters allow us to build a polynomial-like mapping in a neighbourhood of every bitransitive or disjoint swapping parameter. A polynomial-like map is a locally defined map which is conjugate to a polynomial (see Definition 2.3).
Theorem B. Let a 0 be a swapping parameter with an attracting or parabolic cycle of period p . 1. Then, there is an open set W containing a 0 and p 0 . 1 dividing p such that, for every a [ W, there exist two open sets U and V with c þ [ U such that ðB p 0 a ; U; VÞ is a polynomial-like map. Moreover, (a) If a 0 is bitransitive, ðB p 0 a ; U; VÞ is hybrid equivalent to a polynomial of the form p 2 c ðzÞ ¼ ðz 2 þ cÞ 2 þ c. (b) If a 0 is disjoint, ðB p 0 a ; U; VÞ is hybrid equivalent to a polynomial of the form p 2 c ðzÞ ¼ ðz 2 þ cÞ 2 þ c or of the form z 2 þ c.
It is known that the boundary of every bounded Fatou component of a polynomial, with the exception of Siegel discs, is a Jordan curve [30]. This is not true however for rational functions although it can be established under some conditions (e.g. postcritically finite among others [26]). In our case, as a consequence of the two previous constructions, we know that the boundary of every connected component of the basin of attraction of an attracting or parabolic cycle of B a not contained in S 1 and other than z ¼ 0 and z ¼ 1 is a Jordan curve (see Proposition 4.10). Indeed, if B a has such a cycle, the previous constructions provide a conjugation between B a and a polynomial which sends the immediate basin of attraction of the cycle of B a to the immediate basin of attraction of a bounded cycle of the polynomial.
A rational map is hyperbolic if all its critical points are attracted to attracting cycles. A hyperbolic component is a connected component of the open set H ¼ fajB a is hyperbolic}. The parametrization of hyperbolic components of rational functions which depend holomorphically on their parameters is well known (see [5,12]). If the family of functions does not depend holomorphically on parameters, some extra difficulties appear. Nakane and Schleicher [24] study the parametrization of hyperbolic components with cycles of even period for the family of antipolynomials p c;d ðzÞ ¼ z d þ c. We focus on the parametrization of hyperbolic components with disjoint parameters using different methods than the ones of [24]. Notice that, due to the symmetry of B a , disjoint cycles are symmetric with respect to S 1 and therefore have the same period and conjugate multiplier (see Theorem 4.5). Hence, given a hyperbolic component U with disjoint parameters, it makes sense to define the multiplier map L : U ! D as the map which sends every a [ U to the multiplier of the attracting cycle whose basin captures the critical orbit of c þ .
Theorem C. Let U be a disjoint hyperbolic component such that U , fa [ C; jaj . 2}. Then, the multiplier map is a homeomorphism between U and the unit disc.
Since the multiplier of any bitransitive cycle is a non-negative real number (see Proposition 4.11), the previous result does not hold for bitransitive components. This phenomenon had already been noticed in [24] for the polynomials p 2 c;d . In Section 2 we introduce some notation and useful results to prove Theorems A, B and C. In Section 3 we describe the basic properties of the Blaschke family B a and prove Theorem A. In Section 4 we study the parameter plane of the family: we describe the different kinds of hyperbolic dynamics that may occur depending on the behaviour of the free critical points, we introduce the relation with the family of cubic polynomials M b , describe the dynamics that can take place along the swapping regions proving Theorem B and finally we prove Theorem C.

Preliminaries and tools
Given a rational map f, we denote by kz 0 l :¼ fz 0 ; · · ·; f p21 ðz 0 Þ ¼ z p21 } an attracting or parabolic cycle of period p $ 1, where f ðz i Þ ¼ z iþ1 with subindices taken modulus p. We denote with Aðkz 0 lÞ the basin of attraction of the cycle whereas A * ðkz 0 lÞ denotes its immediate basin of attraction, i.e. the connected components of Aðkz 0 lÞ which contain a point of kz 0 l. With A * ðz q Þ we denote the connected component of A * ðkz 0 lÞ containing z q . The marked point z 0 of the cycle is usually taken so that A * ðz 0 Þ contains a critical point.

The Riemann-Hurwitz formula
When dealing with the simple connectivity of open sets, it is useful to consider the Riemann -Hurwitz formula (see [3]). It can be stated as follows.
Theorem 2.1 (Riemann -Hurwitz formula). Let U and V be two connected domains of C of finite connectivity m u and m v and let f : U ! V be a degree k proper map branched over r critical points counted with multiplicity. Then The following corollary is used several times along the paper. Proof. By construction, f j U : U ! V is proper. Let r be the multiplicity of the critical point. Then, f j U has at least degree r þ 1. By The Riemann -Hurwitz formula, since m V ¼ 1, we have m U 2 2 # 2ðr þ 1Þ þ r ¼ 21. Since m U is at least 1, we conclude that it is indeed 1 and U is simply connected. A

Polynomial and antipolynomial-like mappings
The theory of polynomial-like mappings [13], introduced by A. Douady and J. Hubbard, allows us to understand why copies of the filled Julia set of polynomials appear in the dynamical planes of rational maps or even entire transcendental or meromorphic maps.
Definition 2.3. A triple ðf ; U; VÞ is called a polynomial-like (respectively, antipolynomial-like) mapping of degree d if U and V are bounded simply connected subsets of the plane, U , V and f : U ! V is holomorphic (respectively, antiholomorphic) and proper of degree d. Its filled Julia set is defined as Observe that given any polynomial (respectively, antipolynomial) P of degree d, there exists a disc D R of radius R . 0 so that ðP; D R ; PðD R ÞÞ is a polynomial-like mapping (respectively, antipolynomial-like mapping).
Definition 2.4. We say that two (anti)polynomial-like maps ðf ; U; VÞ and ðf 0 ; U 0 ; V 0 Þ are hybrid equivalent if there exist neighbourhoods U f and U f 0 of K f and K f 0 , respectively, and a quasiconformal conjugation f : U f ! U f 0 between f and f 0 such that Polynomial-like mappings, as their name indicates, behave locally as polynomials do. This is the content of the Straightening Theorem [13].
Theorem 2.5 (The Straightening Theorem). Every polynomial-like mapping ðf ; U; VÞ of degree d is hybrid equivalent to a polynomial P of degree d. If K f is connected, P is unique up to affine conjugation.
The antipolynomial-like theory was first introduced by Milnor [21] in order to study why small 'copies' of the Tricorn appear in the parameter plane of real cubic polynomials. Hubbard and Schleicher [16] used this theory afterwards in the study of the Multicorns, the parameter plane of the antipolynomial maps p c;d ðzÞ ¼ z d þ c. They stated the Antiholomorphic Straightening Theorem.
Theorem 2.6 (The Antiholomorphic Straightening Theorem). Every antipolynomial-like mapping ðf ; U; VÞ of degree d is hybrid equivalent to an antipolynomial P of degree d. If K f is connected, then P is unique up to affine conjugation.

Conjugation with the doubling map
Along the paper it is important to know whether the Blaschke products B a are conjugate to the doubling map of the circle u ! 2u ðmod 1Þ, where u [ R=Z (equivalently given by z ! z 2 , where z [ S 1 ). The following result tells us that the lift H of an increasing covering map f of the circle of degree 2 is semiconjugate to the doubling map u ! 2u. See [8,10,22] for some M . 0. It is not difficult to prove that, if jaj . 3, B a j S 1 is an expanding map and, therefore, h a is a quasisymmetric conjugacy (cf. [8]). The next theorem, due to Petersen [25], gives some conditions weaker than expansivity which guarantee that h a is a quasisymmetric conjugacy. Recall that the v-limit set vðzÞ of a point z [ C is defined to be the accumulation set of the orbit of z. A point z is called recurrent if and only if z [ vðzÞ.
Theorem 2.9. Let B :Ĉ !Ĉ be a Blaschke product with poles in D such that the restriction B : S 1 ! S 1 is a (positively oriented) degree d $ 2 covering and such that S 1 contains no non-repelling periodic point and vðcÞ > S 1 ¼ Y for every recurrent critical point c. Then, B :

Dynamical plane of the Blaschke family
We consider the degree 4 Blaschke products of the form where a [ C and t [ R=Z. As all Blaschke products, the family B a;t leaves S 1 invariant and therefore its members are symmetric with respect to the unit circle, i.e. B a;t ðzÞ ¼ I + B a;t + I ðzÞ where I ðzÞ ¼ 1= z. The next lemma tells us that, for the purpose of classification, we can get rid of the parameter t. The proof is straightforward.
Lemma 3.1. Let a [ R and let hðzÞ ¼ e 22pia z. Then h conjugates the maps B a;t and B ae 22pia ;tþ3a . In particular we have that B a;t is conjugate to B ae 2pit=3 ;0 .
Hence, we focus on the study of the family B a ðzÞ ¼ z 3 ððz 2 aÞ=ð1 2 azÞÞ (1) for values a; z [ C.
Given the fact that these rational maps have degree 4, there are six critical points counted with multiplicity. The fixed points z ¼ 0 and z ¼ 1 are critical points of multiplicity 2 and hence superattracting fixed points of local degree 3. The other two critical points, denoted by c^, are given by The critical points c þ and c 2 are free and satisfy jc þ j $ 1 and jc 2 j # 1. If they are not in S 1 (i.e. jaj . 2 or jaj , 1), they and their orbits are symmetric with respect to S 1 . The following result proves that the critical points determine the parameter if jaj $ 2 or jaj , 1.
Lemma 3.2. Given a Blaschke product B a;t as in (2) with jaj $ 2 or jaj , 1, the parameter a is continuously determined by the critical points c^. Moreover, if the image B a ðz 0 Þ Ó f0; 1} of a point z 0 is fixed, the parameter t is continuously determined by a.
Proof. The continuous dependence of t with respect to a is clear. Let a ¼ r a e 2pia , where a [ R=Z and r a $ 2 (respectively, r a , 1). It follows from (3) that the critical points c þ and c 2 have the same argument a as a. It is left to see that r a depends continuously on jc þ j ¼ r c . It follows from symmetry that jc 2 j ¼ 1=r c . Consider Rðr c Þ ¼ r c þ 1=r c . For r c $ 1, R is a strictly increasing function which satisfies Rð1Þ ¼ 2. Using (3) we have and, therefore, r a ÁRðr c Þ ¼ 2ð2 þ r 2 a Þ=3. This quadratic equation yields two solutions r a^¼ ð3R^ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9R 2 2 32 p Þ=4. The solution r a þ ðRÞ takes the value 2 for R ¼ 2 and is strictly increasing and tending to 1 when R tends to 1. The solution r a 2 ðRÞ takes the value 1 for R ¼ 2 and is strictly decreasing and tending to 0 when R tends to 1. Therefore, each critical point c þ [ C, jc þ j $ 1 (respectively, jc þ j . 1Þ, determines continuously a unique parameter a such that jaj $ 2 (respectively, jaj , 1).
A Another relevant point to discuss is what type of dynamics may occur in S 1 . It follows directly from the invariance of S 1 under B a and the fact that B a : S 1 ! S 1 cannot be a degree 1 covering (and hence conjugate to an irrational rotation) that any point z [ S 1 either belongs to the Julia set or is attracted to an attracting or a parabolic periodic orbit For completeness we describe some features of the dynamics of B a which depend on the modulus of a. The first thing to consider is whether there is or not a preimage of 1 in D. This family has a unique pole at z 1 ¼ 1= a and a unique zero z 0 ¼ a. Their position, together with the positions of c^, influences the possible dynamics of B a . We proceed to describe the situation depending on jaj (see Figure 1).
When jaj , 1 we have that both critical points c^lie on the half ray containing a. Moreover, jc 2 j , 1 and jc þ j . 1. The only pole, z 1 ¼ 1= a; has modulus greater than 1.
Hence, B a : D ! D is a holomorphic self-map of D having z ¼ 0 as a superattracting fixed point. Since, by symmetry, there is no preimage of the unit disc outside the unit circle, B a j D is a degree 4 branched covering. By Schwarz lemma we have that z ¼ 0 is the only attracting point of B a in D which attracts all points in D. Summarizing, we have the following.
When jaj ¼ 1 both critical points and the preimages of 0 and 1 collapse at the point z ¼ a, where the function is not formally defined. Everywhere else we have the equality: Hence, by symmetry, jc^j ¼ 1 (see Figure 1(b)). The critical orbits lie in S 1 and are not related to each other. The circle map B a j S 1 has no degree defined. Indeed, it can be proven that some points in S 1 have two preimages under B a j S 1 whereas other points have four preimages. In Figure 2 we show the dynamical planes of three maps B a with 1 , jaj , 2.
When jaj ¼ 2 we have a unique critical point c ¼ a=2 of multiplicity 2 in the unit circle. There are two preimages of S 1 which meet at c (see Figure 1(c)). There may or may not be an attracting or parabolic cycle in S 1 when jaj ¼ 2. The parameter might be, for example, of Misiurewicz type (i.e. the free critical point is preperiodic). In this situation the only Fatou components of B a are the basins of z ¼ 0 and z ¼ 1. We also remark for further use that the map B a j S 1 is 2-to-1. In Figure 3 we show the dynamical planes of two maps B a with jaj ¼ 2.
When jaj . 2, as is the case when jaj , 1, we have that both critical points c^lie on the half ray containing a and are symmetric with respect to S 1 . In this case we have two disjoint preimages of the unit circle: one of them inside D, surrounding the pole z 1 , and the symmetric one outside surrounding the zero z 0 ¼ a (see Figure 1(d)). As in the case jaj ¼ 2, B a j S 1 is 2-to-1. In Figures 4 and 6(a) we show the dynamical planes of three maps B a with jaj . 2.

Connectivity of the Julia set: proof of Theorem A
The goal of this section is to prove Theorem A. Notice that statement (a) has already been proven in Lemma 3.3. In Proposition 3.4 we prove statement (b).    We now begin the proof of statement (c). In Propositions 3.5 and 3.6 we prove that all periodic Fatou components other than A * ð0Þ or A * ð1Þ are simply connected.
Proposition 3.5. Let B a be as in (1). Then B a has no Herman rings.
Proof. Shishikura [31] proved that if a rational map has a Herman ring, then it has two different critical points whose orbits accumulate on the two different components of the boundary of it. If jaj # 1, we have that the Julia set satisfies J ðB a Þ ¼ S 1 (see Lemma 3.3), so B a cannot have Herman rings. If 1 , jaj # 2, the two critical orbits lie in S 1 and, hence, there can be no Herman rings.
We focus now on the case jaj . 2. By Shishikura's result and symmetry, a cycle of Herman rings would have components both inside and outside the unit disc. Hence, it would have at least one component in the preimage of the unit disc V e ¼ B 21 a ðDÞnD and another one in the preimage of the complementary of the unit disc V i ¼ B 21 a ðCn DÞ (see Figure 1(d)). Recall that V e is a simply connected set disjoint from S 1 . Moreover, all its preimages are bounded, none of them can intersect the unit circle and all of them are simply connected by Corollary 2.2. Every component of the cycle of Herman rings is contained in a preimage of V e of some order n $ 0. We claim that such a cycle must have a component which surrounds either the unit disc or V e . If this is so, this component cannot be contained in a simply connected preimage of V e , which leads to a contradiction.
Let I ðzÞ ¼ 1= z be the reflection with respect to S 1 . To prove the claim observe that, due to symmetry, if A is a component of the cycle of Herman rings, then so is I ðAÞ. Moreover, since infinity is a superattracting fixed point, all components are bounded and at least one of them, say A 0 , surrounds the pole z 1 (by the Maximum Modulus Principle). Recall that z 1 is contained in V i and that, again by symmetry, IðV i Þ ¼ V e . Then, either A 0 surrounds the unit disc or surrounds V i or is contained in V i . In the first case we are done. In the second case I ðA 0 Þ surrounds V e and we are also done. In the third case, B a ðA 0 Þ separates infinity and the unit disc and, hence, surrounds the unit disc. This finishes the proof. A Proposition 3.6. Let B a be as in (1) with jaj $ 2. Let kz 0 l be an attracting, superattracting or parabolic p-cycle of B a other than f0} or f1}. Then A * ðkz 0 lÞ is simply connected.
Proof. Case 1: First we consider the case in which each connected component of the immediate basin of attraction contains at most one critical point (counted without multiplicity). For the attracting case consider a linearizing domain A coming from Koenigs linearization around z 0 (see [20], Theorem 8.2). The subsequent preimages U n defined as the components of B 2n a ðAÞ such that z 2n [ U n contain at most one critical point and are hence simply connected by Corollary 2.2. The result follows since the nested subsequence of preimages fU np } covers A * ðz 0 Þ. The parabolic case follows similarly taking a petal instead of a linearizing domain (see [20], Theorem 10.7) whereas in the superattracting case we may use a Böttcher domain (see [20], Theorem 9.1).
Case 2: Now consider the case in which one connected component, say A * ðz 0 Þ, of the immediate basin of attraction contains the two free critical points. This excludes the case jaj ¼ 2 (see Section 3). Without loss of generality we assume that z 0 is a fixed point. Indeed, the first return map from A * ðz 0 Þ onto itself has no other critical points since the other components of the immediate basin of attraction contain none.
Due to symmetry of the critical orbits, the fixed point z 0 lies in S 1 . Hence, A * ðz 0 Þ intersects S 1 , which is invariant. If z 0 is attracting, take the maximal domain A of the Koenigs linearization (see [20], Lemma 8.5). Its boundary ›A contains, due to symmetry, the two critical points. Each critical point has a different simply connected preimage of B a ðAÞ attached to it. Let V be the connected component of B 21 a ðAÞ containing A. The map B ajV : V ! A is proper and of degree 3 since z 0 has three different preimages. Given that V contains exactly two critical points and B ajV is of degree 3, it follows from the Riemann -Hurwitz formula (see Theorem 2.1) that V is simply connected. Using the same reasoning all of its preimages are simply connected. Finally, since A * ðz 0 Þ is covered by the nested sequence of simply connected preimages of A, we conclude that A * ðz 0 Þ is simply connected. The parabolic case is done similarly takingˆto be the maximal invariant petal (see [20], Theorems 10.9 and 10.15). Notice that, due to symmetry, for jaj . 2 we cannot have a superattracting cycle of local degree 2 with an extra critical point in A * ðz 0 Þ. A We now finish the proof of statement (c). Assume that there exists a periodic Fatou component other than A * ð0Þ and A * ð1Þ. Then, such periodic Fatou component has a critical point related to it. Indeed, if it is Siegel disc, there is a critical point whose orbit accumulates on its boundary (see [31]). If it is the basin of attraction of an attracting, superattracting or parabolic cycle kz 0 l, there is a critical point c [ A * ðkz 0 lÞ (cf. [20]). Therefore, there is at most one unoccupied critical point. Hence, by means of the Corollary 2.2 of the Riemann -Hurwitz formula, any preperiodic Fatou component eventually mapped to a periodic component other that A * ð1Þ or A * ð0Þ is also simply connected.
The final statement of the theorem holds since the Julia set of a rational map is connected if and only if all connected Fatou components are simply connected.

Parameter plane of the Blaschke family
The aim of this section is to study the parameter plane of the Blaschke family B a . Figure 5 shows the result of iterating the critical point c þ . Since the two critical orbits of B a are related by symmetry unless 1 , jaj , 2, this information suffices also for c 2 everywhere else. Indeed, if 1 , jaj , 2, the critical orbits may have completely independent behaviour (see Figure 2(c),(d)).
The next lemma explains the observed symmetries on the parameter plane. Its proof is straightforward.
Lemma 4.1. Let j be a third root of the unity. Then, B a and B ja are conjugate by the conformal map tðzÞ ¼ jz. Moreover, B a and B a are conjugate by the anticonformal map I ðzÞ ¼ z.

Hyperbolic parameters
We say that a parameter a [ C is hyperbolic if B a is hyperbolic, that is, if both free critical points c^¼ c^ðaÞ are attracted to attracting cycles. Recall that the two critical orbits are symmetric except for 1 , jaj # 2, in which case they belong to S 1 . Hence, if one critical point is attracted to the superattracting fixed point z ¼ 0 (respectively, z ¼ 1) the other one is attracted to z ¼ 1 (respectively, z ¼ 0). Parameters for which this happens are called escaping parameters. We denote the set of escaping parameters by E. We shall denote the set of non-escaping parameters by B. Observe that B is not the connectedness locus of B a . Indeed, in view of Theorem A, parameters in E have a connected Julia set if the critical points belong to Að0ÞnA * ð0Þ or Að1ÞnA * ð1Þ. On the other hand, some Julia sets for 1 , jaj , 2 (which must belong to B) may, a priori, be disconnected (although we believe that this is never the case). Notice also that, if jaj ¼ 1, the products B a degenerate to degree 3 polynomials without free critical points in which case a is neither escaping nor non-escaping. Proof. The first two statements are already proven. To prove the third one we have to see that if jaj is big enough, then the parameter a is escaping. First we prove that if jzj . lðjaj þ 1Þ with l $ 1, then jB a ðzÞj . ljzj. It follows from the previous hypothesis that jz 2 aj . l and that jzj 2 . jzjðjaj þ 1Þ . j1 2 azj. Therefore, we have jB a ðzÞj ¼ jzj 3 jz 2 aj j1 2 azj . jzj 3 l jzj 2 ¼ ljzj: To finish the proof notice that, as jaj tends to 1, the critical point c þ ðaÞ tends to aÁ2=3. Consequently, it can be checked that the modulus of the critical value v þ ¼ B a ðc þ ðaÞÞ grows as Cjaj 2 for some C . 0 and, for jaj big enough, jv þ j . lðjaj þ 1Þ with l . 1. We conclude that jB n a ðv þ Þj ! 1 when n ! 1. Therefore, for jaj big enough, a [ E. A Following [19,28], we classify hyperbolic non-escaping parameters as follows.  (c) Capture if one of the critical points belongs to the immediate basin of attraction of an attracting cycle and the other one belongs to a preperiodic preimage of it (see Figure 6 (right)). (d) Disjoint if the critical points belong to the immediate basin of attraction of two different attracting cycles (see Figures 2(c),(d), 4 (right) and 11 (right)).
We have omitted the parameters a with jaj ¼ 2 in the previous definition since, for them, the two free critical points collide into a single one.
Because of the non-holomorphic dependence in the parameter a, this parameter slice presents some special hyperbolic components which we call tongues in analogy to the Arnold tongues which appear in the Arnold standard family of perturbations of rigid rotations (see [2,14,15]). In this settings they cannot be defined as the level sets of the rotation number (since there is no rotation number to speak about), but instead, we may use the equivalent characterization of the existence of an attracting periodic orbit in S 1 with given combinatorics. Tongues for the double standard map were studied previously by Misiurewicz and Rodrigues [22,23] and Dezotti [10]. They are defined as follows.
Definition 4.4. We define T as the set of parameters a, jaj $ 2, such that B a j S 1 has an attracting cycle. The connected components of T are called tongues.
Tongues can be classified according to the combinatorics of the attracting cycle, which can be described via the semiconjugacy between B a j S 1 and the doubling map (see Lemma 2.7). A detailed study of T will appear in a forthcoming paper.
We proceed now to describe in which regions the different types of non-escaping hyperbolic parameters belong. Conversely, any parameter a belonging to a tongue is adjacent. (b) If a is a bitransitive parameter, then either 1 , jaj , 2 or c þ and c 2 enter and exit the unit disc infinitely many times. (c) If a is a capture parameter, then 1 , jaj , 2. (d) If a is a disjoint parameter, then either 1 , jaj , 2 or jaj . 2. In the later case the orbits of the two attracting cycles are symmetric with respect to the unit disc and, hence, have the same period. Moreover, if the multiplier of one attracting cycle is l, the multiplier of the other attracting cycle is l.
Proof. We begin with statement (a). If a is an adjacent parameter, then both critical points belong to the same component of the immediate basin of attraction of a periodic cycle. Then, either 1 , jaj , 2 or jaj . 2 and both critical points are attracted to an attracting cycle in S 1 . In this last case, a belongs, by definition, to a tongue. The converse holds by symmetry.
To prove statement (b) notice that, in the bitransitive case, the immediate basin of attraction of the attracting cycle which the critical points are attracted to has at least two different connected components. If jaj . 2, by symmetry, at least one is contained in the unit disc and another one is on its complement. Thus, the critical orbits enter and exit D infinitely many times. Statement (c) follows directly from the fact that, for jaj $ 2, the critical orbits are symmetric.
The first part of (d) follows from symmetry. In order to see that the attracting cycles have conjugated multipliers, we conjugate B a via a Möbius transformation M to a rational mapB a that fixes the real line. The result follows then from the fact thatB 0 a ð zÞ ¼B 0 a ðzÞ and that M preserves the multiplier of the periodic cycles. A

Relation with cubic polynomials
In this section we introduce a quasiconformal surgery which relates, under certain conditions, the dynamics of a Blaschke product B a outside the unit disc with the ones of a cubic polynomial of the form M b ðzÞ ¼ bz 2 ðz 2 1Þ with b [ C (see Figure 7). They have z ¼ 0 as a superattracting fixed point and a second critical point, c ¼ 2=3, which is free. This one-dimensional slice (or a cover thereof) was introduced by Milnor in 1991 in a preliminary version of [21]. They have also been studied by Roesch [29], among others. In Figure 8 we show its parameter plane. We proceed to introduce the quasiconformal surgery which relates the Blaschke products B a with the cubic polynomials M b (cf. [25]). For an introduction to the tools used in quasiconformal surgery we refer to [1,5]. The idea of the surgery is to 'glue' the map R 2 ðzÞ ¼ z 2 inside D keeping the dynamics of B a outside D whenever the parameter a is such that B a j S 1 is quasisymmetrically conjugate to the doubling map u ! 2u ðmod 1Þ.
More precisely, we restrict to the set of parameters a such that jaj $ 2. For these parameters, B a j S 1 is a degree 2 cover of S 1 and is hence semiconjugate to the doubling map by a non-decreasing continuous map h a , not necessarily surjective (see Lemma 2.7). We know, by Theorem 2.9, that if jaj . 2 and the circle map B a j S 1 has neither attracting nor parabolic cycles, then h a is a quasisymmetric isomorphism of the circle. Therefore, we define X to be the set of parameters a, jaj $ 2, such that h a is a quasisymmetric conjugacy between B a j S 1 and R 2 . Let a [ X . The map h :¼ h a is quasisymmetric and conjugates B a j S 1 with the doubling map. Since h is quasisymmetric, it extends to a quasiconformal map H : D ! D (see [4,11], cf. [5]). We define the model map as follows: Proof. The map F is quasiregular since it is continuous inĈ, holomorphic outside D and locally quasiconformal in Dnf0}. Its topological degree is 3 since gluing the map z ! z 2 in D decreases the degree of B a in 1. Indeed, recall from Section 3 that there are three preimages of D and one preimage of CnD in D under B a . Instead, there are only two preimages of D and none of CnD in D under R 2 , and therefore under F.
We now define an F-invariant almost complex structure s, i.e, an almost complex structure such that F * s ¼ s, as ðF m Þ * ðH * s 0 Þ on F 2m ðDÞnF 2mþ1 ðDÞ; for m $ 1; where s 0 denotes the standard complex structure and * denotes the pullback operation. By construction, s has bounded dilatation. Indeed, s is, in D, the pull back of s 0 by a The scaling from green to orange corresponds to parameters for which the critical orbit tends to z ¼ 1.
quasiconformal map. Everywhere else either we pull back sj D by a holomorphic map (so we do not increase the dilatation) or we use the standard complex structure. Let c be the integrating map of s given by the Measurable Riemann Mapping Theorem (see [1], p. 57, [5], Theorem 1.28) such that cðH 21 ð0ÞÞ ¼ 0, cð1Þ ¼ 1 and cðc þ Þ ¼ 2=3. Then, the following diagram commutes.
The composition c + F + c 21 is a quasiregular map preserving the standard complex structure and, by Weyl's Lemma (see [1], p. 16, [5], Theorem 1.14), c + F + c 21 is holomorphic. Since this map has topological degree 3 and no poles it is a cubic polynomial. By the chosen normalization, z ¼ 0 is a superattracting fixed point and it follows that c is a quasiconformal conjugacy between M bðaÞ and B a on this region, and so are all iterates on orbits which never enter D. Therefore, for all parameters a [ X such that the orbit of the exterior critical point Oðc þ Þ never enters D all relevant dynamics are preserved (see Figure 7). The study of the parameters a [ X such that Oðc þ Þ meets D is done in the next section. The surgery described above defines a map F : X ! Y between the subset X of the parameter plane of B a (see Figure 5) and a subset Y of the parameter plane of M b (see Figure 8). By Theorem 2.9 and the fact that every parameter a with jaj . 2 such that B a j S 1 has a parabolic cycle belongs to the boundary of a tongue (which is to be proven in a forthcoming paper), the set X includes all parameters with jaj . 2 such that a does not belong to the closure of any tongue. The image set Y does not include any parameter b in the main capture component (i.e. the set of parameters for which the basin of 0 contains both critical orbits). We conjecture that F is a degree 3 cover between X and Y. Another application of this surgery construction will be explained in Section 4.4 (see Proposition 4.10).

Swapping regions: proof of Theorem B
Definition 4.7. We say that a parameter a, jaj . 2, is a swapping parameter if the orbit of the exterior critical point Oðc þ Þ enters the unit disc at least once (or equivalently if Oðc 2 Þ has points inĈn D) . A maximal open connected set of swapping parameters is called a swapping region.
The goal of this section is to describe the dynamics which may take place for swapping parameters (see Figure 9(b)). Numerical computations show that small 'copies' of the Tricorn, the connectedness locus of the antipolynomials p c ðzÞ ¼ z 2 þ c (see [6,24] and Figure 9(a)) and the Mandelbrot set appear embedded inside swapping regions (see Figures 9(c),(d)). We should not expect these 'copies' to be actual homeomorphic copies of the Tricorn (c.f. [17]). These Tricorn and Mandelbrot-like sets appear as the accumulation set of parameters for which Oðc þ Þ enters and exits the unit disc more and more times. In the limit we may have parameters having attracting cycles which enter and exit the unit disc (see Figure 9(c),(d)). In this situation, we build, in Theorem B, a polynomial-like of degree either 2 or 4. Milnor [19] worked with cubic polynomials with real parameters and described a similar structure as the one that we explain in Lemma 4.9. Using antipolynomial-like mappings (see Section 2) he related the dynamics of the bitransitive parameters with the ones of the Tricorn. We use his ideas to prove that when the polynomial-like map built in Theorem B is of degree 4, it is hybrid equivalent to a polynomial of the form p 2 c ðzÞ ¼ ð z 2 þ cÞ 2 þ c ¼ ðz 2 þ cÞ 2 þ c.  red points correspond to parameters for which Oðc þ Þ ! 1 whereas black points correspond to parameters for which Oðc þ Þ ! 0. Green points correspond to bitransitive parameters (see Figure 11 (left)), whereas yellow points correspond to disjoint parameters (see Figure 11 (right)). The red and black annuli surrounding the Tricorn and Mandelbrot-like sets correspond to parameters for which c þ enters and exists the unit disc a finite number of times before being captured by 1 or 0.
The following lemma tells us that swapping regions are disjoint from tongues (see Definition 4.4).
Lemma 4.8. A parameter a with jaj . 2 such that B a has an attracting or parabolic cycle in S 1 cannot be swapping.
Proof. Assume that B a has an attracting p-cycle kz 0 l in S 1 with multiplier l. By symmetry of B a , there is a point of the cycle, say z 0 , such that c^[ A * ðz 0 Þ. Let A be the maximal domain of the Koenigs linearization of B p a around z 0 and let f be the linearizer (see [20], Theorem 8.2 and Lemma 8.5). Again by symmetry, c^[ ›A. It follows from the invariance of S 1 with respect to B a that fðA > S 1 Þ is invariant under multiplication by l. Since f is injective, multiplication by l maps fðA > DÞ into fðA > DÞ or into fðA > ðĈn DÞÞ. Since B a j S 1 preserves the orientation of S 1 we conclude that lÁfðA > DÞ , fðA > DÞ. Therefore, Oðc 2 Þ cannot exit the unit disc and the parameter is not swapping. The parabolic case is derived similarly takingˆto be the maximal petal having the critical points on its boundary (see [20], Theorems 10.9 and 10.15).
A We are interested in the hyperbolic regions contained in the swapping areas. We shall study them using the theories of polynomial and antipolynomial-like mappings. We focus on the bitransitive parameters which, for jaj . 2, are necessarily inside swapping regions (see Theorem 4.5(b)).
We recall some notation from Section 3. For jaj . 2 the unit circle has two preimages different from itself and not intersecting S 1 , say g i , D and g e , Cn D (see Figure 1(d)). The map B a sends g i and g e bijectively to S 1 . Moreover, let V i be the region bounded by g i and contained in D and let V e be the region bounded by g e and contained in Cn D. The maps B a j V i : V i !Ĉn D and B a j V e : V e ! D are conformal (there is only one preimage of z ¼ 1 in D and one preimage of z ¼ 0 in Cn D), so c^Ó V i < V e . We now prove a lemma which deals with the period of attracting and parabolic cycles for parameters inside swapping regions. Lemma 4.9. Let a, jaj . 2, be a parameter inside a swapping region. If B a has an attracting or parabolic cycle, then its period is at least 3. Moreover, if a is bitransitive, the period is even.
Proof. First of all notice that, from Lemma 4.8 and invariance of S 1 , no component of the basin of attraction of the cycle can intersect neither g i nor g e . A parabolic or attracting cycle needs to have a critical point in its immediate basin of attraction. The component in which the critical point lies is contained neither in V i nor in V e . Moreover, since the periodic cycle needs to enter and exit the unit disc, the immediate basin of attraction of the cycle has a component in V i and another one in V e . Then, the immediate basin of attraction has at least three different components and, hence, the cycle has at least period 3. Now assume that a is bitransitive. Suppose without loss of generality that the component which contains c þ is mapped under k . 0 iterates to the component which contains c 2 . Because of symmetry, the first return map from the component of c 2 to component of c þ also takes k iterates. Hence, the period of the attracting cycle is 2k. A We proceed now to prove Theorem B.
Poof of Theorem B. First of all notice that, due to Lemma 4.8, A * ðkz 0 lÞ neither intersects g e nor g i . Since the cycle enters and exits the unit disc, A * ðkz 0 lÞ has at least one connected component entirely contained in V e . Let A * ðz 0 Þ be the connected component of A * ðkz 0 lÞ containing c þ . Let n 0 [ N be minimal such that B n 0 a 0 ðz 0 Þ ¼ z n 0 [ V e . Let S 0 be the connected component of B 2n 0 a 0 ðV e Þ, containing c þ (and hence A * ðz 0 Þ). The set S 0 is simply connected by Corollary 2.2 since V e is simply connected and B n 0 a 0 j S 0 has a unique critical point. Recursively define S n to be the connected component of B 21 a 0 ðS n21 Þ containing the point z 2n of the cycle (recall that the subindices of the cycle are taken in Z=pZ). Again by Corollary 2.2, the components S n are simply connected for all n . 0. Let p 0 [ N be minimal such that c þ [ S p 0 . Since ›V e > S 1 ¼ Y, we have that ›V e > ›S p 0 2n 0 ¼ Y and that S p 0 2n 0 , V e . Therefore, we have S p 0 , S 0 . Notice that p 0 is a divisor of p.
The map B a 0 j S n : S n ! S n21 is conformal if S n contains no critical point and 2-to-1 if it contains c þ or c 2 (it cannot contain both critical points at the same time since S n > S 1 ¼ Y). Hence, the triple ðB p 0 a 0 ; S p 0 ; S 0 Þ is a polynomial-like map of degree 4 or 2 depending on whether there is some S q 0 containing c 2 or not. As in Lemma 4.9, if such q 0 exists, p 0 ¼ 2q 0 . Notice that, if the parameter is bitransitive, this q 0 exists and, therefore, the degree is 4. Since the condition c þ ðaÞ [ S p 0 ðaÞ , S 0 ðaÞ is open, the polynomial-like map can be defined for an open set of parameters W around a 0 . From now on we consider a [ W.
We now use antipolynomial-like mappings to see that, in the case of degree 4 polynomial-like mapping, the degree 4 polynomial to which ðB 2q 0 a ; S 2q 0 ; S 0 Þ is hybrid equivalent can be taken of the form p 2 c ðzÞ ¼ ðz 2 þ cÞ 2 þ c. See Section 2 for an introduction to antipolynomial-like mappings. We proceed to construct an antipolynomial-like map ðf ; S 2q 0 ; I ðS q 0 ÞÞ of degree 2, where I ðzÞ ¼ 1= z denotes the reflection with respect to S 1 . This antipolynomial-like map is hybrid equivalent to an antipolynomial of the form z 2 þ c. The result then follows if f ðI ðS q 0 ÞÞ ¼ S 0 and ðf 2 ; It is easy to see that S q 0 ,S 0 . Indeed, taking n 0 as in the definition of S 0 , by symmetry, B n 0 a ðS 0 Þ ¼ V i . Since B n 0 a ðS q 0 Þ is contained in V i , we conclude that S q 0 ,S 0 (see Figure 10). From S q 0 ,S 0 we can deduce that S 2q 0 ,S q 0 . Finally, take f ¼ I + B q 0 a . Since B a ¼ I + B a + I we have that f 2 ¼ B 2q 0 a . Then, the antipolynomial-like map ðI + B q 0 a ; S 2q 0 ;S q 0 ¼ I ðS q 0 ÞÞ satisfies the desired conditions. A Theorem B tells us that all bitransitive parameters contained in swapping regions can be related to the dynamics of p 2 c ðzÞ, where p c ðzÞ ¼ z 2 þ c, since the polynomial-like map has degree 4. However, notice that if an antipolynomial p c ðzÞ has an attracting cycle of even period 2q, then p 2 c ðzÞ has two disjoint attracting cycles of period q. Therefore, there are also parameters which appear as disjoint parameters in the Tricorn-like sets in the parameter plane of the Blaschke family (see Figure 9(c) and Figure 11 (right)). These disjoint parameters also lead to degree 4 polynomial-like maps. Finally, the polynomiallike maps of degree 2 obtained from the other disjoint parameters are hybrid equivalent Figure 10. Sketch of the situation described in Theorem B for the degree 4 case. to quadratic polynomials z 2 þ c. These parameters correspond to the ones inside the small Mandelbrot-like sets observed by means of numerical computations (see Figure 9(d)).

Parametrizing hyperbolic components: proof of Theorem C
The aim of this section is to study the multiplier map of the bitransitive and disjoint hyperbolic components of the Blaschke family B a for parameters a such that jaj . 2. Recall that a hyperbolic component is a connected component of the set of parameters for which B a is hyperbolic. The section is structured as follows: first we prove a proposition that is useful later on, then we see that the multiplier map is a homeomorphism between any disjoint hyperbolic component and the unit disc proving Theorem C and finally we study the bitransitive case.
The following proposition tells us that, given B a with jaj . 2, the boundaries of the connected components of the basin of attraction of every attracting cycle contained in C * nS 1 are Jordan curves. The result is a direct consequence of the relation of the family B a with polynomials which has been described in Proposition 4.6 and Theorem B, respectively.
Proposition 4.10. Assume that B a has an attracting cycle kz 0 l which is contained in C * nS 1 : Then, the boundaries of the connected components of the basin of attraction Aðkz 0 lÞ are Jordan curves.
Proof. It follows from the hypothesis of the proposition that jaj . 2 since for 1 , jaj # 2 any attracting cycle other than z ¼ 0 or z ¼ 1 is contained in S 1 and for jaj , 1 there are no attracting cycles in C * . It follows from Proposition 4.6 and Theorem B that the closure of every connected component of A * ðkz 0 lÞ is homeomorphic to the closure of a connected component of the immediate basin of attraction of a bounded attracting cycle of a polynomial. Since the boundary of every bounded Fatou component of a polynomial other than a Siegel disc is a Jordan curve (see [30]), the boundary of every connected component of A * ðkz 0 lÞ is also a Jordan curve. Finally, since all critical points are contained in the immediate basins of attraction of attracting cycles, the closure of every connected component U of Aðkz 0 lÞnA * ðkz 0 lÞ is mapped homeomorphically to the closure of a connected component of A * ðkz 0 lÞ and, therefore, ›U is a Jordan curve too. A We now prove Theorem C. The main idea of the proof is to build a local inverse of the multiplier map L around every Lða 0 Þ [ D. It is done performing a cut-andpaste surgery to change the multiplier of the attracting cycle using a degree 2 Blaschke product with an attracting cycle of the desired multiplier as a model (see [5], Chapter 4.2).
Proof of Theorem C. Consider the family of degree 2 Blaschke products where l [ D. Notice that b l ðDÞ ¼ D. They have 0 and 1 as attracting fixed points of multipliers l and l, respectively. The only other fixed point ðð1 2 lÞ=ð1 2 lÞÞ [ S 1 is repelling. Every degree 2 holomorphic self-map of D having z ¼ 0 as fixed point with multiplier l and a repelling fixed point ð1 2 lÞ=ð1 2 lÞ [ S 1 is of the form b l . Its Julia set satisfies J ðb l Þ ¼ S 1 . Furthermore, if jlj , r , 1, D l ¼ b 21 l ðD r Þ is a simply connected open set which compactly contains the disc of radius r, D r ¼ fz; jzj , r}, whereas b l ðD r Þ is compactly contained in D r .
Let a 0 [ U and let l 0 be the multiplier of the attracting cycle kz 0 l of period p of B a 0 such that c þ [ A * ðz 0 Þ. Since there is no other critical point in A * ðkz 0 lÞnA * ðz 0 Þ and ›A * ðz 0 Þ is a Jordan curve (Proposition 4.10), the map B p a 0 : A * ðz 0 Þ ! A * ðz 0 Þ has degree 2 and a unique fixed point z 0 0 in ›A * ðz 0 Þ. Let R : A * ðz 0 Þ ! D be the Riemann map sending z 0 to 0 and z 0 0 to ð1 2 l 0 Þ=ð1 2 l 0 Þ. The map R + B p a 0 + R 21 is, by construction, the restriction to D of the Blaschke product b l 0 ðzÞ. Fix r 0 and r so that 0 # jl 0 j , r 0 , r , 1. We proceed now to perform a surgery to the product b l 0 which changes the multiplier of the attracting fixed point 0 to l for any jlj , r 0 . Let D l 0 ¼ b 21 l 0 ðD r Þ and let A l 0 denote the annulus D l 0 nD r . Define g l : D ! D as where h l is chosen to be a quasiconformal map which interpolates b l and b l 0 depending continuously on l. Such an interpolating map can be taken since the boundary maps g l j ›A l 0 are degree 2 analytic maps on analytic curves. The inner boundary map depends continuously on l whereas the outer map is independent of it. Therefore, the map h l : A l 0 ! A l , where A l denotes the annulus D r nb l ðD r Þ, can be chosen to be a quasiconformal covering map of degree 2 which depends continuously on l (see Exercise symmetric with respect to S 1 which has z ¼ 0 and z ¼ 1 as superattracting cycles of local degree 3. Therefore,B l is a Blaschke product of the form Bã ðlÞ;tðlÞ (see Equation (2)). Since F l has a unique critical point in CnD which depends continuously on l and f l ðzÞ depends continuously on l for each z [Ĉ, Bã ;t has a unique critical point c þ ðlÞ [ CnD which depends continuously on l. Therefore, since Bã ðlÞ;tðlÞ fixes x 0 [ S 1 , we have by Lemma 3.2 thatãðlÞ andtðlÞ depend continuously on l. Finally, by lemma 3.1, Bã ðlÞ;tðlÞ is conjugate to a Blaschke product B aðlÞ (see Equation (1)), where aðlÞ ¼ãðlÞe 2pitðlÞ=3 depends continuously on l.
To finish the proof we check that aðl 0 Þ ¼ a 0 and, therefore, every aðlÞ belongs to the same hyperbolic component U as a 0 . We have not justified that the quasiconformal interpolating map h l 0 equals b l 0 and, hence, B a 0 and Bã ðl 0 Þ;t 0 might be distinct. However, the integrating map f l 0 is a conformal conjugacy between them inĈnAðkz 0 lÞ < AðkIðz 0 ÞlÞ and is a quasiconformal conjugacy in a neighbourhood of their Julia sets. Definef l 0 to be the conformal map from Aðkz 0 lÞ < AðkIðz 0 ÞlÞ to Aðkf l 0 ðz 0 ÞlÞ < AðkIðf l 0 ðz 0 ÞÞlÞ such that, restricted to every connected component, coincides with the Riemann map normalized so that the attracting cycles kz 0 l, kI ðz 0 Þl, kz 0 0 l and kI ðz 0 0 Þl are mapped to kf l 0 ðz 0 Þl, kI ðf l 0 ðz 0 ÞÞl, kf l 0 ðz 0 0 Þl and kI ðf l 0 ðz 0 0 ÞÞl and their preimages are in correspondence. Since B p a 0 is conjugate to b l 0 (respectively, b l 0 ) in A * ðz 0 Þ (respectively, A * ðI ðz 0 ÞÞ) and so is B p aðl 0 Þ;t 0 in A * ðf l 0 ðz 0 ÞÞ (respectively, A * ðI ðf l 0 ðz 0 ÞÞÞ), the conformal mapf l 0 is a conjugacy. Moreover, it extends to the boundary of every connected component of the basins of attraction since they are Jordan domains by Proposition 4. 10. Given that f l 0 and f l 0 conjugate B a 0 and Bã ðl 0 Þ;t 0 in ›Aðkz 0 lÞ < ›AðkIðz 0 ÞlÞ , JðB a 0 Þ they coincide since they map periodic points to periodic points. Consequently, the map w l 0 defined as f l 0 in CnðAðkz 0 lÞ < AðkIðz 0 ÞlÞÞ andf l 0 in Aðkz 0 lÞ < AðkIðz 0 ÞlÞ is a global conjugacy. Moreover, since w l 0 is quasiconformal inĈnð›Aðkz 0 lÞ < ›AðkIðz 0 ÞlÞÞ, coincides with f l 0 in J ðB a 0 Þ and f l 0 is quasiconformal in a neighbourhood of J ðB a 0 Þ, w l 0 is quasiconformal by Rickman's Lemma (cf. [5,13]). Since w l 0 is conformal a.e. inĈ, it is 1-quasiconformal and therefore conformal map ofĈ by Weyl's Lemma. Since w l 0 fixes 0 and 1, leaves S 1 invariant and fixes x 0 [ S 1 , we conclude that w l 0 is the identity and Bã ðl 0 Þ;t 0 ¼ B a 0 .
For every a 0 [ U we have constructed a continuous local inverse to the multiplier map L : U ! D. Therefore, L is a homeomorphism. A We finish this section giving some ideas of what happens with bitransitive parameters (see Figure 9(c)). It follows from Theorem B that these parameters are strongly related to the quadratic antiholomorphic polynomials p c ðzÞ ¼ z 2 þ c. Indeed, the polynomial-like map constructed in Theorem B is hybrid equivalent to a degree 4 polynomial of the form p 2 c ðzÞ with a bitransitive attracting cycle. Therefore, the polynomial p c ðzÞ also has an attracting cycle of odd period since, otherwise, the polynomial p 2 c ðzÞ would have two disjoint attracting cycles. Nakane and Schleicher [24] studied the parameter plane of the antipolynomials p c;d ¼ z d þ c and, in particular, p c;2 ðzÞ ¼ p c ðzÞ. If the period of the cycles of a hyperbolic component was even, they proved a result analogous to Theorem C. They also showed that the multiplier map is not a good model for the odd period hyperbolic components. The reason why the multiplier map is not good for this case is the fact that the antiholomorphic multiplier ð›=› zÞf k ðz 0 Þ of a cycle kz 0 l of odd period k of an antiholomorphic map f ðzÞ is not a conformal invariant, only its absolute value is. They proved that the multiplier of the period k cycle kz 0 l of the holomorphic map f 2 ðzÞ equals the square of the absolute value of the previous antiholomorphic multiplier. Given a bitransitive hyperbolic component U of p 2 c ðzÞ, it also follows from their work that the set of parameters c [ U for which the attracting cycle has multiplier l [ ð0; 1Þ is a Jordan curve and that U contains a unique parameter c 0 for which the cycle is superattracting. We expect a similar result for bitransitive hyperbolic components of the Blaschke family B a , but we only prove, for the sake of completeness, the following result.
Proposition 4.11. Let kz 0 l be a bitransitive cycle of a Blaschke product B a as in (1) with jaj . 2. Then it has non-negative real multiplier.
Proof. By Lemma 4.9, the cycle kz 0 l has even period 2q. Let I ðzÞ ¼ 1= z. By symmetry, I ðB q a ðz 0 ÞÞ ¼ z 0 . Therefore, z 0 is a fixed point of the antiholomorphic rational map f ¼ I + B q a . Moreover, B 2q a ¼ f 2 . Therefore, the multiplier of the cycle is given by