Singular perturbations of z n with a pole on the unit circle

We consider the family of complex maps given by where n, d ≥ 1 are integers, and a and λ are complex parameters such that |a| = 1 and |λ| is sufficiently small. We focus on the topological characteristics of the Julia and Fatou sets of .


Introduction
In the last few years, a number of papers have appeared that deal with the dynamics of functions obtained by a perturbation of the complex function z 7 ! z n by adding a pole at the origin [3,[5][6][7]. These rational functions are of the form f l ðzÞ ¼ z n þ l=z d . When jlj p 1, we consider this function as a singular perturbation of z n . The reason for this terminology is that when l ¼ 0, the map is z n and the dynamical behaviour is well understood. When l -0, however the degree jumps to n þ d and the dynamical behaviour changes significantly. The interest in this type of perturbation arises from the application of Newton's method to find the roots of a family of polynomials that, at one particular parameter value, has a multiple root. At this parameter value, the Newton iteration function undergoes a similar type of singular perturbation.
In [8], we investigated a more general class of functions for which the pole is not located at the origin but rather is located at some other point in the complex plane that does not lie on the unit circle. In particular, we considered the family of functions given by f l;a ðzÞ ¼ z n þ l ðz 2 aÞ d ; where n $ 2 and d $ 1 are integers, and a and l are complex parameters where jaj -0; 1 and jlj is sufficiently small. In this paper, we continue the study of the family f l;a . In the first part, we study the dynamics of equation (1) when the pole a is on the unit circle and jlj is sufficiently small.
In the second part, we focus on the dynamics of equation (1) when n ¼ 1, d $ 1 and a; l [ C. Our goal is to describe the topology and dynamics of the Julia set of f l;a , i.e. the set of points where the family of iterates of f l;a is not a normal family in the sense of Montel. Equivalently, the Julia set is the closure of the set of repelling periodic points of f l;a . We denote the Julia set by J ¼ Jðf l;a Þ. The complement of the Julia set is called the Fatou set.
We first consider the case when n $ 2. When l ¼ 0, infinity and the origin are superattracting fixed points and the Julia set is the unit circle. When we add the perturbation by setting l -0 but very small, several aspects of the dynamics remain the same, but others change dramatically. For example, when l -0, the point at 1 is still a superattracting fixed point and there is an immediate basin of attraction of 1 that we call B ¼ B l . On the other hand, there is a neighbourhood of the pole a that is now mapped d-to-1 onto B. When this neighbourhood is disjoint from B we call it the trap door and denote it by T ¼ T l . Every point that escapes to infinity and does not lie in B has to do so by passing through T. Since the degree of f l;a changes from n to n þ d, 2d additional critical points are created. The set of critical points includes 1 and a whose orbits are completely determined, so there are n þ d additional 'free' critical points. The orbits of these points are of fundamental importance in characterizing the Julia set of f l;a .
When l is sufficiently small and a -0, we may find d 1 . 0 such that, if jlj , d 1 , f l;a still has an attracting fixed point q ¼ q l near the origin. Throughout this paper, we assume that jlj , d 1 . Let Q ¼ Q l denote the immediate basin of attraction of q. The set of n þ d 'free' critical points may be divided into two groups: the first group consists of n 2 1 critical points that are attracted to q. These are the critical points that bifurcate away from the origin when l becomes nonzero. The remaining d þ 1 critical points surround the pole a and, for jlj p 1, they are mapped close to a n . It follows that the dynamics of this family of functions is determined by the behaviour of this set of d þ 1 critical points and the position of a when jlj is small.
We first review the case when jaj -1. When jlj p 1 and 0 , jaj , 1, the orbits of the d þ 1 critical points that lie around a converge to the fixed point q near the origin, and when jaj . 1 they converge to 1. The following theorem summarizes some of the known results studied in [5,8,11].
The case a ¼ 0 with n ¼ d ¼ 2 is very different. In this case, there are infinitely many open sets in any neighbourhood of l ¼ 0 in which the Julia sets corresponding to these parameters are all Sierpinski curves, but any two such maps whose parameters are drawn from different open sets have non-conjugate dynamics (see [6]). Moreover, in this case, when l ! 0 the Julia sets of f l;a converge to the unit disk (see [9]). The cases when a ¼ 0; d ¼ 1 and n $ 2 are also very different and are still under study. Figures 1 and 2 show examples of each of the cases discussed above. The differences between the cases jaj -1 and jaj ¼ 1 can be explained as follows. For sufficiently small l -0 and outside a small neighbourhood of the pole a the map f l;a ðzÞ behaves approximately like z n , since the distance between them is small. Then, the set of d þ 1 critical points that surround a is mapped close to a n . This implies that when jaj -1 the orbits of these critical points behave as 'one' critical orbit. Instead, when jaj ¼ 1 the critical points that surround the pole a behave independently. Some of them can converge to q, some of them can converge to 1 and some of them may be related to a Fatou component different from B and Q, or even belong to the Julia set of f l;a . Hence, a complete description of the Julia set can be challenging. However, when these critical points belong to B and Q we can give a detailed description of the Julia and Fatou sets of these maps. Let S a denote the set of d þ 1 critical points that surround the pole a when jlj is small.
In Theorem 1.2, we describe some important components of the Fatou set, namely the basins of attraction of q and 1. These results need no assumptions on the behaviour of the critical points in S a since the order d of the pole a is enough to assure that some of these critical points belong to B. An important consequence of Theorem 1.2 is the following. When d . 1, the pole a lies in B, that is, for jlj sufficiently small these maps have no trap door as in the case when jaj . 1 (see [8]). Figure 3 displays the dynamical plane of f l;a corresponding to Theorem 1.2.
If the critical points in S a are distributed between B and Q and, for jlj sufficiently small the number of critical points in B and Q remains constant, then we can understand the structure and dynamics on the Julia set of f l;a . Let jS a > Bj and jS a > Qj denote the number of critical points from S a that lie in B and Q, respectively. There are two possibilities shown in the next theorem.  Figure  2. These plots represent the typical case when jaj ¼ 1 and d $ 5. In this case, B is completely invariant and infinitely connected and the basin of attraction of q has infinitely many simply connected components. (b) Magnification around the pole a. Theorem 1.3 (Structure of the Julia and Fatou sets for jaj ¼ 1). Let n $ 2, d $ 1, jaj ¼ 1 and suppose that for l sufficiently small S a , B < Q and that jS a > Bj and jS a > Qj remain constant, then either (a) Exactly one critical point from S a belongs to B and the Julia set J is a quasi-circle that surrounds the origin where f l;a : J 7 ! J is conjugate to z 7 ! z nþd on the unit circle. The Fatou set consists of two completely invariant discs, namely B and Q; or else, (b) the Julia set J consists of countably many simple close curves and uncountably many point components that accumulate on each one of these curves. Only one of these curves surrounds the origin. The Fatou set consists of one infinitely connected component and infinitely many discs.
Notice that if S a , B (resp. S a , Q) then we are in part (b) of the above theorem. This is exactly what happens in the case when jaj . 1 (resp. jaj , 1) described in Theorem 1.1. For this reason, in the case jaj -1, the situation described in Theorem 1.3 part (a) is not observed. This new possibility when jaj ¼ 1 is allowed by the fact that the critical points in S a behave independently and in a very specific manner. We also have Theorem 1.4 (Dynamics on the Julia set). Suppose f l 1 ;a 1 and f l 2 ;a 2 are two functions such that they both lie in one of the cases distinguished in Theorem 1.3. In other words, for jlj sufficiently small, the set S a , B < Q and the number of critical points in B and Q coincide for both functions but the exact position of the pole a or of these critical points is arbitrary. Then, there exists 1 . 0 such that, for jl 1 j, jl 2 j , 1, these maps are conjugate on their Julia sets. Moreover, the dynamics are determined by a specific quotient of a subshift of finite type.
We can actually prove the existence of the Julia sets described in Theorem 1.3. Some of the results in the next theorem hold only for sectors of values of l in the parameter l-plane.
Let Arg(z) denote the argument of the complex number z. Then, given two real numbers a and b such that 0 # a , b # 2p, we define a sector S a;b of values of the parameter l in the usual way, that is, S a;b ¼ {l; a , ArgðlÞ , b}. Moreover, inside each one of these sectors Theorem 1.4 holds.
The case when d ¼ 2 is very interesting since for some values of l sufficiently small we can obtain very different topological and dynamical behaviour.
The fact, that for jlj sufficiently small we have that q and 1 are attracting fixed points implies that the Julia set cannot be totally disconnected. In other words, the Fatou set consists of at least two disjoint open sets. The minimum of two is attained by part (a) of Theorem 1.5 and, in this case, the Julia set is the common boundary of Q and B. Remark 1. As we mentioned, when d . 1 the basin of attraction of infinity is completely invariant (that is, there is no trap door). By the above theorem, we also know that when d ¼ 1 there is a sector of parameters in the l-plane for which this is also the case. Numerical experiments suggest that when jaj ¼ 1 and jlj ! 1, the basin of attraction of 1 is always completely invariant. Figures 4 and 5 display the dynamical plane of f l;a corresponding to the different cases that appear in Theorem 1.5.
Notice that very interesting bifurcations happen when we fix n and d so that ðð1=nÞ þð1=dÞÞ , 1 and we also fix l sufficiently small and let the parameter a vary. The structure of the Julia set changes dramatically when the pole a moves away from the origin. When a ¼ 0, we have that the Julia set of f l;a is a Cantor set of simple closed curves that surrounds the origin (see Figure 1(a)). When 0 , jaj , 1, there is a neighbourhood of the origin in the l-plane where the Julia set of f l;a consists of countably many simple closed curves only one of which surrounds the origin (namely, ›B) and uncountably many point components that accumulate on these curves. The preimages of ›B lie inside ›B (see Figure 2(a)). When jaj ¼ 1, we see that there is a sector of parameters in the l-plane for which the Julia set of f l;a has the same topology as in the previous case but the only curve that surrounds the origin is now ›Q and the rest of the curves lie outside ›Q (see Figure 3). When jaj ¼ 1 and for some values of n and d (see Theorem 1.5) there is also a sector of parameters in the l-plane for which the Julia set becomes a simple closed curve that surrounds the origin (see Figure 4). Finally, when jaj . 1 there is a neighbourhood of the origin in the l-plane for which the structure of the Julia set again consists of countably many simple closed curves only one of which surrounds the origin (namely, ›Q) and uncountably many point components that accumulate on these curves (see Figure 2(b)). In between each one of these states, the Julia set suffers great transformations due to the fact that the critical points in S a are now acting independently. A complete description of these transitions between states goes beyond the scope of this paper.  In the second part of the paper, we focus on the family given by equation (1.1), when n ¼ 1. In this case, the point at infinity is always in the Julia set and this causes major changes in the dynamical behaviour of f l;a . Also, when n ¼ 1, we can conjugate f l;a via a Möbius map to make it completely independent of the parameters a and l. For these reasons, the behaviour of the map f l;a with n ¼ 1 is completely different from the previous cases and the characteristics of the Julia and Fatou sets of f l;a reflect these changes. We have Theorem 1.6. Let n ¼ 1 and d $ 1 then for all parameters a; l [ C, the map f l;a is conformally conjugate to z þ 1=z d . In particular, the Julia set of f l;a is connected and the Fatou set contains all the points attracted to the unique parabolic fixed point at infinity. When d ¼ 1, the Fatou set consists of two simply connected regions; otherwise, it consists of infinitely many simply connected components. The rest of the paper is organized as follows. In Section 2, we obtain some basic results about the function f l;a when n $ 2. In Section 3, we prove Theorems 1.2 -1.5. Finally, in Section 4, we study the dynamics of f l;a when n ¼ 1 and prove Theorem 1.6.

Preliminaries
Let n $ 2, d $ 1 and jaj ¼ 1. A straightforward computation shows that, when l -0, f l;a has n þ d critical points that satisfy the equation When l ¼ 0, this equation has n þ d roots, the origin with multiplicity n 2 1 and a with multiplicity d þ 1. By continuity, for small enough jlj, these roots become simple zeros of f 0 l;a that are approximately symmetrically distributed around the origin and the pole a.
As a consequence, when jlj is small, n 2 1 of the critical points of f l;a are grouped around 0, near the fixed point q, while d þ 1 of the critical points are grouped around the pole a.
Let c ¼ c l be one of the n þ d critical points of f l;a given by equation (2). Replacing z by c in equation (2), we find lðc 2 aÞ 2d ¼ ðn=dÞc n21 ðc 2 aÞ and so the critical value v corresponding to c is given by Note that as l ! 0, the fixed point q l ! 0 as well. From equation (3) it follows that, if c ! 0, then v ! 0. Similarly, if c ! a, then v ! a n . We use S a to denote the set of d þ 1 critical points around a and we use S q to denote the set of n 2 1 critical points around q. We have Lemma 2.1. When jlj tends to zero, the critical values corresponding to the critical points in S q tend to q and the critical values corresponding to the critical points in S a tend to a n .
To describe the structure of the Julia set, we first need to give an approximate location for this set. Roughly speaking, when jlj is small, the Julia set of f l;a lies in a small annulus around the unit circle.  Proof. Fix n, d and a. For the first part, we have jz 2 aj $ jaj 2 jzj ¼ s . 0, so that j f l;a ðzÞj # jzj n þ jlj jz 2 aj d # ð1 2 sÞ n þ jlj s d : Let jlj , s d ½ð1 2 sÞ 2 ð1 2 sÞ n . Then, j f l;a ðzÞj , ð1 2 sÞ n þ s d ½ð1 2 sÞ 2 ð1 2 sÞ n s d ¼ 1 2 s: As a consequence, j f l;a ðzÞj , jzj and so the orbit of z converges to the fixed point q near the origin. Therefore, z lies in Q.
For the second part, we have jz 2 aj $ jzj 2 jaj ¼ s . 0, so that Let jlj , s d ½ð1 þ sÞ n 2 ð1 þ sÞ. Then, Hence, j f l;a ðzÞj . jzj and the orbit of z converges to 1 so that z [ B. A The following result gives us a simple procedure to verify when a point belongs to Q or B.
Then, for all s * , s , 1 we have that Proof. For the first part, notice that for s . 0, ð1 2 sÞ n , 1 2 ns. Then, simple computations show that ðn 2 1Þs dþ1 , s d ½ð1 2 sÞ 2 ð1 2 sÞ n : The condition s . s * is equivalent to jlj , ðn 2 1Þs dþ1 and the result follows from Proposition 2.2. The second part follows in a similar way by noticing that for s . 0, we have 1 þ ns , ð1 þ sÞ n . A In order to prove our main theorems, we need to obtain more precise results regarding the location of the critical points in S a and their corresponding critical values. To simplify the notation, we introduce two new variables d ¼ ðld=ðna n21 ÞÞ 1=ðdþ1Þ and 1 ¼ ðjljd=nÞ ð1=ðdþnÞÞ , the first one is a multivalued complex function of l and a, and the second one is a real function of jlj. Both parameters play a major role in the rest of this paper.
Let c be a critical point of f l;a in S a and let v ¼ f l;a ðcÞ be its corresponding critical value. We need four lemmas all of which hold for jlj sufficiently small. In Lemma 2.4, we prove that the distance between the critical point c and the pole a is bounded by 1. In Lemma 2.5, we find an approximation for c that we denote byc. In Lemma 2.6, we obtain an approximation for v that we denote byṽ; the distance between v andṽ will be proved to be smaller or equal to 1jdj. Finally, in Lemma 2.7, we find a criterion to prove when the critical value v belongs to B or Q.
Proof. Fix n; d and a. Let R 0 . 0 and R a . 0 be two real numbers such that R n21 0 R dþ1 a ¼ jljd=n. Consider the closed disc of radius R 0 centred at the origin, that is, D 0 ¼ {z : jzj # R 0 }, and the closed disc of radius R a centred at a, that is, D a ¼ {z : jz 2 aj # R a }. The critical points of f l;a belong to D 0 < D a since all points outside this union verify jzj n21 jz 2 aj dþ1 . jljd=n (see equation (2)). By hypothesis c [ S a and since jaj ¼ 1, for jlj sufficiently small, we have that jcj . R 0 obtaining thus that jc 2 aj # R a . Now, let R 0 ¼ ðjljd=nÞ 1=ðdþnÞ and R a ¼ ðjljd=nÞ 1=ðdþnÞ and the lemma follows. A For jlj sufficiently small, we have that the d þ 1 critical points in S a can be approximated bỹ c ¼ a þ d. These points are the vertices of a regular polygon of d þ 1 sides centred at a. Moreover, if c andc are a critical point and its approximation we have jc 2cj # n 2 1 d þ 1 2 ðnþdÞ=ðdþ1Þ 1 jdj: Proof. Fix n; d and a. To find approximations of the critical points in S a , we use the fact that solving equation (2) is equivalent to computing fixed points of the multivalued function T(z) defined by We remark that there are d þ 1 possible different choices for the function T that are the ðd þ 1Þ branches of the map given by equation (4). Starting with the initial point a we find an approximate value ofc given bỹ : It is clear that the values ofc form the vertices of a regular polygon with d þ 1 sides centred at a (Figure 7). Since c is the fixed point of T, we can obtain an upper bound for the distance between the critical point c and the approximate valuec. We have jc 2cj ¼ jTðcÞ 2 TðaÞj # jT 0 ðjÞkc 2 aj; where j is a point in the segment joining c and a. From Lemma 2.4, we know that the critical points in S a tend to a when jlj tends to zero. Then, let jlj be small enough so that jjj $ 1=2. We get  Proof. Fix n; d and a. For part (a), we first prove thatṽ belongs to B, and then we verify that this implies that v also belongs to B. To prove thatṽ belongs to B, we show that for jlj sufficiently small the condition jṽj . 1 þ s * with s * ¼ ðjlj=ðn 2 1ÞÞ 1=ðdþ1Þ , is satisfied. Then, Corollary 2.3 implies the result. From the definitions of jdj and s * it follows that when n, d $ 2, we have jdj $ s * . Then, it is enough to show that jṽj . 1 þ jdj; from hypothesis, this reduces to show jṽj $ 1 þ kjdj^Oðjdj 2 Þ . 1 þ jdj: For jlj sufficiently small and k . 1, we have that ðk 2 1Þjdj^Oðjdj 2 Þ . 0. Thus, we conclude thatṽ belongs to B. Finally, we need to show that v ¼ f l;a ðcÞ andṽ ¼ f l;a ðcÞ are close enough to assure that v also belongs to B. It follows from Lemma 2.6 that the distance between v andṽ is bounded by 1jdj. Hence, we have that jvj $ jṽj 2 jv 2ṽj $ 1 þ kjdj^Oðjdj 2 Þ 2 1jdj: Using the fact that k . 1 we conclude that, for jlj sufficiently small, v also belongs to B and the first part follows. The second part follows in a similar way. The details are left to the reader. A In the next proposition, we prove that controlling the behaviour of some critical points we can obtain information about the topology of the basin of attraction of 1 and q. We have Proof.
(a) We prove the result by contradiction. Let c be a critical point in S a such that its corresponding critical value v ¼ f l;a ðcÞ belongs to B. Assume that B is not completely invariant. Then, there exists a preimage of B disjoint from B; we call this preimage T. Since the only preimage of 1 different from itself is a, we conclude that a [ T. Now, B is mapped to itself at least n-to-1 because 1 [ B, and T is mapped to B at least d-to-1 because a [ T. Since the map is of degree n þ d we conclude that c Ó T; B. This is a contradiction to the fact that v [ B.
Then, B is completely invariant as we wanted to show. (b) First assume that B and Q are completely invariant and simply connected sets. This implies that the pole a belongs to B. We have that f l;a maps Q to itself, and B to itself both in an ðn þ dÞ-to-1 fashion. On the one hand, the connectivity of Q is m ¼ 1, the degree of the map is s ¼ n þ d and the number of critical points in Q is N ¼ n 2 1 þ x; that is, n 2 1 close to 0 and x close to a. By the Riemann -Hurwitz formula, we get m 2 2 ¼ sðm 2 2Þ þ N; so that x ¼ d:

A. Garijo and S.M. Marotta
On the other hand, the connectivity of B is m ¼ 1, the degree of the map is s ¼ n þ d and the number of critical points in B is N ¼ n 2 1 þ d 2 1 þ y; that is, n 2 1 at 1, d 2 1 at the pole a and y close to a. By the Riemann -Hurwitz formula, we get m 2 2 ¼ sðm 2 2Þ þ N; so that y ¼ 1; and one direction of the implication follows.
For the other direction of the implication, assume that one critical point in S a belongs to B and the other d critical points in S a belong to Q. It follows from part (a) that B is completely invariant. The fact that B is completely invariant implies that Q is simply connected. To see this notice that when B is completely invariant then the Julia set is equal to the boundary of B. Now, consider any Jordan curve g completely contained in Q. Let U 0 and U 1 be the two components of Cng. Without loss of generality, we can assume that B is contained in U 0 . By construction, U 1 cannot contain points in the Julia set, hence U 1 is contained in Q proving thus that Q is simply connected.
Now, consider f l;a : Q ! Q. The connectivity of Q is m ¼ 1, the number of critical points in the domain is N ¼ n 2 1 þ d; that is, n 2 1 close by 0 and d close to a, and the degree of the map is s. By the Riemann -Hurwitz formula, we get and then Q is completely invariant. Finally, since Q is completely invariant we have that B is simply connected and the result follows. (c) Since B contains at least two critical points from S a , it follows from part (a) that B is completely invariant. Then, Q is simply connected. Since B is the immediate basin of attraction of a superattracting fixed point, then B is either simply connected or infinitely connected. This follows from a well known result in complex dynamics (see Theorem 5.2.1 in [1]). It follows from part (b) that B is infinitely connected and Q is not completely invariant. Therefore, the basin of attraction of q has infinitely many simply connected components. A

The case n $ 2
In this section, we prove the results concerning the topological characteristics of the Julia and the Fatou sets as well as the dynamics of f l;a on its Julia set when n $ 2. These results are Theorems 1.2 -1.5 stated in Section 1.

Proof of Theorem 1.2
Theorem 1.2 shows that in most cases, the basin of attraction of infinity is completely invariant, that is, there is no trap door.
Proof. (a) The idea of the proof is to find a critical point c in S a such that its corresponding critical value v ¼ f l;a ðcÞ belongs to B. To do this, we prove that for jlj sufficiently small, jvj . 1 þ s * with s * ¼ ðjlj=ðn 2 1ÞÞ 1=ðdþ1Þ , and then the result follows from Corollary 2.3. Finally, by Proposition 2.8 part (a), we conclude that B is completely invariant. Since we do not know the values of c and v, we use the approximationsc andṽ ¼ f l;a ðcÞ defined in Lemmas 2.5 and 2.6. Fix n; d $ 2 and a. We denote by c m the critical point in S a with the largest magnitude; that is, jc m j $ jcj for all c [ S a . Let v m ¼ f l;a ðc m Þ be the critical value corresponding to c m . Also, letc m andṽ m ¼ f l;a ðc m Þ be approximations of c m and v m , respectively. As we have shown in the proof of Lemma 2.5, the d þ 1 values ofc are given bỹ The fact that the values ofc are located at the vertices of a regular polygon centred at a and the condition that c m is the value of c with the largest modulus imply that (Figure 8) To computeṽ m ¼ f l;a ðc m Þ ¼c n m þ l=ðc m 2 aÞ d , we note that Then,ṽ m can be written as By definition,c m is taken so that it has the largest modulus and from the above expression the same happens with the corresponding value ofṽ m . Then, we have When n; d $ 2, we have that nð1 þ ð1=dÞÞ cos ðp=ðd þ 1ÞÞ . 1. Then, using Lemma 2.7, we conclude that v m belongs to B as we wanted to show. (b) Fix n $ 2, d $ 5 and a with jaj ¼ 1. By Proposition 2.8 part (c), we only need to prove that two critical values corresponding to critical points in S a belong to B. We denote by c m and c l , the two critical points in S a with the largest magnitudes; that is, jc m j $ jc l j $ jcj for all c [ S a . We also denote by v m ¼ f l;a ðc m Þ and v l ¼ f l;a ðc l Þ their corresponding critical values. From Theorem 1.2 part (a) we conclude that, for jlj sufficiently small, v m belongs to B. Then, we only have to prove that v l also belongs to B. Letc l andṽ l ¼ f l;a ðc l Þ denote approximations of c l and v l , respectively. From Lemma 2.5, we know thatc ¼ a þ d are the vertices of a regular polygon of d þ 1 sides centred at a. The definition of c m and c l implies thatc l is adjacent toc m and then they are separated by an angle of 2p=ðd þ 1Þ measured from a. So, ifc m and a have the same argument, then the modulus ofc m is equal to 1 þ jdj and Instead, ifc m and a have different arguments, then one of the vertices adjacent toc m has modulus larger than the above value. Then, in general, the modulus ofc l satisfies Same computations as in the proof of part (a) show that It is easy to check that when n . 1 and d . 4 (or if n . 2 and d . 3), we have that nð1 þ ð1=dÞÞ cos ð2p=ðd þ 1ÞÞ . 1. Then, using Lemma 2.7, we conclude that v l also belongs to B as we wanted to show. The hypothesis of Theorem 1.3 states that n $ 2, d $ 1, jaj ¼ 1 and for l sufficiently small S a , B < Q. Also, the number of critical points in Q and B is fixed for jlj sufficiently small. Moreover, we know that either S a > B ¼ Y or there are at least two critical points from S a in B and the rest lie in Q. From Theorem 1.2, we conclude that the first situation can only happen in the very special case when d ¼ 1. Numerical experiments suggest that this rarely happens (if it happens at all). See Remark 1. In this particular case, we would have the existence of a disjoint preimage of B, that is, a trap door T, and all the critical points in S a would lie in Q. The proof that the structure of the Julia set is as stated in Theorem 1.3 follows the same lines as in [8] in the case when jaj , 1. In this section, we focus on the second case, that is, when more than one critical point from S a belongs to B and the rest belong to Q.
We shall prove that for any a with jaj ¼ 1, there exists 1 a . 0 such that if jlj , 1 a , then the Julia set of f l;a consists of a countable collection of simple closed curves together with an uncountable collection of point components that accumulate on these curves. Only one of these curves surrounds the origin while all others bound disjoint discs that are eventually mapped onto Q. Moreover, any two such maps are topologically conjugate on their Julia sets.
By Proposition 2.8 part (c), we know that for sufficiently small l, a lies in B, B is infinitely connected and Q is simply connected. Let ›Q denote the boundary of Q. The holes in B are due to the preimages of Q and uncountably many point components that accumulate on the boundaries of these discs.
Let d * ¼ jS a > Bj # d and assume that d * $ 2. Since there are d þ 1 2 d * critical points from S a in Q and f l;a is of degree n þ d, it follows by the Riemann -Hurwitz formula that there are d * 2 1 disjoint discs that are preimages of Q in the complement of Q. Let r be a simple closed curve that lies in B and such that the Julia set of f l;a lies in the bounded component surrounded by r. This component is an open neighbourhood of Q. Consider the d * 2 1 preimages of this neighbourhood that contain the preimages of Q. We denote these sets by I 1 ; I 2 ; . . . ; I d * 21 and notice that their boundaries are preimages of the curve r. For jlj small enough, we can choose r so that the I j 's are pairwise disjoint. The set of points whose orbits remain for all iterations in the union of the I j forms a Cantor set on which f l;a is conjugate to the one-sided shift map on d * 2 1 symbols. This follows from standard arguments in complex dynamics [4,12]. This produces an uncountable number of point components in the Julia set. However, there are many other point components in J as we show below.
Let g be a simple closed curve that lies in B and surrounds Q so that the pole a and the critical points in S a > B lie outside g. Consider also a simple closed curve t that lies inside Q and surrounds the origin and such that all the critical points in S q and the critical points in S a > Q lie in the bounded component surrounded by t. Notice that, jS a > Qj ¼ d 2 d * $ 0. Let A be the annulus bounded by t and g. Then, ›Q , A and notice that each I j contains a copy of Q and a copy of each one of the I j 's. See Figure 9. To understand the complete structure of the Julia set, we show that J is homeomorphic to a quotient of a subset of a space of one-sided sequences of finitely many symbols. Moreover, we show that f l;a on J is conjugate to a certain quotient of a subshift of finite type on this space. Since this is true for l sufficiently small, this will prove our main results.
To begin the construction of the sequence space, we first partition the annulus A into n 'rectangles' that are mapped over A by f l;a . a r I 2 I 1 0 g ∂Q Figure 9. Sketch for the proof of Theorem 1.3. Region A and the I j 's for j ¼ 1; . . . ; d * 2 1 are depicted. The annulus A is bounded by the curve t and the curve g and contains the boundary of Q. Proposition 3.1. There is an arc j lying in A and having the property that f l;a maps j 1-to-1 onto a larger arc that properly contains j and such that one of its endpoints lies in t and the other one in g. Moreover, j meets ›Q at exactly one point, namely one of the fixed points in ›Q. With the exception of this point, all other points on j lie in the Fatou set.
Proof. Let p ¼ p l;a be one of the repelling fixed points in ›Q. Note that p varies analytically with both l and a. As it is well known, there is an invariant ray in Q extending from p to q ( [12]). Define the portion of j in Q > A to be the piece of this ray that lies in A.
To define the piece of j lying outside ›Q, let U be an open set that contains p and meets some portion of g and also has the property that the branch of the inverse of f l;a that fixes p is well-defined on U. Let f 21 l;a denote this branch of the inverse of f l;a . Let w [ g > U and choose any arc in U that connects w to f 21 l;a ðwÞ. Then, we let the remaining of the curve j be the union of the pullbacks of this arc by f 2k l;a for all k $ 0. Note that this curve limits on p as k ! 1. A We now partition A into n rectangles using the n preimages of f l;a ðjÞ that lie in A. Denote these preimages by j 1 ; . . . ; j n where j 1 ¼ j and the remaining j j 's are arranged counterclockwise around A. Let A j denote the closed region in A that is bounded by j j and j jþ1 , so that A n is bounded by j n and j 1 . By construction, each A j is mapped 1-to-1 over A except on the boundary arcs j j and j jþ1 , which are each mapped 1-to-1 onto f l;a ðj 1 Þ . j 1 .
The only points whose orbits remain for all iterations in A are those points on the simple closed curve ›Q. Let z [ ›Q. We may attach a symbol sequence SðzÞ to z as follows. Consider the n distinct symbols a 1 ; . . . ; a n taken from Zn{1; 2; . . . ; d * 2 1}. Define SðzÞ ¼ ðs 0 s 1 s 2 . . . Þ where each s j is one of the symbols a 1 ; . . . ; a n and s j ¼ a k if and only if f j l;a ðzÞ [ A k . Note that there are two sequences attached to p, the sequences ða 1 Þ and ða n Þ. Similarly, if z [ j k > ›Q, then there are also two sequences attached to z, namely ðs 0 s 1 . . . s j21 a k21 a n Þ and ðs 0 s 1 . . . s j21 a k a 1 Þ.
Note that if we make the above identifications in the space of all one-sided sequences of the a j 's then this is precisely the same identifications that are made in coding the itineraries of the map z 7 ! z n on the unit circle. So this sequence space with these identifications and the usual quotient topology is homeomorphic to the unit circle and the shift map on this space is conjugate to z 7 ! z n .
Finally, we extend the definition of S(z) to any point in J that remains in the union of the I j 's by introducing the symbols 1; . . . ; d * 2 1 and defining S(z) in the usual manner. We identify the sequences of the form ðja 1 Þ and ðja n Þ as well as ðja k a n Þ and ðja k a 1 Þ.
Let S 0 denote the space of one-sided infinite sequences of symbols a 1 ; . . . ; a n ; 1; . . . ; d * 2 1. Let S denote the space S 0 with all of the identifications described above and endow S with the quotient topology. Then, by construction, the Julia set of f l;a is homeomorphic to S and f l;a jJ is conjugate to the full shift map on S.
This finishes the proof of Theorems 1.3 and 1.4.

Proof of Theorem 1.5
Theorem 1.5 shows the existence of the Julia sets described in Theorem 1.3.
Proof. (a) The idea of the proof is the following. We can choose the parameter l, so that a and d are parallel vectors in the plane (see Figure 10(a)). For this, let l ¼ l 0 then, Argðl 0 Þ has to verify ArgðaÞ ¼ Argðl 0 Þ 2 ðn 2 1ÞArgðaÞ 2 ðmod 2pÞ; or equivalently Argðl 0 Þ ¼ ðn þ 1ÞArgðaÞ ðmod 2pÞ: For l ¼ l 0 , we have that jc^j ¼ 1^ðjl 0 j=nÞ 1=2 . Let S a 1 ;b 1 be the sector of parameters l with a 1 ¼ Argðl 0 Þ 2 p=10 and b 1 ¼ Argðl 0 Þ þ p=10. Then, for l [ S a 1 ;b 1 we have jc þ j . 1 þ cos ðp=20Þjdj; jc 2 j , 1 2 cos ðp=20Þjdj: Simple computations show thatṽ^¼ f l;a ðc^Þ ¼c n þ l=ðc^2 aÞ can be written as v^¼ a n21 ða þ 2ndÞ þ Oðd 2 Þ: For l in the sector S a 1 ;b 1 , we have that jṽ þ j . 1 þ cos ðp=20Þ2njdj^Oðjdj 2 Þ; jṽ 2 j , 1 2 cos ðp=20Þ2njdj^Oðjdj 2 Þ: To show thatṽ þ belongs to B andṽ 2 belongs to Q it is enough to prove that jṽ þ j .  For n $ 2, we have cos 2 ðp=20Þ4nðn 2 1Þ . 1, and then the above inequality is satisfied. Then, by Lemma 2.7 it follows that v þ also belongs to B and v 2 belongs to Q. The case when d ¼ 2 is very similar and then we briefly explain the main changes in the above argument. In this case, we have to show that one critical point in S a belongs to B and the other two critical points belong to Q. For this, we pick a value of the parameter l, that we call l 0 , such that there is only onec with modulus larger than 1 and such that a and d are parallel vectors in the plane (see Figure 10(b)). The three values ofc are given bỹ ; for i ¼ 1; 2; 3: The value of Argðl 0 Þ is the solution of ArgðaÞ ¼ ArgðdÞ, so it has to satisfy ArgðaÞ ¼ Argðl 0 Þ 2 ðn 2 1ÞArgðaÞ 3 ðmod 2pÞ: Thus, we have that Argðl 0 Þ ¼ ðn þ 2ÞArgðaÞ ðmod 2pÞ. Now, we define the sector of parameters S a 2 ;b 2 given by a 2 ¼ Argðl 0 Þ 2 p=10 and b 2 ¼ Argðl 0 Þ þ p=10. Then, when l [ S a 2 ;b 2 , we have jc 1 j . 1 þ cos ðp=30Þjdj; jc i j , 1 2 cos ð11p=30Þjdj for i ¼ 2; 3: We can rewriteṽ i ¼ f l;a ðc i Þ ¼c n i þ l=ðc i 2 aÞ 2 as v i ¼ a n21 a þ 3 2 nd þ Oðd 2 Þ for i ¼ 1; 2; 3: Hence, for l [ S a 2 ;b 2 , we obtain When n $ 2, we have that ð3=2Þn cos ðp=30Þ . 1 and also ð3=2Þn cos ð11p=30Þ . 1, and Lemma 2.7 implies that v 1 belongs to B and v 2 and v 3 belong to Q as we wanted to show.
(b) The idea to prove this part is the following. When d ¼ 2; 3; 4, we can choose ArgðlÞ such that two critical points in S a belong to B and the rest belong to Q. Then, from Proposition 2.8 part (c), we conclude that B is completely invariant and the basin of attraction of q has infinitely many simply connected components. Then, the structure of the Julia and Fatou sets is as in Theorem 1.3 part (b).
In this case,c is given byc : If we impose that ArgðdÞ ¼ ArgðaÞ þ ðp=ðd þ 1ÞÞ for some value l 0 , we obtain that Argðl 0 Þ verifies ArgðaÞ þ p d þ 1 ¼ Argðl 0 Þ 2 ðn 2 1ÞArgðaÞ d þ 1 ð mod 2pÞ; or equivalently Argðl 0 Þ ¼ ðn þ dÞArgðaÞ þ p ð mod 2pÞ: For l ¼ l 0 , there are two values ofc that we denote byc 1 andc 2 , such that jc 1 j ¼ jc 2 j and this is equal to the largest value of the d þ 1 possible values ofc (see Figure 11). We have jc 1 j; jc 2 j ¼ 1 þ cos p d þ 1 jdj: Let S g d ;d d be the sector of parameters l with g d ¼ Argðl 0 Þ 2 p=10 and d d ¼ Argðl 0 Þ þ p=10. Then, when l [ S g d ;d d we have, jc 1 j; jc 2 j . 1 þ cos 11p 10ðd þ 1Þ jdj: Simple computations show that, for i ¼ 1; 2;ṽ i can be written as v i ¼ a n21 a þ n 1 þ 1 d d þ Oðjdj 2 Þ for i ¼ 1; 2:  Therefore, we study the map given by where d $ 1 is an integer. There is no dependence on complex parameters, so for each value of d we have just a unique representative of the family f l;a . The following theorem is due to Shishikura [15] (see also [10]) and gives a connection between the number of weakly repelling fixed points of a rational map and the connectivity of the Julia set. Recall that a weakly repelling fixed point is a fixed point that is either repelling or parabolic of multiplier 1. It is easy to check that gðzÞ has only one parabolic fixed point at infinity and then Theorem 1.6 follows as a corollary of Theorem 4.2. In the following paragraphs, we describe the symmetries and dynamical behaviour of the function gðzÞ.
When d ¼ 1, we have that the Julia set of gðzÞ is the imaginary axis. This follows since the imaginary axis is the smallest closed set with more than two points that is completely invariant under the map. It is easy to check that jg 0 ðzÞj . 1 for z [ iR and that every point that is not in the imaginary axis moves away from it under iteration. The Fatou set consists of the two completely invariant half-planes Re ðzÞ . 0 and Re ðzÞ , 0 for z [ C. Each half of the real axis is forward invariant and every orbit in one of the two half-planes approaches the real axis under iteration and converges to infinity.
When d . 1, the Julia set is still connected as we have already shown; however, the Fatou set now consists of infinitely many simply connected components. The degree of gðzÞ is d þ 1 so the map has 2d critical points counted with multiplicity. The pole 0 is a critical point of order d 2 1 and then, there are d þ 1 critical points symmetrically distributed around the origin. The critical points c of g are given by Infinity and its preimages lie in the Julia set of gðzÞ. This set includes the prepoles, that is, the preimages of the pole at the origin. The prepoles p of gðzÞ are also symmetrically distributed around the origin and are given by p ¼ ð21Þ ð1=ðdþ1ÞÞ . The critical points c of gðzÞ are mapped to the critical values v. We have A straightforward computation shows that each line of the form vt with v dþ1 ¼ 1 and t . 0 is forward invariant under gðzÞ. Moreover, every point in one of these lines converges monotonically to infinity under iteration. From equations (7) and (8), it follows that the critical points lie in these lines where the orbit of every point converges to infinity. In other words, each one of the critical points lies in a different petal of the flower around infinity. Figure 6 displays some examples of the Julia sets studied in this section.