Singular Perturbations in the McMullen Domain

Transcription

1 Singular Perturbations in the McMullen Domain Robert L. Devaney Sebastian M. Marotta Department of Mathematics Boston University January 5, 2008 Abstract In this paper we study the dynamics of the family of complex maps given by f λ (z) = z n + λ/(z h (z a) d ) where n, h and d are integers such that 1/n + 1/h < 1 and d 1 and the pole a lies outside the unit circle. The parameter λ is complex and arbitrarily small. In spite of the large number of possibilities we prove that the Julia set of f λ (z) consists of a countable collection of Cantor sets of circles only one of which surrounds the origin and uncountably many point components that accumulate on each of these curves. Please address all correspondence to Robert L. Devaney, Department of Mathematics, Boston University, 111 Cummington St., Boston, MA 02215, USA. address: bob@bu.edu. 1

2 1 Introduction In the last few years a number of papers have appeared that deal with the dynamics of functions obtained by perturbing a complex polynomial by adding a pole at some point that is not in the Julia set of the polynomial [1, 2, 3, 4, 7]. This type of perturbation of a polynomial is called singular perturbation since the degree of the map jumps when the pole is added. As a consequence, the structure of the Julia sets often changes dramatically after the addition of the pole. Many of the results concerning singular perturbations deal with the special case where the polynomial is z n with n 2. In this case, before the perturbation, the Julia set is the unit circle. For example, in [8], McMullen considers the family of maps f λ (z) = z n + λ/z h when n and h satisfy 1/n + 1/h < 1. For these maps, as long as λ 0 is sufficiently small, the Julia set is a Cantor set of simple closed curves (i.e., homeomorphic to the cross-product of a Cantor set and a circle), each of which surrounds the origin, and the dynamics on each of these sets is the same. The case where n = h = 2 is very different. In [3] it is shown that, in any neighborhood of λ = 0 in the parameter plane, there are uncountably many maps whose dynamics are different from one another. Indeed, it is known that there are infinitely many open sets in any neighborhood of λ = 0 in which the Julia sets corresponding to these parameters are all Sierpinski curves, but any two maps whose parameters are drawn from different open sets have non-conjugate dynamics. More generally, one can consider singular perturbations of z n where the pole is placed at some point away from the origin, i.e., maps of the form f λ (z) = z n + λ/(z a) d. In this case, the Julia sets are quite different from the ones discussed above. In [4] it is shown that these Julia sets now consist of only countably many simple closed curves and uncountably many point components that accumulate on these curves. Unlike the case a = 0, only one of these simple closed curves surrounds the origin; all of the other curves bound disks that are disjoint from one another. See Figure 1. In this paper we perturb z n in a different manner using a combination of the previous cases. That is, we consider the case where the singular perturbation consists of two poles, one at the origin and the other one outside the unit circle, so we consider the family of maps f λ (z) = z n + 2 λ z h (z a) d

3 Figure 1: Top left: the Julia set for z /z 3 (a Cantor set of simple closed curves). Top right: the Julia set for z /(z 1.2) 4 (countably many simple closed curves and uncountably many points) and bottom: the Julia set for z 2 1/(16z 2 ) (a Sierpinski curve). The colored regions are the immediate basin of attraction of infinity and its preimages. Different shades represent different escape times. For the picture in the top right corner the white regions are in the basin of attraction of an attracting fixed point near the origin. 3

4 Figure 2: The Julia set of f λ (z) = z /(z 3 (z a) 2 ) and a magnification about the pole at a = 0.75(1 + i). where n, h and d are integers such that 1/n + 1/h < 1 and d 1. The pole a is complex and such that a > 1 and λ is an arbitrarily small complex parameter. In this case, the Julia set is a combination of the above types of Julia sets. We shall prove: Theorem 1.1. (Structure of the Julia set) Let 1/n + 1/h < 1, d 1 and a > 1. Then, for λ sufficiently small, the Julia set of f λ consists of 1. an invariant Cantor set of simple closed curves, each of which surrounds the origin; 2. a countable collection of disjoint preimages of this Cantor set of circles, none of which surrounds the origin; 3. and uncountably many point components that accumulate on both sides of every curve in these Cantor sets. See Figure 2. The case when a 1 is very different and it is not studied in this paper. The differences arise because when λ is small, the new critical points that lie near the pole a do not, in general, behave as one as in the case when a > 1. 4

5 We shall also show that, as long as the parameter λ is sufficiently small, any two such maps are conjugate as long as the nonzero pole lies outside the unit circle. Theorem 1.2. (Dynamics on the Julia set) Fix n, h and d with 1/n+1/h < 1 and d 1 and let a, b > 1. Then there exists ɛ > 0 such that, if λ, µ < ɛ, the maps f λ,a = z n +λ/(z h (z a) d ) and f µ,b = z n +µ/(z h (z b) d ) are conjugate on their Julia sets. Moreover, the dynamics on the Julia sets of these maps is determined by a specific quotient of a subshift of finite type. The above theorem implies that if we fix n, h, d and a and look in the parameter λ-plane, we find a neighborhood of λ = 0 where each map has the Julia set described in Theorem 1.1 and is conjugate on the Julia set to any other map in this neighborhood. It also implies that if we fix n, h, d and λ satisfying the above hypothesis and look in the parameter a-plane we see one region a > 1 where all these maps again have the same Julia set described in Theorem 1.1 and are conjugate to each other on their Julia sets. 2 Preliminaries Our goal in this paper is to investigate the Julia set of f λ. Recall that the Julia set of a complex analytic function is the set of points at which the family of iterates of the map is not a normal family in the sense of Montel. Equivalently, the Julia set is the closure of the set of repelling periodic points for the map as well as the set of points on which the map behaves chaotically. We denote the Julia set of f λ by J = J(f λ ). When λ = 0 the dynamical behavior of f λ is well understood. In this case and 0 are superattracting fixed points and the Julia set is the unit circle. When we add the perturbation by setting λ 0 but very small, several aspects of the dynamics remain the same, but others change dramatically. For example, when λ 0, the point at is still a superattracting fixed point and there is an immediate basin of attraction of that we call B = B λ. As a consequence the Julia set of f λ is also the boundary of the full basin of attraction of. The function f λ takes B n to 1 onto itself. On the other hand, there is a neighborhood of the origin that is mapped h to 1 onto B. When this neighborhood is disjoint from B we call it the trap door T = T λ. Also, there is a neighborhood of a that is now mapped d to 1 onto B. It turns out that when a > 1 this neighborhood of a is contained in B and every point that escapes to infinity 5

6 and does not lie in B must do so by passing through the trap door T. Since the degree of f λ changes from n to n+h+d when λ becomes nonzero, 2(h+d) additional critical points are created. We shall show that the dynamics of f λ are completely determined by the orbits of these critical points. Let 1/n + 1/h < 1, d 1 and a > 1. The condition on n and h is equivalent to the requirement that n and h be greater than or equal to 2 but not simultaneously equal to 2. A straightforward calculation shows that the critical points of f λ satisfy nz 2h+n 1 (z a) 2d = λ ( hz h 1 (z a) d + d(z a) d 1 z h). (2.1) Notice that there is a critical point of order h 1 at the origin and a critical point of order d 1 at the pole a. If we remove these solutions from Equation (1) and let c = c λ be any other critical point of f λ, then c must satisfy nc n+h (c a) d+1 = λ ( h(c a) + dc ). (2.2) When λ = 0 this equation has n+h+d+1 roots, the origin with multiplicity n + h and the pole a with multiplicity d + 1. By continuity, for small enough λ, these roots become simple roots of f λ that lie in the plane approximately symmetrically distributed around the origin and the pole a. Indeed, when λ is arbitrarily small, n + h of the critical points of f λ are grouped around the origin while d + 1 of the critical points are grouped around the pole a. The critical value corresponding to c is given by and, using (2.2), we have v = f λ (c) = c n + λ c h (c a) d v = c n + ncn (c a) h(c a) + dc. (2.3) From Equation 2.3 it follows that, if c 0, then v 0. Similarly, if c a, then v a n. Let S a = {d + 1 critical points around a} and S 0 = {n + h critical points around 0}. We have: Proposition 2.1. (Location of the critical values) For λ sufficiently small the set S 0 is mapped arbitrarily close to the origin and the set S a is mapped arbitrarily close to a n. 6

7 When λ = 0, f λ is hyperbolic on the unit circle. Thus, for small enough λ 0 there exists a simple closed curve Γ close to the unit circle such that the map f λ acts on Γ as z z n. The curve Γ is a boundary component of B. The following proposition gives a bound on the location of the Julia set. Proposition 2.2. Suppose 1/n + 1/h < 1, d 1 and a > 1. Let z = r. If r > a then, for sufficiently small λ, z B. Proof: Fix n, h, d and a such that 1/n + 1/d < 1, d 1 and a > 1. We write z a z a = r a > 0. Then we have f λ (z) z n Let λ < r h (r a ) d (r n r). Then λ z h z a λ d rn r h (r a ). d f λ (z) > r n rh (r a ) d (r n r) r h (r a ) d = r n r n + r = r. Hence f λ (z) > z and the orbit of z converges to so that z B. Now we give some estimates for the size of the trap door T and the approximate position of the critical points in S 0 and their corresponding critical values. Proposition 2.3. (Estimates for T and S 0 ) When λ << 1 we have that: 1. the critical points in S 0 are arbitrarily close to the (n + h)-roots of λh/(n( a) d ) and, 2. there is a simple closed curve γ around the origin that is a preimage of Γ and lies arbitrarily close to a circle of radius ( λ / a d ) 1/h. It is easy to see that when d = 0 the above estimates reduce to the ones studied in [1]. We will show later that the trap door as well as all components of the Fatou set are infinitely connected. All the punctures and holes in T are bounded inside γ, so the curve γ is the outer boundary of the trap door T. 7

8 Proof: Fix n, h, d and a with 1/n + 1/h < 1, d 1 and a > 1. For the first part let z = (λh/(n( a) d )) 1/(n+h) and replace c by z in Equation (2.2) to get hλ ( λh ( ( a) d n( a) d )1/(n+h) a ) d+1 ( ( λh = λ h ( n( a) d )1/(n+h) a ) λh +d( n( a) )1/(n+h)). d Cancel λ on both sides and then let λ go to zero. It is easy to see that, when λ 0, then z 0 and so (z a) ( a). Thus, when λ is arbitrarily small, we can simplify both sides of the above equation to get an identity and the result follows. For part 2 we have that, when λ is arbitrarily small, the curve Γ is arbitrarily close to the unit circle. Since γ is a preimage of Γ we then have that the curve γ is mapped arbitrarily close to the unit circle. Let z = ( λ / a d ) 1/h e iθ and let λ = λ e iφ. Then apply f λ to z, i.e., f λ (z) = ( λ / a d ) n/h e inθ + λ e iφ ( λ / a d )e ihθ (( λ / a d ) 1/h e iθ a) d. Cancel λ in the second term and then let λ 0. e i(φ hθ) = 1 as we wanted to show. We get f λ (z) Part 2 of Proposition 2.3 implies that when λ << 1 there are h preimages of the exterior of the curve Γ inside the curve γ. This in turn implies that outside Γ the map f λ has degree n + d. Also, the map is of degree n in B so that there are d preimages of the interior of Γ outside Γ and near the pole a. Therefore, the map is of degree n + h in the annulus between γ and Γ. To show the existence of a Cantor set of simple closed curves that surrounds the origin we show (as in the case when d = 0) that when λ is arbitrarily small the critical values corresponding to the critical points in S 0 lie inside the trap door T. The following proposition implies this result. Proposition 2.4. When λ 0 the critical values v that correspond to the critical points in S 0 are such that v 0 faster than λ 1/h. λh Proof: Let z = ( ) 1/(n+h) and compute f n( a) d λ (z). We have, v = f λ (c) f λ (z) = ( λh ) n/(n+h) + n( a) d λ ) n( a) 1/(n+h) λh (( ) d n( a) 1/(n+h) a) d d ( λh 8

9 that is, v λ n/(n+h) K(λ) where K(λ) tends to a finite value when λ 0. Since 1/n + 1/h < 1 is equivalent to n/(n + h) > 1/h, we get the result. Corollary 2.5. When λ is arbitrarily small the critical values corresponding to the critical points in S 0 lie in the trap door T and the critical values corresponding to the critical points in S a lie in B. 3 The structure of J Let λ << 1 so that we can define a disk τ that is bounded by the curve γ (the outer boundary of the trap door T ) and such that τ contains the critical values corresponding to the critical points in S 0. The fact that τ T will be clear after we prove the next two propositions. Proposition 3.1. There is a preimage of τ that is an annulus A that is contained in between γ and Γ. The annulus A contains the set S 0. All the preimages of A are annuli and, by construction, the union of all these annuli forms the complement of a Cantor set of simple closed curves. Each one of these curves is in J. We remark that the existence of the Cantor set of circles for the family z n + λ/z h with 1/n + 1/h < 1 is due to McMullen [8]. Proof: Restricted to the region between T and Γ the map f λ is of degree n + h. Let σ be the connectivity of the preimage of τ that lies in this region. The connectivity of τ is t = 1. There are n+h critical points in the preimage of τ. Then, applying the Riemann-Hurwitz formula, we get σ 2 = (n + h)(t 2) + n + h. This implies that σ = 2. Thus, the preimage of the disk τ has two boundary components and it is thefore an annulus. Taking preimages of A we get the result. Figure 3 shows the first step in the construction of the Cantor set of simple closed curves that surrounds the origin. Notice that the above Proposition proves that when λ << 1, T and B are disjoint. Now we prove that there are preimages of the Cantor set of simple closed curves outside Γ. 9

10 Γ γ τ PSfrag replacements A Figure 3: The gray disk is τ, is bounded by γ and contains all the critical values corresponding to the critical points in S 0. The curve γ is the outer boundary component of the trap door T. The grey annulus A is the preimage of the disk τ and contains the n + h critical points in S 0. Proposition 3.2. For sufficiently small λ the critical points in S a are contained in B and B is not simply connected. Proof: Let η be a simple closed curve that separates the pole a from the point a n and let λ be sufficiently small so that outside of η we have all the critical values corresponding to the critical points near a. There is a curve µ that is a preimage of η and such that the pole a and the set of critical points around a lie in the region bounded by η and µ. Then we have a function of degree n + d that maps the exterior of µ (connectivity σ) to the exterior of η (connectivity t = 1) and there are N = 2d + n 1 critical points in the domain (d + 1 critical points around a, a and ). By the Riemann-Hurwitz formula, we have σ 2 = (n + d)(t 2) + N, so we get σ = d + 1 and thus there are d holes in the region bounded by η and µ. These holes contain preimages of the interior of η. It follows that these holes contain the other d preimages of the interior of Γ and thus, B is not simply connected. 10

11 PSfrag replacements I 1 a n 0 µ a η Γ I 2 Figure 4: The curve η separates the pole a from a n. The curve µ is a preimage of η. The disks I i for i = 1,..., d are mapped 1 to 1 onto the region bounded by η. The curve Γ is a boundary component of B λ and it bounds the invariant Cantor set of simple closed curves that surrounds the origin. The gray disks inside the I i s contain preimages of the Cantor set of circles that surround the origin. The above proposition shows the existence of preimages of the Cantor sets of circles that surround the origin in the exterior of Γ. Taking preimages of the exterior of the curve µ we find countably many Cantor sets of simple closed curves in the Julia set of f λ that accumulate on the exterior of Γ. Since the trap door contains a preimage of the exterior of Γ we get another set of Cantor sets of circles inside the disk τ and then it follows that every preimage of T contains countably many Cantor sets of circles. We see that every component of the Fatou set is infinitely connected. Part of our main results have been proven. There are, however, other components of the Julia set. Every one of the Cantor sets of circles described above is a preimage of the one that surrounds the origin, so those curves cannot contain periodic points. Repelling periodic points form a countable dense subset of the Julia set J of f λ and so there must be other components of the Julia set accumulating on every one of the curves in the Cantor sets of circles. We show next that these other components of J are points. The d holes found in Proposition 3.2 contain preimages of the interior of η. We denote these disks by I 1, I 2,..., I d. It follows that each of these disks is mapped 1 to 1 over the union of all of them. Thus, the set of points whose orbits remain for all iterations in the union of the I j form a Cantor set on which f λ is conjugate to the one-sided shift map on d symbols. This produces 11

12 an uncountable number of point components in J. However, there are many other point components in J, as any point whose orbit eventually lands in this Cantor set of points is also in J. There are, however, still other points in J, as we show below. To understand the complete structure of the Julia set, we show that J is homeomorphic to a quotient of a subset of the space of one sided sequences on n + h + d symbols. Moreover, we show that f λ on J is conjugate to a certain quotient of a subshift of finite type on this space. Since this is true for any a > 1 and λ sufficiently small, this will complete the proof of our main results. In order to describe the dynamics on the whole Julia set we do the following. Consider the curve η that is completely contained in B λ and separates a from a n. Notice that every point outside η is in the immediate basin of attraction of. As before, let µ be the preimage of η that lies in B λ and is mapped n to 1 onto η. To construct the symbolic dynamics we need to consider other preimages of η. Let s be the preimage of η that surrounds the origin and is contained in the trap door T. The curve s is mapped h to 1 onto η. Fix a with a > 1 then by Proposition 2.4 we can take λ sufficiently small so that the forward images of the critical values corresponding to the critical points in S 0 are outside the curve η. Then, the region bounded by the curve s contains the critical values corresponding to the critical points in S 0. The annulus between s and η is denoted by Ã. Recall that there is an annulus A that contains the n + h critical points in S 0 and lies between the curves γ (exterior boundary of T ) and Γ (interior boundary of B λ ). The two preimages of s inside A are disjoint (since the critical points are mapped inside of s). We denote by r 1 the preimage of s (a simple closed curve) that lies inside A that surrounds the origin and such that all the critical points in S 0 lie inside r 1. That is, r 1 is mapped n to 1 onto s. Likewise, let r 2 denote the other preimage of s (a simple closed curve) that lies in A and surrounds the origin and such that all the critical points in S 0 lie outside r 2. That is, r 2 is mapped h to 1 onto s. Let A 1 denote the annulus between the curves r 1 and µ and let A 2 denote the annulus between the curves r 2 and s. The annulus A 1 is mapped as an n to 1 covering onto à and so A 1 covers A 1, A 2, and the disks I i for i = 1,..., d. That is, the image of A 1 contains the Julia set of f λ. Similarly, the image of A 2 has this property; that is, A 2 is mapped as an h to 1 coverging onto Ã. See Figure 5. To begin the construction of the sequence space, we first partition the 12

13 PSfrag replacements µ r 1 r 2 s γ Γ I 1 a I 2 a n A 1 A 2 η 0 Figure 5: Definition of the annuli A 1 and A 2. The annulus A 1 is bounded by the curves µ and r 1 and the annulus A 2 is bounded by the curves r 2 and s. The curve η separates the pole a from a n. The curves µ and s are preimages of η. The disks I i with i = 1,..., d are mapped 1 to 1 to the interior of η. annulus A 1 into n rectangles that are mapped over à by f λ. Proposition 3.3. There is an arc ξ lying in A 1 and having the property that f λ maps ξ 1 to 1 onto a larger arc that properly contains ξ and connects the curve µ and the curve r 1. Moreover, ξ meets Γ at one of the repelling fixed points on Γ. Proof: Let q = q λ be one of the repelling fixed points on Γ. Note that q varies analytically with both λ and a. As is well known, there is an invariant external ray in B λ extending from q to. Define the portion of ξ in B λ A 1 to be the piece of this external ray that lies in A 1. To define the piece of ξ lying inside Γ, let U be an open set that contains q and meets some portion of r 1 and also has the property that the branch of the inverse of f λ that fixes q is well-defined on U. Let f 1 λ denote this branch of the inverse of f λ. Let w r 1 U and choose any arc in U that connects w to f 1 (w). Then we let the remainder of the curve ξ be the union of the λ 13

14 pullbacks of this arc by f k λ for all k 0. Note that this curve limits on q as k. We now partition A 1 into n rectangles. Consider the n preimages of f λ (ξ) that lie in A 1. Denote these preimages by ξ 1,..., ξ n where ξ 1 = ξ and the remaining ξ j s are arranged counterclockwise around A 1. Let Rj 1 denote the closed region in A 1 that is bounded by ξ j and ξ j+1, so that Rn 1 is bounded by ξ n and ξ 1. By construction, each Rj 1 is mapped 1 to 1 over A 1 except on the boundary arcs ξ j and ξ j+1, which are each mapped 1 to 1 onto f λ (ξ 1 ) ξ 1. Now recall that the only points whose orbits remain for all iterations in A 1 are those points on the simple closed curve Γ. Let z Γ. We may attach a symbolic sequence S(z) to z as follows. Consider the n symbols α 1,..., α n. Define S(z) = (s 0 s 1 s 2...) where each s j is one of the symbols α 1,..., α n and s j = α k if and only if f j λ (z) R1 k. Note that there are two sequences attached to q, the sequences (α 1 ) and (α n ). Similarly, if z ξ k Γ, then there are also two sequences attached to z, namely (α k 1 α n ) and (α k α 1 ). Finally, if f j λ (z) ξ k, then there are again two sequences attached to z, namely (s 0 s 1... s j 1 α k 1 α n ) and (s 0 s 1... s j 1 α k α 1 ). Note that if we make the above identifications in the space of all one-sided sequences of the α j s, then this is precisely the same identifications that are made in coding the itineraries of the map z 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 z n. Now we partition the annulus A 2 into h rectangles that are mapped over à by f λ. Since f λ A 2 covers itself we see that there is a simple closed curve inside A 2 that is fixed by f λ. We denote this curve by Γ and note that Γ is mapped by f λ as a d to 1 covering onto itself. Proposition 3.4. There is an arc ξ lying in A 2 and having the property that f λ maps ξ 1 to 1 onto a larger arc that properly contains ξ and connects the curve s and the curve r 2. Moreover, ξ meets Γ at one of the repelling fixed points on Γ. Proof: Let p = p λ be one of the repelling fixed points on Γ. Note that p varies analytically with both λ and a. Consider a small neighborhood V of p completely contained in A 2. Since p is repelling we have that there is a forward image of V, that we denote V, such that V intersects s and r 2 and contains V. Consider two points in these intersections and their 14

15 corresponding preimages in V. Join each one of these points to p with simple arcs. Then, pull these arcs back towards p. This curve is the arc ξ. We now partition A 2 into h rectangles. Consider the h preimages of f λ (ξ ) that lie in A 2. Denote these preimages by ξ 1,..., ξ h where ξ 1 = ξ and the remaining ξ j s are arranged counterclockwise around A 2. Let Rj 2 denote the closed region in A 2 that is bounded by ξ j and ξ j+1, so that R2 h is bounded by ξ h and ξ 1. By construction, each Rj 2 is mapped 1 to 1 over A 2 except on the boundary arcs ξ j and ξ j+1, which are each mapped 1 to 1 onto f λ(ξ 1 ) ξ 1. As before, the only points whose orbits remain for all iterations in A 2 are those points on the simple closed curve Γ. Let z Γ. We may attach a symbolic sequence S(z) to z as follows. Consider the h symbols β 1,..., β h. Define S(z) = (s 0 s 1 s 2...) where each s j is one of the symbols β 1,..., β h and s j = β k if and only if f j λ (z) R2 k. Note that there are two sequences attached to p, the sequences (β 1 ) and (β h ). Similarly, if z ξ k Γ, then there are also two sequences attached to z, namely (β k 1 β h ) and (β k β 1 ). Finally, if f j λ (z) ξ k, then there are again two sequences attached to z, namely (s 0 s 1... s j 1 β k 1 β h ) and (s 0 s 1... s j 1 β k β 1 ). In this case we have that if we make the above identifications in the space of all one-sided sequences of the β j s, then this is precisely the same identifications that are made in coding the itineraries of the map z z h 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 z h. We may thus attach a symbol sequence S(z) to any point in J as follows. Let S(z) = (s 0, s 1, s 2,..., s j,...). Let α = {α i with i = 1,..., n} be a set of n symbols that are to be attached to any point in the Julia set that at some iterate lands on one of the n subrectangles inside A 1 ; and let β = {β i with i = 1,..., h} be a set of h symbols that are to be attached to any point in the Julia set that at some iterate lands on one of the h subrectangles inside A 2. Finally, let D = {1, 2,..., d} be a set of d symbols that are to be attached to points that at some iterate land on one of the disks I i with i = 1,..., d. We let s k α, β or D if the k th iterate of z belongs to A 1, A 2 or I i for i = 1,..., d, respectively. The points in the invariant Cantor set of circles that surrounds the origin have itineraries that consists of sequences of α i s and β i s. For points on these curves we need to extend the identifications mentioned above on the space of sequences. 15

16 Every point in a preimage of the Cantor set of circles that surrounds the origin that is inside one of the disks I i with i = 1,..., d has a sequence that starts with an element in D and then is followed by elements in α and β. Every point that is in a preimage of the Cantor set of circles that surrounds the origin that is in the trap door T has a sequence that starts with a β i then it is followed by an element in D and then consists of elements in α and β. We extend the identifications mentioned above to every curve in every Cantor set of circles in the Julia set of f λ. To extend this definition to all of J, we let Σ denote the set of sequences involving all the symbols from the sets α, β and D with no restrictions; that is, we let any symbol be followed by any other. Let Σ denote the space Σ where we extend the above identifications to any pair of sequences that ends in a pair of sequences identified earlier. We endow Σ with the quotient topology. Then, by construction, the Julia set of f λ is homeomorphic to Σ and f λ J is conjugate to the shift map on Σ. This finishes the proof of our main results. 3.1 Example In this section we discuss a particular case when n = 3, h = 5, d = 4 and a > 1. Then we have f λ (z) = z 3 + λ z 5 (z a) 4. Notice that the degree of f λ is 12 and so there are 22 critical points counted with multiplicity. Infinity is a critical point of order 2. Thus, there are 20 critical points of f λ that satisfy the equation 3c 12 (c a) 8 = λ(5c 4 (c a) 4 + 4(c a) 3 c 5 ). As can be seen in this equation, the origin is a critical point of order 4 and the pole a is a critical point of order 3. If we remove these solutions we find 3c 8 (c a) 5 = λ(5(c a) + 4c) and we see that, when λ 0, there are 8 critical points that approach 0 (the set S 0 ), and 5 critical points that approach a (the set S a ). The critical values corresponding to these free critical points are determined by v = f λ (c) = c 3 + 3c3 (c a) 5(c a) + 4c. 16

17 Figure 6: Top left: the Julia set of f λ (z) = z 3 + λ/(z 5 (z a) 4 ) when the pole a = 1.2 and λ = Top right: magnification about the pole a and, bottom: magnification of Cantor set of circles inside the trap door T. The colored regions are B and its preimages. The different shades represent different escape times. 17

18 Note that if c 0 then v 0 and if c a then v a 3. The preimage of the disk bounded by γ is the annulus A that contains the 8 critical points in S 0 and whose closure lies between γ and Γ. The preimages of this annulus are two annuli and, taking preimages of these annuli, we construct a Cantor set of simple closed curves that surrounds the origin. There are 4 holes near the pole a. This holes are outside Γ in B that contain preimages of the Cantor set of circles that surrounds the origin. Since the trap door T is a preimage of the region outside Γ, it also contains preimages of those holes and so does every preimage of T. It follows that each annulus in the Cantor sets of simple closed curves contains preimages of the Cantor set of circles. All of these new Cantor sets of simple closed curves are preimages of the one that surrounds the origin and so they cannot contain periodic points. Repelling periodic points are dense in the Julia set and so there must be other components in the Julia set accumulating on each curve of these Cantor sets. These other components of the Julia set are points. Figure 6 shows the Julia set of f λ (z) when a = 1.2 and λ = References [1] Devaney, R. L., Look, D. M. and Uminsky, D. The Escape Trichotomy for Singularly Perturbed Rational Maps. Indiana University Mathematics Journal 54 (2005), [2] Devaney, R. L., Holzer, M., Look, D. M., Moreno Rocha, M. and Uminsky, D. Singular Perturbations of z n. To appear in Transcendental Dynamics and Complex Analysis, Cambridge University Press, eds. P. Rippon and G. Stallard. [3] Devaney, R. L., Blanchard, P., Look, D. M., Seal, P. and Shapiro, Y. Sierpinski Curve Julia Sets and Singular Perturbations of Complex Polynomials. Ergodic Theory and Dynamical Systems 25 (2005), [4] Devaney, R. L. and Marotta, S. M. Evolution of the McMullen Domain for Singularly Perturbed Rational Maps. To appear in Topology Proceedings. 18

19 [5] Devaney, R. L. and Marotta, S. M. The McMullen Domain: Rings Around the Boundary. Transactions of the American Mathematical Society 359 (2007), [6] Devaney, R. L. Structure of the McMullen Domain in the Parameter Planes for Rational Maps. Fundamenta Mathematicae 185 (2005), [7] Marotta, S. M. Singular Perturbations of z n with multiple poles. To appear in International Journal of Bifurcation and Chaos. [8] McMullen, C. Automorphisms of Rational Maps. Holomorphic Functions and Moduli. Vol. 1. Math. Sci. Res. Inst. Publ. 10. Springer, New York, [9] Milnor, J. Dynamics in One Complex Variable. 3rd Edition. Princeton University Press, [10] Morosawa, S., Nishimura, Y., Taniguchi, M. and Ueda, T. Holomorphic Dynamics. Cambridge University Press,