This is a preprint of: “Generalized Rings Around the McMullen Domain”, Antoni Garijo, HyeGy- ong Jang, Sebastian M. Marotta, Qual. Theory Dyn. Syst., vol. 18, 233–264, 2019. DOI: [10.1007/s12346-018-0287-y] Generalized Rings Around the McMullen Domain Antonio Garijo∗, HyeGyong Jang†, and Sebastian M. Marotta‡ August 8, 2018 Abstract We consider the family of rational maps given by Fλ (z) = z n +λ/z d where n, d ∈ N with 1/n + 1/d < 1, the variable z ∈ C b and the pa- rameter λ ∈ C. It is known that when n = d ≥ 3 there are infinitely many rings S k with k ∈ N, around the McMullen domain. The Mc- Mullen domain is a region centered at the origin in the parameter λ-plane where the Julia sets of Fλ are Cantor sets of simple closed curves. The rings S k converge to the boundary of the McMullen do- main as k → ∞ and contain parameter values that lie at the center of Sierpiński holes, i.e., open simply connected subsets of the parameter space for which the Julia sets of Fλ are Sierpiński curves. The rings also contain the same number of superstable parameter values, i.e., parameter values for which one of the critical points is periodic and correspond to the centers of the main cardioids of copies of Mandelbrot sets. In this paper we generalize the existence of these rings to the case when 1/n+1/d < 1 where n is not necessarily equal to d. The number of Sierpiński holes and superstable parameters on S 1 is τ1n,d = n − 1, and on S k for k > 1 is given by τkn,d = dnk−2 (n − 1) − nk−1 + 1. ∗ Dep. d’Eng. Informàtica i Matemàtiques, Universitat Rovira i Virgili, Tarragona, Spain. Email: antonio.garijo@urv.cat † Faculty of Mathematics, University of Science Pyongyang D.P.R. of Korea. ‡ Corresponding author: Sebastian M. Marotta, Department of Mathematics, Boston University, Boston, MA, USA. Email: smmarotta@gmail.com 1 1 Introduction In this paper we consider the family of complex rational maps1 Fλ : C b→C b given by λ Fλ (z) = z n + d z where n, d ∈ N with 1/n + 1/d < 1 and the parameter λ ∈ C. Let m = n + d denote the degree of Fλ . McMullen introduced Fλ in [1] where he shows that when λ 6= 0 is suf- ficiently small, the Julia set of Fλ is a Cantor set of simple closed curves, see also [2]. The condition 1/n + 1/d < 1 is equivalent to nd > m and this defines the set of values n, d ≥ 2 with m ≥ 5. The dynamics of some of these maps, the topological structures of their Julia sets and the structure of the parameter λ−planes have been widely studied by several authors, see for example [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. A recent survey of results involving some of the maps in this family is given in [19]. The structure of the parameter planes of these maps is furthered studied in [20, 21, 22]. In the dynamical plane the object of principal interest is the Julia set of Fλ , which we denote by J(Fλ ), and is the set of points at which the family of iterates {Fλn } fails to be a normal family in the sense of Montel. It is known that J(Fλ ) is also the closure of the set of repelling periodic points for Fλ as well as the boundary of the set of points whose orbits escape to ∞, or the set of points where the map presents chaotic behavior. For background results in complex dynamics see for example [23, 24, 25, 26, 27]. The function Fλ has 2(n + d) − 2 critical points counted with multiplicity, namely, n − 1 at ∞, d − 1 at 0 and n + d additional critical points whose orbits depend on the value of the parameter λ. One of the reasons this family of maps has gained so much attention is the fact that these “free” critical points all behave symmetrically. This implies that there is essentially one critical orbit and then the λ−plane is a natural parameter plane for each of these families. Since n ≥ 2, the point at ∞ is a superattracting fixed point for any value of λ. Let Bλ denote the immediate basin of attraction of ∞, and notice that Bλ is mapped to itself at least in an n−to−1 fashion. When all the critical points are in Bλ the Julia set of the map is a Cantor set of points 1 b = C ∪ {∞} for the Riemann sphere. We use C for the complex plane and C 2 and Bλ is mapped to itself in an m−to−1 fashion. Let Tλ be the preimage of Bλ that contains the origin. When Tλ is disjoint from Bλ then Tλ is a simply connected set that is mapped d−to−1 onto Bλ . We call Tλ the trap door, since every point that escapes to infinity and it is not in Bλ must fall throuth Tλ along its orbit. When the orbits of all the critical points of Fλ are attracted to ∞, the Julia set of Fλ can have exactly three different topological structures. The following result is included in [2], where a more general result is proved. Theorem 1.1 (The Escape Trichotomy). Fix n, d ∈ N with 1/n + 1/d < 1 then, (a) If the critical values of Fλ lie in Bλ , then the Julia set is a Cantor set of points. (b) If the critical values of Fλ lie in Tλ 6= Bλ , then the Julia set is a Cantor set of simple closed curves. (c) If the critical values of Fλ lie in any other preimage of Tλ , then the Julia set is a Sierpiński curve. Case 1 corresponds to λ in the Cantor set locus, that is, an open connected set that surrounds infinity in the λ−plane where the Julia set of Fλ is a Cantor set of points. In this case the Fatou set consists of one infinitely connected region, i.e., Bλ = Tλ . Case 2 corresponds to the McMullen domain, that is, a punctured (at the origin) open disk that is bounded by a simple closed curve in the λ−plane where all the maps Fλ have Julia sets that are Cantor sets of simple closed curves. Then the Fatou set consists of 2 simply connected domains (Bλ and Tλ ) and infinitely many concentric annuli that are preimages of Tλ . Case 3 is very different; the parameter plane of the family Fλ shows infinitely many Sierpiński holes, that is, disjoint simply connected domains with parameters for which the Julia set of Fλ is a Sierpiński curve. A Sierpiński curve is a set homeomorphic to the well-known Sierpiński carpet fractal, see [28]. For convenience, when comparing different maps from the family Fλ , sometimes we use < n, d > to denote z n + λ/z d . Figure 1 illustrates the Escape Trichotomy Theorem 1.1. Unlike cases 1 and 2, Sierpiński curve Julia sets happen also when the free critical points do not escape to infinity, see for example [29, 30, 31]. In this paper we discuss some of the properties of the parameter plane for these maps. The following result can be found in [6]. 3 Bλ Bλ Tλ = Bλ Tλ Tλ B B B M M M Figure 1: The top line of pictures shows Julia sets for the family Fλ (z) that illustrate the three topological structures presented in the Escape Trichotomy Theorem 1.1. From left to right: a Cantor set of points for the case < n, d >=< 5, 6 > with λ = 1 + i, a Cantor set of simple closed curves for the case < 6, 5 > with λ = 0.001(1 + 2i) and a Sierpiński curve for the case < 7, 5 > with λ = 13i/50. The basin of attraction of ∞ is Bλ and its preimage containing the origin is the trap door Tλ . The maps < 5, 6 > and < 6, 5 > have degree m = 11 and therefore there are 11 sectors of the sphere that are equal to each other under rotation by 2π/11. The map < 7, 5 > has degree m = 12 and therefore there are 12 sectors of the sphere that are equal to each other under rotation by π/6. The bottom line of pictures shows (from left to right) the parameter λ−planes for < 5, 6 >, < 6, 5 > and < 7, 5 >. Notice the (n − 1)−fold symmetry of these planes illustrated by the white baby Mandelbrot sets symmetrically distributed around the origin λ = 0. The unbounded region is the Cantor set locus B, the disk centered at the origin is the McMullen domain M and the other shaded disks in the connectedness locus correspond to Sierpiński holes. 4 Theorem 1.2 (Principal baby Mandelbrot sets with halos) Fix n, d ∈ N with 1/n + 1/d < 1, then there exists a small copy of the Mandelbrot set in pa- rameter λ−plane for Fλ in each of the n − 1 sectors of the form (2j − 1)π (2j + 1)π < Arg λ < , j = 0, 1, 2, ..., n − 2. n−1 n−1 Each of these baby Mandelbrot sets have infinitely many “halos” attached, i.e., infinitely many points on the boundary of Sierpinki holes. Notice that the cases with n = 2 and d > 2 show only one principal baby Mandelbrot set straddling the real axis. These babies are slightly different from the ones in the rest of the family in the sense that the “tail” (the portion of the set to the left of the period-2 bulb) of these Mandelbrot sets and the period-2 bulbs are smaller. When n = d = 2, a map that is not in our family, the tail seems to completely disappear. See Figure 2. It is known that there are infinitely many disjoint Sierpiński holes for each of these families (see, [10] and [32]) and that there is a unique parameter in each Sierpiński hole for which the orbit of the critical point lands on 0 at some iteration and therefore on ∞ at the next iteration, say at iteration k > 2. We then call this λ-value the center of the Sierpiński hole and k the escape time of the hole. All other parameters in a given Sierpiński hole have the property that the orbit of the critical value lands in Bλ at the escape time iterate. By Whyburn’s result, the Julia sets corresponding to any two parameters drawn from a Sierpiński hole are homeomorphic. However, as shown in [2], there exist Sierpiński holes corresponding to each escape time k ≥ 3, and these have the property that if λ1 and λ2 lie in Sierpiński holes with different escape times, then Fλ1 and Fλ2 are not topologically conjugate on their Julia sets. Much more information about the λ−planes for the case n = d ≥ 3 is given in [20, 21, 22]. Our goal in this paper is to investigate further properties of the param- eter plane for these maps and, in particular, to extend a result from [3] that explains the structure of the parameter plane in a neighborhood of the McMullen domain. Figure 3 displays magnifications of the region around the McMullen do- main in the case n = 5 and d = 6. In the first image, note that there are four large Sierpiński holes symmetrically placed around the McMullen domain. These Sierpiński holes all have escape time 3. Between the Sierpiński holes 5 Figure 2: The top line of pictures shows the parameter planes for the family z n + λ/z d from left to right we have < n, d >=< 2, 2 >, < 2, 3 > and < 2, 4 >. The bottom line of pictures shows magnifications in the region −0.03 < 1, τkn,d = dnk−2 (n − 1) − nk−1 + 1. 7 (c) The curve S k also passes through τkn,d superstable parameter values where a critical point of Fλ is periodic of period qk for some q ∈ N. The ideas behind the proof are similar to the ones used in [3]. However, some important differences are the fact that in the general case there is no involution of the form z 7→ λ1/n /z that conjugates Fλ to itself as in the case n = d. Also, the proof in the more general case requires the use of generalized symmetries and ideas that where developed in [6, 33]. The paper is organized as follows. In §2 we explain the main dynamical properties of the family Fλ and describe part of the general structure of the parameter λ-plane. In §3 we present the proof of Theorem A. In §3.1 we prove the base case, that is, the existence of the curve S 1 . In §3.2 and §3.3 we present the mapping properties of Fλ . The existence of the rest of the rings S k for k > 1 is proven in §3.4 and §3.5. The rings are first constructed in the dynamical plane (§3.4) and then translated to the parameter plane (§3.5). 2 Preliminaries and previous results In this section we explain the main properties that arise in the dynamical and the parameter planes of the family Fλ , given by λ Fλ (z) = z n + zd where n, d ∈ N with n, d ≥ 2 and the parameter λ ∈ C. 2.1 Dynamical plane Note that when λ 6= 0, Fλ has degree m = n + d and so it has 2m − 2 critical points counting multiplicities. Infinity and the origin are critical points with multiplicities n − 1 and d − 1, respectively. As we mentioned before, ∞ is a superattracting fixed point of Fλ and 0 is the only finite preimage of it. Thus, the remaining m “free” critical points, are given by  1/m λd cλ = . (2.1) n We see that the critical points all lie on a circle, the critical circle, and are symmetrically distributed around the origin. The associated critical values 8 vλ = Fλ (cλ ) are also distributed in a circle, the critical value circle, and simple computations show that  n/m m λd vλ = Fλ (cλ ) = . (2.2) d n For later reference, we also point out that there are m prepoles pλ , verifying Fλ (pλ ) = 0, so pλ = (−λ)1/m (2.3) that lie on a circle, the prepole circle, and are also symmetrically located around the origin. We use Cλ , Vλ and Pλ to denote the critical circle, the critical value circle and the prepole circle, respectively. The three circles are centered at the origin with radius |cλ |, |vλ | and |pλ |, respectively. We can compare any two of these three circles in the natural way by comparing their corresponding radii. We introduce the following notation, • Cλ ≺ Vλ if and only if |cλ | < |vλ |, • Cλ = Vλ if and only if |cλ | = |vλ |, • Cλ  Vλ if and only if |cλ | > |vλ |. When the context is clear we may drop the subindex λ of the critical 2π points, critical values and prepoles. Let ν = ei m be a primitive m−th root of unity, we write cj = ν j cλ for j = 0, 1, · · · , m − 1, where cλ is a concrete critical point. The same applies to the critical values and the prepoles. We restrict the use of the letters c, v and p for critical points, critical values and prepoles, respectively. We turn our attention to the symmetries that arise in the dynamical plane of the family Fλ . It is easy to check that Fλ (νz) = ν n Fλ (z) and then for all k ∈ N, k Fλk (νz) = ν n Fλk (z). (2.4) j Hence the orbits of points of the form ν z all behave “symmetrically” under iteration of Fλ . For example, if Fλk (z) → ∞, then Fλk (ν j z) also tends to ∞ for each j. If Fλk (z) tends to an attracting cycle, then so does Fλk (ν j z). Note, however, that the cycles involved may be different depending on j and, indeed, they may even have different periods. Nonetheless, all points lying on these attracting cycles are of the form ν j z0 for some z0 ∈ C. 9 Theorem 2.1 (Symmetries in dynamical plane, see [6]) Let n, d ∈ N with 2π 1/n + 1/d < 1 and let ν = ei m , then there exist r, q ∈ N such that for all k ≥ r, k Fλq+k (νz) = ν n Fλq+k (z). (2.5) Moreover, (a) If every prime factor of m is a prime factor of n, then there exists k r ∈ N such that ∀k ≥ r, ν n = 1 and the orbits of νz and z coincide, that is, Fλk (νz) = Fλk (z). q (b) If m and n are relatively prime, then there exists q ∈ N such that ν n = ν and Fλq is conjugate to itself under z 7→ νz, that is, Fλq (νz) = νFλq (z). This implies that the dynamics and the Julia sets of Fλ are symmetric under rotation by 2π/m, see Figure 1. We call the straight line connecting the origin to ∞ and through one of the critical points (resp., prepoles) a critical point ray (resp., prepole ray). The critical point rays Cj are defined by tcj with 0 < t < ∞ and j = 0, 1, . . . , m − 1. It follows from Equation (2.4) that for all j ∈ N we have Fλ (ν j z) = ν jn Fλ (z) and then the image of the ray Cj is given by Fλ (tcj ) = t0 vj 0 , where vj 0 = Fλ (cj ) and j 0 = jn (mod m) and t0 > 0. Similarly, let the prepole ray Pj be given by tpj with 0 < t < ∞ and j = 0, 1, . . . , m − 1. Then the image of the ray Pj is given by Fλ (tpj ) = t0 pnj , where t0 = t−d (tm − 1) is a real number. Notice that if t > 1, then t0 > 0, and if t < 1, then t0 < 0. It follows that the straight ray connecting prepole pj to ∞ is mapped onto a straight ray connecting the origin to ∞, and the segment connecting the origin and the prepole pj is mapped onto the opposite straight ray to that of t > 1. Hence each of the critical point rays is mapped in a two-to-one fashion onto a ray connecting vj 0 to ∞. Similarly, each of the m prepole rays is mapped in a one-to-one fashion onto the straight line defined by t0 pnj , where t0 is now any real number. 2.2 Parameter plane Let C denote the connectedness locus, that is, the set of λ-values for which the Julia set of Fλ is a connected set. The complement of this set in C∗ = C−{0} has two components each homeomorphic to a disk with a puncture, namely, 10 the Cantor set locus, that is the set of λ-values for which the Julia set is a Cantor set of points, that we denote by B; and the McMullen domain, that is, the set of λ-values for which the Julia set is a Cantor set of simple closed curves, that we denote by M. See Figures 1 and 3. The following bounds of the regions C, B and M in the λ-plane are found in [6],   nd−m md n d If λ ≤ λmin = d 2m then λ ∈ M.   mn (2.6) n 2d If λ ≥ λmax = d m then λ ∈ B. We turn now our attention to the relative position of the three concentric circles defined in the dynamical plane: the critical circle Cλ , the critical value circle Vλ and the prepole circle Pλ . The relative position between these circles depends on the values of n, d and λ. We first analyze the relative position between Cλ and Pλ , since it does not depend on λ. Thus, for example when n = d then Pλ = Cλ , when n > d then Pλ  Cλ and, finally when n < d then P λ ≺ Cλ . Finally, we also define two auxiliary real values λc and λp . The first one, λc , is the real value of λ such that Vλ = Cλ . The second one, λp , is such that Vλ = Pλ . Simple computations show that   n−1 m  m   n−1  n n d d n n−1 λc = and λp = . (2.7) d m m d The circle of radius λc plays an important role in the parameter plane, for if λ lies inside the circle of radius λc , that is, |λ| < λc , then Vλ ≺ Cλ . Instead, if λ lies in the circle of radius λc , that is, |λ| = λc , then Vλ = Cλ . Finally, if |λ| > λc then Vλ  Cλ . We call the circle of radius λc in parameter plane the dividing critical circle. In a similar fashion, if λ lies inside the circle of radius λp then Vλ ≺ Pλ . We call the circle of radius λp in parameter plane the dividing prepole circle. It is easy to see that if n > d then λp > λc and this implies that Pλ  Cλ , that is, the prepole circle lies outside the critical point circle. If n = d then λp = λc and Pλ = Cλ . Finally, if d > n then λc > λp and Cλ  Pλ . The parameter plane also possesses several symmetries. First of all, we have Fλ (z) = Fλ (z) so that Fλ and Fλ are conjugate via the map z 7→ z. Therefore the parameter plane is symmetric under the map λ 7→ λ. We also have (n − 1)−fold symmetry in the parameter plane for Fλ . See Figure 1. 11 Theorem 2.2 (Symmetries in parameter λ−plane, see [6]) Let n, d ∈ N 2π with 1/n + 1/d < 1. Let ω = ei n−1 , then there are k 0 , q ∈ N such that Fλq is k0 q conjugate to Fωλ under z 7→ ω q z, that is, k0 k0 q Fωλ (ω q z) = ω q Fλq (z). It follows that the parameter λ-plane is symmetric under the map λ 7→ ωλ. 3 Proof of Theorem A In this section we prove the main result of this paper. The proof of the existence of the rings S k for k ∈ N when n, d ∈ N, and verifying n, d ≥ 5, is done by induction. 3.1 The curve S 1 In the case that n = d the first ring S 1 , is a round circle in the parameter plane such that for all λ values in this circle, the critical points, the critical values and the prepoles belong to the same circle. Since, when changing the argument of λ, the critical points and prepoles rotate at a rate n-times slower than the critical value, we get the results of [3]. In the case that n 6= d the critical circle Cλ and the prepole cicle Pλ have different radii, and this fact makes the proof of S 1 necessary. However, in this section we show that there is a natural curve, denoted by Γλ , in the dynamical plane that goes through each critical point and prepole of Fλ . See Figure 5. We translate this curve Γλ to the parameter plane to find the curve S 1 that goes through the centers of Sierpinski holes and the centers of the main cardioids of the principal baby Mandelbrot sets. We split the proof of the existence of the curve S 1 in the parameter plane into two parts. In the first part, Propositions 3.1, 3.2 and 3.3, we prove the existence of a Jordan curve Γλ that surrounds the origin in dynamical plane such that it contains every critical point and prepole and is symmetric under rotation by 2π/m. We also show that the curve Γλ is contained in between the critical circle Cλ and the prepole circle Pλ , except at the intersection points where they agree. In the second part, Proposition 3.4, we transfer this curve to the parameter λ−plane, using the Schwarz Lemma, to show the existence of S 1 . This new curve is also a Jordan curve that surrounds the 12 origin and contains the centers of the main cardioids of (n−1) principal baby Mandelbrot sets and the same number of centers of Sierpinski holes in the case n = d. Indeed, the existence of these (n − 1) principal baby Mandelbrot set was proved in [6]. We recall that Vλ denotes the critical value circle, that is the circle of radius |vλ | centered at the origin. Proposition 3.1 The preimage of the critical value circle Vλ consists of two simple closed curves ΓI and ΓE that surround the origin and intersect exactly at the m critical points. There is a third simple closed curve Γλ that goes through the critical points and prepoles and is contained in between ΓI and ΓE . Moreover, Fλ maps the curve Γλ inside the closed disk bounded by Vλ and for all z ∈ Γλ we have that min{|cλ |, |pλ |} ≤ |z| ≤ max{|cλ |, |pλ |}, (3.1) and ΓI , ΓE , and Γλ are symmetric with respect to rotations by 2π/m. Proof. We first explain the idea of the construction and later on we compute explicitly these preimages of the critical value circle Vλ . We observe that Vλ contains all the critical values of Fλ except the critical value located at infinity. We recall that ∞ is a critical point with multiplicity n − 1 and 0 is also a critical point with mutlitplicity d − 1. Thus the set C \ Vλ has two connected components, the first one is a punctured annulus A between Vλ and ∞ and, the second one, a round disk D bounded by Vλ (see Figure 5). We claim that the preimage of the punctured annulus A are formed by two annuli, A∞ and A0 , the first one punctured at ∞ and the second punctured at 0. To see the claim, we notice that by the Riemann-Hurwitz formula the preimage of an annulus without critical values is always an annulus and the only preimages of ∞ by Fλ are ∞ and 0. Moreover, the boundaries of the punctured annulus A∞ are ∞ and a Jordan curve, denoted by ΓE , and by construction Fλ (ΓE ) = Vλ with degree n. In a similar way, the boundaries of the punctured annulus A0 are 0 and a Jordan curve, denoted by ΓI such that Fλ (ΓI ) = Vλ with degree d. Finally, since by the Riemann-Hurwitz formula the preimage of a disk without critical values is always a simply connected region and the map between these two simply connected regions is conformal, the preimage of the disk D consists of m simply connected regions, denoted by Ωj for 0 ≤ j ≤ m − 1, each one containing one of the m prepoles and such that Fλ maps each Ωj conformally onto the disk D. See Figure 5. 13 Now we can compute explicitly the two curves ΓE and ΓI preimages of the critical value circle Vλ . For this, we write vλ = µλn/m , so that µ = m/d(d/n)n/m , see Equation (2.2). We search for the values of z such that 1 |Fλ (z)| = |vλ |, since the radius of Vλ is |vλ |. We write z = r|λ| m eiϕ with r ≥ 0, and using the definition of vλ = Fλ (cλ ) (see, Equation (2.2)) we have that n |Fλ (z)| = |λ| m rn |1 + r−m ei(Arg (λ)−mϕ) | = |vλ | = µ|λ|n/m , where λ = |λ|eiArg (λ) . We define the auxiliary function H(r, ϕ) = r2n |1 + r−m ei(Arg (λ)−mϕ) |2 = r2n + 2rn−d cos(Arg (λ) − mϕ) + r−2d , (3.2) and now the preimage of Vλ writes as H(r, ϕ) = µ2 . We obviously have that qmin (r) = r2n − 2rn−d + r−2d ≤ H(r, ϕ) ≤ r2n + 2rn−d + r−2d = qmax (r). It is easy to see that qmin has minimum at r = 1 with  qmin (1) = 0 and qmax 1 1 has a minimum at r = (d/n) m with qmax (d/n) m = µ2 . See Figure 4. Computing the derivative of H(r) with respect to r gives ∂H (r, ϕ) = 2r−2d−1 (nr2m + (n − d)rm cos(Arg (λ) − mϕ) − d). ∂r Now let ρ = rm and let g(ρ) = nρ2 + (n − d)ρ cos(Arg (λ) − mϕ) − d. We see that there exists a unique ρ∗ (ϕ) > 0 for which g(ρ∗ (ϕ)) = 0 given by,  s  2 1 d − n n−d cos(Arg (λ) − mϕ) + cos2 (Arg (λ) − mϕ) + 4d/n 2 n n (3.3) 1 Then, for each ϕ the function H(r, ϕ) has a minimum at r∗ (ϕ) = ρ∗ (ϕ) m . By (3.2) we see that for each ϕ, lim H(r, ϕ) = lim H(r, ϕ) = ∞. r→0+ r→∞ Hence we conclude that H(r∗ , ϕ) ≤ µ2 . See Figure 4. Then there exists three unique functions rI (ϕ), r∗ (ϕ) and rE (ϕ), such that rI (ϕ) ≤ r∗ (ϕ) ≤ rE (ϕ) and H(rI (ϕ), ϕ) = H(rE (ϕ), ϕ) = µ2 . Thus we can define 14 qmax (r) qmin (r) µ2 H(r, ϕ) r r∗ ρ∗ ρ = rm 0 g(ρ) Figure 4: Diagram for the proof of Proposition 3.1, for fixed ϕ. 1 • ΓE (ϕ) = |λ| m · rE (ϕ) · eiϕ 1 • ΓI (ϕ) = |λ| m · rI (ϕ) · eiϕ 1 • Γλ (ϕ) = |λ| m · r∗ (ϕ) · eiϕ By construction we have that Fλ (ΓE ) = Fλ (ΓI ) = Vλ and Fλ (Γλ ) ⊂ D. We now turn our attention to the curve Γλ . First, we observe that 1 if Arg (λ) − mϕ = 0 (mod 2π), then rI (ϕ) = rE (ϕ) = r∗ (ϕ) = (d/n) m 1 and Γλ (rE (ϕ)) = Γλ (rI (ϕ)) = Γλ (r∗ (ϕ)) = (λd/n) m , thus the three curves pass trough the critical points cλ . If Arg (λ) − mϕ 6= 0 (mod 2π), then rI < r∗ < rE . Finally, if Arg (λ) − mϕ = π(mod 2π) then we obtain r∗ = 1 1 and Γλ (ϕ) = (−λ) m , proving that Γλ passes trough the prepoles pλ . The maximum and minimum of ρ∗ (ϕ), see Equation (3.3), are determined by the maximum and minimum of the term d−n n cos(Arg (λ) − mϕ), that depends on the values of d and n. When n ≥ d we see that the maximum is obtained for Arg (λ) − mϕ = 0 (mod 2π), and the minimum for Arg (λ) − mϕ = π (mod 2π), while for d > n the opposite happens. It follows that 1 1 min{1, (d/n) m } ≤ r∗ (ϕ) ≤ max{1, (d/n) m } 15 Thus, if z lies in Γλ we see that min{|cλ |, |pλ |} ≤ |z| ≤ max{|cλ |, |pλ |}. Finally, by construction these three curves are symmetric with respect to rotations around the origin by z → νz, with ν = ei2π/m . 2 c2 c1 c3 Ωj Fλ c0 c4 Γλ ΓE A Vλ ΓI D c10 c5 c9 c6 c7 c8 Figure 5: The curves ΓI (grey), ΓE (solid black) and Γλ (dashed) when n = 5, d = 6 and then m = 11. These three curves intersect at the critical points. The region Ωj between ΓI and ΓE contains a prepole that lies in the curve Γλ , and it is mapped by Fλ in a one-to-one fashion over D. For n = d ≥ 3 the curve Γλ coincides with the critical circle. Figure 5 illustrates the curves when λ is small so that |vλ | < min{|cλ |, |pλ |}. If n = d then critical points and prepoles are at the same distance from the origin. Instead, if n > d we see that the critical points are closer to the origin than the prepoles, and if d > n then the prepoles are closer to the origin than the critical points. The image of Γλ is a “flower” that passes successively through the critical values and the origin and it is contained in the closure of the critical value circle. It follows from the previous proposition that there are m domains Ωj with j = 0, 1, . . . , m − 1, whose boundaries consist of ΓI and ΓE in between two consecutive critical points. Each of these domains Ωj contains a prepole inside and is mapped in a one-to-one fashion onto D. The following result is a direct consequence of Proposition 3.1. 16 Proposition 3.2 The curves ΓI and ΓE that are the preimages of Vλ are such that (a) Fλ takes the curve ΓE passing through two consecutive critical points one-to-one onto a sector of Vλ whose argument is 2πn m . Fλ takes the curve ΓI passing through two consecutive critical points one-to-one onto the remainder sector of Vλ whose argument is 2πd m . (b) Fλ maps the domain Ωj that is enclosed by the curves ΓI and ΓE between two consecutive critical points in a one-to-one fashion onto the domain D. (c) Fλ maps the curve ΓE in an n-to-one fashion, and the curve ΓI in a d-to-one fashion, both onto the critical value circle Vλ . In the parameter λ−plane we restrict our attention to the region O given by π O = {λ ∈ C | λmin < |λ| < λmax and |Arg (λ)| < }. n−1 See subsection 2.6 for the definition and properties of λmin and λmax . For λ ∈ O, we denote by c0 = c0 (λ) the unique critical point with |Arg (c0 )| < π/m. We notice that if λ ∈ R+ then c0 ∈ R+ . Hereafter we parametrize Γλ (θ) for θ ∈ R and we recall that this curve joins critical points and prepoles in the counterclockwise direction. So Γλ (θ) is 2π-periodic in θ and depends analytically on λ for λ ∈ O. We fix Γλ (0) = Γλ (2π) = c0 (λ). In the next proposition we bound the position of the curve Γλ when the parameter λ belongs to O, see Equation (2.6). Proposition 3.3 The curve Γλ is contained in the region {z ∈ C ; vλmin ≤ nd m |z| ≤ vλmax } for all λ-values in O, where vλmin = (1/2) nd−m (d/m) nd−m and vλmax = 2. Proof. Let λ be a parameter in O then we have that cλmin ≤ |cλ | ≤ cλmax , vλmin ≤ |cλ | ≤ vλmax and pλmin ≤ |pλ | ≤ pλmax , see Equations (2.1) and (2.3). Using the definition of vλ = Fλ (cλ ) we have that, nd m vλmin = Fλmin (cλmin ) = (1/2) nd−m (d/m) nd−m (3.4) vλmax = Fλmax (cλmax ) = 2 17 We recall that for any parameter λ the critical circle Cλ , the critical value circle Vλ and the prepole circle Pλ have radii |cλ |, |vλ | and |pλ |, respectively. Moreover, for all z in the curve Γλ , see Proposition 3.1, min{|cλ |, |pλ |} ≤ |z| ≤ max{|cλ |, |pλ |}. First we show that vλmax > max{cλmax , pλmax }. On the one hand, since  1/n d/m < 1, we have that cλmax = 2d m < 2 = vλmax . On the other hand, we have pλmax = (n/d)1/m 21/n (d/m)1/n and we need to analyze different cases. If d ≥ n then by the above inequality we conclude that pλmax ≤ cλmax < 2. If n > d we see that  1/n  1/n  1/n 1/m 1/n d 1 1/m 1/n d 1 n pλmax = n 2 d nm = vλmin . 2m 2 m 2 m For the inequality that corresponds to pλmin = (n/d)1/m (d/2m)d/(nd−m) , we compute vλmin /pλmin and show it is smaller than 1. We start with   d(n−1)  1   n vλmin 1 nd−m d m d nd−m = , pλmin 2 n m and notice that, since d(n − 1)/(nd − m) > 1 we have   m1   nd−m n vλmin 1 d d < . pλmin 2 n m Then, it is enough to show that   m1   nd−m n d d < 2, n m or, equivalently, that 18   nd−m mn d m < 2m . n d But this is clear, since d < 2n+d . n This last inequality follows easily from elementary calculus. Then vλmin < pλmin , as we wanted to show. 2 Now, we are ready to prove the existence of the first ring S 1 in the param- eter plane. To do so, we transfer the curve Γλ constructed in the dynamical plane to the parameter λ−plane, using the Schwarz Lemma. Proposition 3.4 The ring S 1 in parameter space is a simple closed curve surrounding the McMullen domain of Fλ , such that it passes through n − 1 centers of Sierpiński holes with escpape time equal to 3, and n − 1 superstable parameter values where a critical point of Fλ is periodic of period q for some q ∈ N. Proof. We consider the simply connected region defined in the parameter plane O given by, π O = {λ ∈ C | λmin < |λ| < λmax and |Arg (λ)| < }. n−1 Given a parameter λ ∈ O, we denote by c0 (λ) the critical point given by  1/m d|λ| c0 (λ) = eiArg (λ)/m . n We claim that c0 (λ) is a holomorphic  map of λ. To see the claim we notice 1 that c0 (λ) = exp m log(dλ/n) , where log denotes the principal value of the logarithm map and since n ≥ 2, then |Arg (λ)| < π/(n − 1) ≤ π. We define the map G : O 7→ W defined by G(λ) = v0 (λ), where v0 (λ) is the critical value of Fλ corresponding to the critical point c0 (λ). This map takes O onto the set W . Both sets O and W are simply connected. It is easy to compute the set W , the precise values of vλmin and vλmax are given by Equation (3.4). We have, 19 nπ W = {z ∈ C ; vλmin < |z| < vλmax and |Arg (z)| < }. m(n − 1) Also, by Proposition 3.3 the map Γλ : O 7→ W defined by λ 7→ Γλ (θ) takes O inside W. This function is analytic on its domain, and its image is bounded from the origin. We finally define, for each value of θ, the new map Q(λ) = G−1 (Γλ (θ)). Then applying the Schwarz Lemma to the map Q we obtain a fixed point inside O so that there exists a unique value of λ, that we denote λθ , such that for each θ the point vλθ ∈ Γλ , i.e., Γλθ (θ) = v0 (λθ ). The set of λθ ’s for which this happens varies continuously with θ and defines a simple closed curve in the parameter plane that is symmetric under rotation by 2π/m and such that the critical value is mapped to the curve Γλ . This curve is the first curve S 1 . π O nπ W n−1 m(n−1) G λmin λmax vmin vmax = 2 Γ S1 π nπ − n−1 − m(n−1) Figure 6: The mapping from set O to set W in the parameter λ-plane. For all points z in W we have that π n |Arg (z)| < · , m n−1 and this is larger than 2π/m. Then we see that in the region O we have exactly two parameters values λθ1 and λθ2 , such that v0 (λθ1 ) = c0 (λθ1 ) and v0 (λθ2 ) = p0 (λθ2 ). The first one corresponding to the center of the main cardiod of the principal Mandelbrot set, and the second one corresponding to the center of a main Sierpiński hole. Then the critical value vλ coincides with one critical point and one prepole. Finally, using the symmetry of the 20 parameter plane, see Theorem 2.2, we get n − 1 cases where λθ corresponds to vλθ = cλθ and vλθ = pλθ . These values locate the centers of the main cardioids of the n − 1 principal Mandelbrot sets alternating the centers of n − 1 Sierpinski holes. 2 Using the curve S 1 we can divide the parameter λ-plane into two pieces bounded by S 1 . We denote by I(S 1 ) and E(S 1 ) the bounded and unbounded component of C \ S 1 , respectively. By Proposition 3.1 the image of Γλ under Fλ is contained in the closed disk bounded by the critical value circle Vλ centered at the origin with radius |vλ |. Therefore, we have the following result. Proposition 3.5 The set I(S 1 ) contains all the parameters λ such that the critical values vλ lie in the region bounded by Γλ . Moreover, when λ ∈ I(S 1 ) then Fλ maps Γλ strictly inside itself. 3.2 Elementary Mapping Properties In the rest of the paper we restrict, for technical reasons, to the parameter values λ in the following simply connected region depending on d and n. Recall that if n ≤ d then Cλ  Pλ , and if n > d then Pλ  Cλ , see §2.2. • If n ≤ d then O0 = {λ ∈ C ; |λ| > λmin , λ ∈ I(S 1 ), and |Arg λ| < π}. (3.5) • If n > d then d−1 O0 = {λ ∈ C ; |λ| > λmin , λ ∈ I(S 1 ), and |Arg λ| < π}. n−1 (3.6) For example, in the case n = 5 and d = 6, as well as for all cases where n = d, we get that |Arg λ| < π. Instead, for example, when n = 6, d = 5 we get |Arg λ| < 4π/5. We recall that when |λ| < λmin , see Equation (2.6), then the Julia set of Fλ is a Cantor set of circles. In addition, we observe that when λ ∈ O0 then Fλ maps Γλ strictly inside of itself, see Proposition 3.5. As before, for λ ∈ O0 , we denote by c0 = c0 (λ) the unique critical point with |Arg (c0 )| < π/m, where c0 ∈ R+ if λ ∈ R+ . 21 Fλ (B) 2πn/m 2πd/m Arg c0 + π/m Fλ (A) A c0 B Arg c0 − π/m ∂Vλ Fλ (B) ∂Vλ Figure 7: In this picture we illustrate the critical sectors A, B and their images for the case d > n (e.g., n = 5, d = 6), and then Fλ (A) ⊂ Fλ (B), with |λ| small so that |vλ | < |cλ |. The critical point sector has argument 2π/m. The image of A (dark grey) lies in a sector with argument 2πn/m, and the image of B (light grey) lies in a sector with argument 2πd/m. Let A be a sector bounded by two prepole rays corresponding to adjacent prepoles in the exterior of the curve ΓE , and let B be a sector bounded by two prepole rays corresponding to adjacent prepoles in the interior of the curve ΓI , i.e., A is an exterior sector, and B is an interior sector in the dynamical plane with argument 2π/m. We call A a critical point exterior sector and B a critical point interior sector since A and B contain at the “center” of their boundary a unique common critical point c0 (λ) of Fλ . See Figure 7. We also define the following auxiliary sets, n o A− = z ∈ A : Arg c0 − π/m ≤ Arg z ≤ Arg c0 , n o B+ = z ∈ B; Arg c0 ≤ Arg z ≤ Arg c0 + π/m . It follows that, Proposition 3.6 Let Fλ (z) = z n + λ/z d , with 1/n + 1/d < 1. For λ ∈ O0 we have that, 22 Arg c0 + π/m Arg c0 A− B+ ∂Tλ Arg c0 − π/m Figure 8: The regions A− and B+ . (a) Fλ maps A in a one-to-one fashion onto the sector with argument 2πn m , bounded by the images of two prepole rays, i.e., two half straight lines in the exterior of the critical value circle Vλ . (b) Fλ maps B in a one-to-one fashion onto the sector bounded by the opposite straight rays to the line boundaries of Fλ (A) with argument 2πd/m, in the exterior of Vλ . The circular boundaries of Fλ (A) and Fλ (B) both contain vλ = Fλ (cλ ), the common boundary point of A and B. (c) Moreover, either Fλ (A) or Fλ (B) is contained in the other, that is, A ⊂ Fλ (A) ⊂ Fλ (B) when d ≥ n, and A ⊂ Fλ (B) ⊂ Fλ (A) when n > d. (d) If Arg λ > 0 then A− ⊂ Fλ (A− ) ∩ Fλ (B+ ). Proof. The first two statements are clear from Proposition 3.2. We first prove the third one. Let λ ∈ O0 . The straight line boundaries of A and B are the prepole rays and their arguments are denoted by Arg c0 ± π/m. Letting z be a point of the images of the straight line boundaries, we have Arg z = n(Arg c0 ± π/m) or Arg z = d(Arg c0 ∓ π/m). See §2.1. 23 π Using the definition of the critical value c0 , we have that |Arg (c0 )| < m and we see that     π π π π n Arg c0 − < Arg c0 − < 0 < Arg c0 + < n Arg c0 + , m m m m and     π π π π d Arg c0 − < Arg c0 − < 0 < Arg c0 + < d Arg c0 + . m m m m Since 2πd/m + 2πn/m = 2π we see that, either Fλ (A) or Fλ (B) is con- tained in the other. If d ≥ n we obtain that A ⊂ Fλ (A) ⊂ Fλ (B), and if d < n we obtain the symmetric result A ⊂ Fλ (B) ⊂ Fλ (A). Now we prove the fourth statement. Let λ ∈ O0 with Arg λ > 0. The curve ΓE boundary of A− lies outside the circular boundaries Vλ of Fλ (A− ) and Fλ (B+ ). On the one hand, we know that the arguments of the line bound- aries of A− are Arg c0 − π/m and Arg c0 . In a similar way, the arguments of the line boundaries of Fλ (A− ) are Arg v0 = nArg c0 and Arg v0 − πn/m. So we have that Arg v0 − nπ/m < Arg c0 − π/m < 0 < Arg c0 < Arg v0 since Arg λ > 0. This shows that A− ⊂ Fλ (A− ). On the other hand, the arguments of the line boundaries of Fλ (B+ ) are dArg c0 and dArg c0 − dπ/m. Now using the different definition of O0 de- pending on d and n, see Equations (3.5) and (3.6), we have that • If n ≤ d then Arg c0 < π/m and (n − 1)Arg c0 − π(d − 1)/m < 0. • If n > d then Arg c0 < π(d − 1)/(m(n − 1)) and (n − 1)Arg c0 − π(d − 1)/m < 0, Thus, since πd π π(d − 1) Arg v0 − = Arg c0 − + (n − 1)Arg c0 − , m m m we get πd π Arg v0 − < Arg c0 − < 0 < Arg c0 < Arg v0 . m m Hence A− ⊂ Fλ (B+ ), and the result follows. 2 24 3.3 Some Special Cases In this section we discuss the dynamics of several special cases of Fλ that help define the rings around the McMullen domain. First suppose that λ > 0 and such that the critical point c0 is a fixed point, see §2.2. If   n−1 m n d λ= > 0, d m then   mn   m1 m d n d vλ = Fλ (cλ ) = λm = λ = cλ , d n n so that cλ ∈ R+ is a super-attracting fixed point. By the symmetry of the parameter plane, it follows that for n − 1 other parameters ω j λ with j = 0, 1, . . . , n − 2, where ω = ei2π/(n−1) , there is an iterate Fλq for some q ∈ N with a super-attracting fixed point, see Theorem 2.2. We next restrict attention to values of λ lying in R+ . Following [3], we can consider that there is a Mandelbrot set whose central spine lies along the interval [λ− , λ+ ], where λ− and λ+ correspond to the interval [−2, 1/4] of the actual Mandelbrot set, contained in R+ . The existence of these and other copies of Mandelbrot sets in the parameter plane of Fλ is given in [6, 20, 21, 22]. Proposition 3.7 (Superstable parameters for λ ∈ R+ ) There is a decreasing sequence of parameters in R+ , such that λ1 > λ2 · · · converging to λ− , and for λ = λk , the critical point c0 is periodic with period k, and the critical orbit in R+ has the special form when k ∈ N and k ≥ 2 : 0 < vλ = Fλ (c0 ) < c0 = Fλk (c0 ) < Fλk−1 (c0 ) < · · · < Fλ3 (c0 ) < Fλ2 (c0 ). In particular, λk is a superstable parameter value of period k, and the orbit of Fλ2k (c0 ) is monotonically decreasing for k − 1 iterations along R+ . We shall show below that each λk lies on S k . Because of the (n − 1)-fold symmetry in the parameter plane, we have a similar sequence of superstable parameter values along the ray λ = ω · R+ , for Fλq (z) and some q ∈ N with 2π ω = ei n−1 . 25 Proposition 3.8 (Superstable parameters for λ ∈ ω · R+ ) . Let λ1 > λ2 · · · be the decreasing sequence in R+ defined in Proposition 3.7. Then there exists natural numbers k 0 , q and r with r ∈ {0, 1, ..., m − 1} such for λ = ωλk the orbit of the critical point cr is periodic with period kq and the critical orbit k0 along the line ω q · R+ has the special form when k ≥ 2 : (k−1)q Fλq (cr ) < cr = Fλkq (cr ) < Fλ (cr ) < · · · < Fλ3q (cr ) < Fλ2q (cr ). 2π 2π Proof. Let ν = ei m , ω = ei n−1 , and λ = aω with a ∈ R+ . Then c0 = cλ = 1 (dωa/n) m and cj = cλ ν j , where j = 0, 1, . . . , m − 1. By Theorem 2.2, with k0 a in the role of λ, we get that there exist k 0 , q ∈ N such that Faω q (ω q z) = k0 ω q Faq (z). Then, for some r, s ∈ N we have that k 0 s = r(n − 1) + 1 and k0 1 k0 k0 k0 cr = ν r c0 = ω q da/n m ∈ ω q · R+ . Since Fλq : ω q · R+ → ω q · R+ , the orbit k0 of the critical point cr lies on ω q · R+ . For a = λk , then the critical point k0 cr is periodic with period kq, the critical orbit for Fλq along the line ω q · R+ has the special abovementioned form when k ≥ 2, as we wanted to show. 2 3.4 Rings in Dynamical Plane In the rest of the paper we restrict, for technical reasons, to the parameter values λ in the simply connected region O0 depending on d and n, as defined in (3.5) and (3.6). Recall that when λ ∈ O0 then Fλ maps Γλ strictly inside itself, see Proposition 3.5. For λ ∈ O0 , we denote by c0 = c0 (λ) the unique critical point with |Arg (c0 )| < π/m. We notice that c0 ∈ R+ if λ ∈ R+ and thus the critical 2π poins of Fλ are located at cj = ν j c0 with ν = ei m for j = 0, 1, . . . , m − 1. In this section we prove the existence of infinitely many rings γλk and ξλk for k = 0, 1, 2, . . . in the dynamical plane. Each ring γλk (resp., ξλk ) is a simple closed curve that is mapped nk -to-1 (resp., dnk−1 -to-1) onto the curve γλ0 by Fλk . We shall use these rings in the next section to construct the rings S k in the parameter plane, for k > 1. We begin by defining γλ0 to be the curve Γλ , defined in Proposition 3.1. Let E and I be the exterior of ΓE and interior of ΓI , respectively. If λ ∈ O0 , then Fλ maps γλ0 strictly inside itself. By Proposition 3.2, it follows that Fλ maps the exterior of the simple closed curve ΓE and the interior of 26 ΓI as n-to-1 and d-to-1 covering, respectively, onto the exterior of the critical value circle Vλ . Therefore there are two preimages of γλ0 , that we denote by γλ1 and ξλ1 that lie in E and I, respectively. The curve γλ1 lies outside of γλ0 and the curve ξλ1 lies inside of γλ0 . The function Fλ maps the curve γλ1 (resp., ξλ1 ) in an n-to-1 (resp., d-to-1) fashion onto γλ0 . See Figure 9. Since Fλ is a covering map in these regions, it follows that γλ1 and ξλ1 must be disjoint simple closed curves. Then Fλ maps the exterior of γλ1 , n- to-1 onto the exterior of γλ0 , and there is a preimage of γλ1 lying in this region and mapped n-to-1 onto γλ1 . Call this simple closed curve γλ2 . Similarly, Fλ maps the interior of ξλ1 as a d-to-1 covering onto the exterior of γλ0 , then there is a preimage of γλ1 lying in the interior of ξλ1 and mapped d-to-1 onto γλ1 . Call this simple closed curve ξλ2 . Continuing inductively, we find a collection of simple closed curves γλk and ξλk for k ≥ 1 having the properties that: 1. γλk+1 (resp., ξλk+1 ) lies in the exterior of γλk (resp., interior of ξλk ); 2. Fλ takes γλk+1 as an n-to-1 covering onto γλk and Fλ takes ξλk+1 as a d-to-1 covering onto γλk ; 3. so Fλk+1 takes γλk+1 as an nk+1 -to-1 covering of γλ0 and Fλk+1 takes ξλk+1 as a dnk -to-1 covering of γλ0 ; 4. γλk+1 converge outward to the boundary of Bλ as k → ∞ and ξλk+1 converge inward to the boundary of Tλ as k → ∞. We now construct a parametrization of each of the γλk and ξλk . The curve γλ0 is the curve Γλ defined in Proposition 3.1. We parametrize it by γλ0 (θ) for θ ∈ R and we recall that this curve joins critical points and prepoles in the counterclockwise direction. So γλ0 (θ) is 2π-periodic in θ and depends analytically on λ for λ ∈ O0 . We fix γλ0 (0) = γλ0 (2π) = c0 (λ). By construction of Γλ (see Proposition 3.1) this curve is symmetric with respect to rotations around the origin by z → νz, with ν = ei2π/m . So, we have that ν j γλ0 (θ) = γλ0 (θ + 2πj/m) for all j and all θ, in particular the rest of the critical points located at ν j c0 correspond to γλ0 (2πj/m). To parametrize γλk , consider the critical point exterior sector A and the critical point interior sector B in Proposition 3.6. We assume A and B contain the critical point c0 (λ) in their boundaries, see Figure 7. It follows that there is a unique point in A mapped to c0 by Fλ ; call this point γλ1 (0). 27 ξλ1 2 ξλ2 γλ1 γλ ∂Bλ γλ0 Tλ Figure 9: Schematic picture that shows the positions of the curves γλ0 , ∂Bλ and ∂Tλ (thick) and how they are mapped by Fλ . The curve γλ0 is mapped inside of itself, the curves γλ2 , and ξλ2 are mapped to γλ1 , and the curves γλ1 and ξλ1 are mapped to γλ0 . There is also a unique point ξλ1 (0) ∈ B mapped to c0 . Then we define γλ1 (θ) and ξλ1 (θ) by requiring that Fλ (γλ1 (θ)) = Fλ (ξλ1 (θ)) = γλ0 (θ) and such that γλ1 (θ) and ξλ1 (θ) vary continuously with θ. Note that γλ1 (θ) is 2πn-periodic and ξλ1 is 2πd-periodic, respectively. The direction of ξλ1 (θ) is the opposite of γλ1 (θ). Since γλ1 (0) ∈ A and ξλ1 (0) ∈ B, we proceed inductively to define γλk (θ) and ξλk (θ) such that γλk (0) ∈ A and ξλk (0) ∈ B are the unique points that are mapped by Fλ to γλk−1 (0), and requiring that Fλ (γλk (θ)) = Fλ (ξλk (θ)) = γλk−1 (θ). (3.7) For each k, γλk (θ) is 2πnk periodic and ξλk (θ) is 2πdnk−1 periodic in θ, and both depend analytically on λ. The following proposition shows the relationship between the parametrization and the symmetry.  Proposition 3.9 For each k ∈ N and θ ∈ R, ν j γλk (θ) = γλk θ + 2πjnk /m and ν j ξλk (θ) = ξλk θ − 2πjdnk−1 /m . Proof. First when k = 1, using Equations (2.4) and (3.7) we have 28  Fλ (ν j γλ1 (θ)) = ν jn Fλ (γλ1 (θ)) = ν jn γλ0 (θ) = γλ0 θ + 2πjn/m  = Fλ (γλ1 θ + 2πjn/m ).  Then it follows that ν j γλ1 (θ) = γλ1 θ + 2πjn/m . For the induction argu-  ment, we assume that ν j γλk−1 (θ) = γλk−1 θ + 2πjnk−1 /m . Then, ν 2j γλk−1 (θ) =   ν j γλk−1 θ + 2πjnk−1 /m = γλk−1 θ + 4πjnk−1 /m , and ν jn γλk−1 (θ) = γλk−1 (θ + 2πjnk /m). (3.8) Hence, using Equations (2.4) and (3.7), we have Fλ (ν j γλk (θ)) = ν jn γλk−1 (θ) = γλk−1 (θ + 2πjnk /m) = Fλ (γλk (θ + 2πjnk /m)).  We conclude that ν j γλk (θ) = γλk θ + 2πjnk /m . Similarly, the result for ξλk is proved. 2 In the next proposition we study the location of the rings ξλk (θ) for certain values of θ. Proposition 3.10 If λ ∈ O0 , then for each k ≥ 1, if n ≤ d the portion of the ring ξλk (θ) with |θ| ≤ 2πdnk−1 /m lies in the  |Arg z| < 4π/m, and  region n d−1 if n > d ≥ 4 it lies in the region |Arg z| < π m n−1 . Proof. First we assume that λ ∈ O0 and Arg λ > 0. Since c0 ∈ A− it follows, from Proposition 3.6, that ξλk (0) ∈ B+ . Then, since the arguments of the line boundaries of B+ are Arg c0 and Arg c0 + π/m, we obtain Arg c0 < Arg ξλk (0) < Arg c0 + π/m. See Figure 8. We deal first with the case −2πdnk−1 /m ≤ θ ≤ 0. The direction of ξλk (θ) is opposite to that of θ, then Arg ξλk (0) ≤ Arg ξλk (θ) ≤ Arg ξλk (−2πdnk−1 /m). Since by Proposition 3.9, νξλk (0) = ξλk (−2πdnk−1 /m), it follows that   0 < Arg c0 < Arg ξλk (0) ≤ Arg ξλk (θ) ≤ Arg ξλk k−1 −2πdn /m = Arg νξλk (0) 2π 2π π 3π Arg λ = + Arg ξλk (0) < + Arg c0 + = + . (3.9) m m m m m Using the different definition of O0 depending on d and n, see Equations (3.5) and (3.6), we obtain that 29 4π • For n ≤ d, since 0 < Arg λ < π, then 0 < Arg ξλk (θ) < m . Then we have that |Arg ξλk (θ)| < 4π/m. • For n > d ≥ 4, since 0 < Arg λ < π(d−1)/(n−1), then 0 < Arg ξλk (θ) < 3π π d−1 n d−1 n d−1 m +m n−1 ≤ πm n−1 . So, we conclude that |Arg ξλk (θ)| < π m n−1 . k−1 Now consider the case 0 ≤  θ ≤k 2πdn /m. By Proposition 3.9, we have νξλk (θ) = ξλ θ − 2πdn /m = ξλ (t), where we let t = θ − 2πdnk−1 /m and k k−1 −2πdnk−1 /m ≤ t ≤ 0. It follows that ξλk (θ) = ν −1 ξλk (t) and Arg ξλk (θ) = Arg ξλk (t) − 2π/m. Then, since −2πdnk−1 /m ≤ t ≤ 0, it follows from Equation (3.9) that 3π Arg λ 0 < Arg ξλk (t) < + . m m Hence 2π π Arg λ − < Arg ξλk (θ) < + . m m m • For n ≤ d, since 0 < Arg λ < π, we have − 2π m < Arg ξλk (θ) < 2π m , and k then it follows that |Arg ξλ (θ)| < 4π/m. d−1 • For n > d ≥ 4, since 0 < Arg λ < π n−1 , we have − 2π m < Arg ξλk (θ) < π π d−1 n d−1 m +m n−1 ≤ πm n−1 . It is easy to see that 2π m < πmn d−1 n−1 , so we conclude k n d−1 that |Arg ξλ (θ)| ≤ π m n−1 . Finally, since 3π m < (3n−2)π m(n−1) , we have |Arg ξλk (θ)| < (3n−2)π m(n−1) . If Arg λ < 0, we invoke the z → z symmetry in the parameter plane. Since Fλ is conjugate to Fλ via z → z, it follows that the curves ξλk (θ) are mapped to ξλk (−θ) by the conjugacy. Hence these curves lie in the same region when Arg λ < 0. This concludes the proof. 2 We have shown the existence of the curves ξλk (θ), we have studied their parametrizations and positions. In the next section we translate these curves to the parameter plane to obtain the curves S k to obtain our main result. 30 3.5 Rings in Parameter Plane Before turning to the proof of the existence of the rings in the parameter plane, we need to examine more carefully the parametrizations of the rings in the dynamical plane in two of the special cases discusses earlier, namely 2π when λ ∈ R+ and λ ∈ ω · R+ , where ω = ei n−1 . First suppose that λ ∈ R+ . For the special parameters λk among the superstable parameters in R+ , we have seen that Fλk (c0 ) always lies in R+ and satisfies (see Proposition 3.7), 0 < Fλk (c0 ) < c0 = Fλkk (c0 ) < Fλk−1 k (c0 ) < · · · < Fλ3k (c0 ) < Fλ2k (c0 ). Hence, Fλ2k (c0 ) lies on γλk−2k ∩ R+ , and Fλjk (c0 ) lies on γλk−j k ∩ R+ , for j = 2, ..., k. In particular, since the definition of the parametrization requires that Fλ (γλj (θ)) = Fλ (ξλj (θ)) = γλj−1 (θ), it follows that, for the special parameter value λk , we have γλ0k (0) = c0 , ξλk−1 k (0) = Fλk (c0 ), γλk−2 k (0) = Fλ2k (c0 ), γλk−3 k (0) = Fλ3k (c0 ), .. . γλ1k (0) = Fλk−1 k (c0 ), Fλk (γλ1k (0)) = Fλkk (c0 ) = c0 . Next we turn our attention to the special parameter value ωλk lying along the line ω · R+ in the parameter plane. For this we first need a result for λ = ωa, with a ∈ R+ . We recall that the critical point ray passing through the critical point cj is denoted by Cj and it is defined by the set of points tcj with 0 < t < ∞. See §2.1. If vj = Fλ (cj ) denotes the corresponding critical value, then the 31 critical point ray Cj is mapped under Fλ in a two-to-one fashion onto a ray connecting vj to ∞. We also introduce the following notation, if the critical point ray Cj1 is mapped under Fλ to the critical point ray Cj2 and the ray Cj2 is rotated by an α angle α from ray Cj1 in the counterclockwise direction, we write Cj1 −−→ Cj2 . 2π Proposition 3.11 Let λ = ωa with a ∈ R+ , where w = ei n−1 . Then the following conditions hold, (a) Fλ maps critical point rays onto critical point rays, and if Cj is a critical point ray then, m( 2π j(n−1)+1) Cj −−−−−−−−−−→ Cj(n−1)+1 . It follows then that critical point rays are eventually periodic under Fλ . (b) For each l ∈ N, Fλl maps the critical point ray C0 onto the critical point ray C1+n+...+nl−1 , and the rotation angle from ray C0 to ray C1+n+...+nl−1 is 2π(1 + n + · · · + nl−1 )/m. The next diagram follows, 2π 2πn 2πn2 m m m C0 −−−→ C1 −−−→ Cn+1 −−−−→ · · · 2πnl−1 2πnl −−−m−−→ C1+n+...+nl−1 −−− m −→ C1+n+...+nl · · · . Notice that in part (a) the points in the rays may not be periodic, just the rays themselves are. Proof. Let tcj with 0 < t < ∞ be the critical point ray passing through 2π the critical point cj where cj = ν j c0 = ν j (dλ/n)1/m with ν = ei m . Its image is given by Fλ (tcj ) = (tn + n/dt−d )cnj = (tn + n/dt−d )cj cjn−1 with cn−1 j = (ν j c0 )n−1 = (ν j (dωa/n)1/m )n−1 . Then we get Fλ (tcj ) = t0 cjn+1 , n−1 n−1 where t0 = (tn + n/dt−d )(d/na) m , where we have used the fact that ω m = ν. Hence the critical point ray Cj is mapped by Fλ onto the critical point ray Cjn+1 . Moreover, the angle from the ray Cj to ray the Cjn+1 is Arg ν j(n−1)+1 = 32 2π(j(n − 1) + 1)/m. Since the number of critical point rays is finite then they are eventually periodic by Fλ , proving the first statement. For the second statement we proceed by induction. If j = 0 then C0 is mapped onto C1 by Fλ and the angle from ray C0 to ray C1 is 2π/m. So, 2π C0 −−m−→ C1 . We assume that Fλl−1 maps the critical point ray C0 onto the critical point ray C1+n+···+nl−2 and then, using the first statement of this proposition and the fact that (1 + n + · · · + nl−2 )n + 1 = 1 + n + · · · + nl−1 , then the image of the ray C1+n+···+nl−2 is the critical point ray C1+n+···+nl−1 . Then Fλl maps ray C0 onto ray C1+n+···+nl−1 and the rotation angle from ray C1+n+···+nl−2 to ray C1+n+···+nl−1 is 2πnl−1 /m. We can see that for each l ∈ N the critical point ray C0 is mapped by Fλl onto the critical point ray C1+n+...+nl−1 with rotation angle 2π(1 + n + ... + nl−1 )/m, as we wanted to show. 2 In §3.4 we showed the existence of infinitely many rings γλk and ξλk for k = 0, 1, 2, . . . in the dynamical plane. Each ring γλk is a simple closed curve such that Fλk : γλk 7→ γλ0 with degree nk -to-1, and in a similar way Fλk : ξλk 7→ γλ0 with degree dnk−1 -to-1, where γλ0 is the curve Γλ defined in Proposition 3.1. Thus, for k = 0, 1, 2, . . . the point Fλk (γλk (2π(1 + n + · · · + nk−1 )/m)) = γλ0 (2π(1 + n + · · · + nk−1 )/m) lies on the critical point ray C1+n+...+nk−1 . It follows that the point γλk (2π(1 + n + · · · + nk−1 )/m) lies on the critical point ray C0 and then, the point Fλ (γλk (2π(1 + n + · · · + nk−1 )/m)) = γλk−1 (2π(1 + n + · · · + nk−1 )/m) lies on the critical point ray C1 . On the other hand, by Proposition 3.9, for all θ we have νγλk (θ) = γλk (θ + 2πnk /m), setting βk = 2π(1 + n + · · · + nk−1 − nk )/m, we have γλk (βk ) = ν −1 γλk (βk +2πnk /m). It follows that the point γλk (βk ) lies on the critical point ray C−1 . Since Fλk (ξλk (βk + 2πnk /m)) = γλ0 (βk + 2πnk /m), it follows that the point ξλk (βk + 2πnk /m) lies on the critical point ray C0 . See Figure 10. By the same result, for all θ we have νξλl (θ) = ξλk (θ − 2πdnk−1 /m), so ξλk (βk + 2πnk /m − 2πdnk−1 /m) = νξλk (βk + 2πnk /m). Hence the point ξλk (βk + 2πnk /m − 2πdnk−1 /m) lies on the critical point ray C1 . Similarly, the point ξλk (βk + 2πnk /m + 2πdnk−1 /m) lies on the critical point ray C−1 . 33 Figure 10: Parametrization of γλ (θ) when λ = ωa, a ∈ R+ , where γ12 = γλ2 (2π(1 + n + n2 )/m), γ02 = γλ2 (2π(1 + n)/m), γ−1 2 = γλ2 (2π(1 + n − n2 )/m). γ1 = γλ (2π(1 + n)/m), γ0 = γλ (2π/m), and γ−1 = γλ1 (2π(1 − n)/m). Also, 1 1 1 1 1 ξ11 = ξλ1 (2π(1 − d)/m), ξ01 = ξλ1 (2π/m), and ξ−1 1 = ξλ1 (2π(1 + d)/m). Finally, 2 2 2 2 2 ξ1 = ξλ (2π(1 + n − nd)/m), ξ0 = ξλ (2π(1 + n)/m), and ξ−1 = ξλ2 (2π(1 + n + nd)/m). We now turn to the proof of the existence of the rings S k in the parameter plane for k > 1. For technical reasons, we consider only the case when n, d ≥ k0 5. Set a = λk+1 . It follows from Proposition 3.8 that, along the line ω q · R+ , for some k 0 , q ∈ N there is a critical point cr for some r ∈ {0, 1, ..., m − 1} such that its orbit has the special form when k ≥ 2 : (k−1)q Fλq (cr ) < cr = Fλkq (cr ) < Fλ (cr ) < · · · < Fλ3q (cr ) < Fλ2q (cr ). Since cr ∈ γλ0 , by Equation (3.7), we have that the set of curves given by {γλ , ξλ , γλk−1 , γλk−2 , . . . , γλ1 } is k + 1 periodic. Then, Fλq maps the curves in the 0 k following way, γλ0 → ξλk → γλk−1 → γλk−2 → · · · → γλ1 → γλ0 . Since c0 ∈ γλ0 and Fλq (c0 ) ∈ ξλk , then by Proposition 3.11, Fλq (c0 ) ∈ C1 , 34 that is, Fλq (c0 ) ∈ ξλk ∩ C1 . To simplify the notation we define, αk = 2π(1 + 2 + · · · + nk−1 − dnk−1 )/m. (3.10) The point ξλk (αk ) lies on the critical point ray C1 , and then vλ = v0 = Fλ (c0 ) = ξλk (αk ). Theorem 3.12 Let n, d ≥ 5. For each k ≥ 1 and every θ satisfying |θ| ≤ 2πdnk−1 /m, there exists a unique parameter λ = λθ,k such that vλ = Fλ (c0 ) = ξλk (θ). Proof. We consider the simply connected region defined in the parameter plane O0 depending on the values of n and d given by, • If n ≤ d then O0 = {λ ∈ C ; |λ| > λmin , λ ∈ I(S 1 ), and |Arg λ| < π}. • If n > d then O0 = {λ ∈ C ; |λ| > λmin , λ ∈ I(S 1 ), and |Arg λ| < d−1 n−1 π}. Given a parameter λ ∈ O0 , we denote by c0 (λ) the critical point given by  1/m d|λ| c0 (λ) = eiArg (λ)/m . n We claim that c0 (λ) is a holomorphic  map of λ. To see this, we notice 1 that c0 (λ) = exp m log(dλ/n) , where log denotes the principal value of the logarithm map. We observe that, in both cases, the critical point c0 is such that |Arg c0 | < π/m. We define the map G : O0 7→ W 0 defined by G(λ) = v0 (λ), where v0 (λ) is the critical value of Fλ corresponding to the critical point c0 (λ). This map takes O0 onto the set W 0 . Both sets O0 and W 0 are simply connected. The set W 0 is given in the following way, • If n ≤ d then W 0 = {z ∈ C ; |z| > vλmin , v0 (λ) with λ ∈ S 1 and |Arg z| < n m π}. • If n > d then W 0 = {z ∈ C ; |z| > vλmin , v0 (λ) with λ ∈ S 1 and |Arg z| < d−1 n n−1 m π}. 35 The precise values of vλmin are computed in Equation (3.4). The function n G(λ) = vλ = (m/n)(dλ/n) m , see Equation (2.2), takes the subset O0 of the parameter plane univalently onto the open set W 0 in the dynamical plane. Also, for fixed θ, the function λ → ξλk (θ) is analytic on O0 and we claim that for each θ such that |θ| ≤ 2πdnk−1 /m, then the set of points ξλk (θ) lies inside a compact set in W 0 . To see the claim we deal first with the arguments of the boundaries. We notice that in Proposition 3.10 we precisely computed the arguments of the set of points in ξλk (θ) for |θ| ≤ 2πdnk−1 /m. Thus, if z is ξλk (θ) for |θ| ≤ 2πdnk−1 /m it follows that (see Proposition 3.10), • If d ≥ n with n ≥ 5 then |Arg z| < 4π/m < nπ/m. • If n > d with n ≥ 5 then |Arg z| < (n/m)(d − 1)/(n − 1). We notice that by definition ξλk (θ) lies inside the open set bounded by Γλ (θ). Hence, for each θ with |θ| < 2πdnk−1 /m, the set of points ξλk (θ) lies inside a compact set in W 0 . Hence we may consider the composition Q(λ) = G−1 (ξλk (θ)) acting on a region of the parameter space. As a function of λ, Q is analytic and maps the simply connected region O0 inside itself. By the Schwarz Lemma, Q has a unique fixed point in this set or on its boundary. But the fixed point cannot lie at λ = 0 since 0 is surrounded by the McMullen domain so that the curves ξλk are bounded away from the origin. Hence there must be a unique fixed point in the interior of O0 . This fixed point is λθ,k . 2 Notice that for each k, the fixed point λθ,k varies continuously with θ, so θ → λθ,k is a curve in the parameter plane. The following proposition iden- tifies the specific values of λθ,k corresponding to the special cases considered earlier. In the case θ = 0, the parameters λk+1 ∈ R+ and ξλkk+1 (0) = vλk+1 , so k that λ0,k = λk+1 . When θ = αk , if a = λk+1 , it follows that ξaω (αk ) = v0 . So λθ,k = ωλk+1 . We write this result as a proposition for future reference. Proposition 3.13 When θ = 0 and k ≥ 1, the parameter values λ0,k are given by the parameters λk+1 ∈ R+ . When θ = αk , then λθ,k is given by ωλk+1 on the symmetry line ω · R+ . The parameters in Proposition 3.13 are the unique parameters, on the corresponding lines in parameter space, for which the orbit of the second 36 iterate of the appropriate critical point monotonically decreases along the corresponding line for k−1 iterations, before returning to itself and becoming periodic. So the curve θ → λθ,k meets each of these symmetry lines only once. Hence the portion of this curve defined for 0 ≤ θ ≤ αk either lies outside the sector 2π 0 ≤ Arg λ ≤ , n−1 for all values of θ, or else this entire curve lies inside the sector. But the former cannot occur since this would imply that some λθ,k would lie in R− , contradicting the fact that each λθ,k lies in O0 . Hence the portion of the curve λθ,k defined for 0 ≤ θ ≤ αk is a continuous arc connecting θ = 0 and θ = 2π/(n − 1). It then follows by the (n − 1)-fold symmetry that, for each k ≥ 1, then λθ,k is a simple closed curve in parameter λ-plane which is periodic of period (n − 1)αk = (n − 1)( 2π m (1 + n + · · · nk−1 − dnk−1 )) 2π k = (n − 1 − dnk + dnk−1 ). m We therefore define the ring S k+1 to be the simple closed curve θ → λθ,k . That is, S k+1 consists of parameter values for which the critical orbit has the following behavior: 1. the critical values lie inside the curve γλ0 ; 2. Fλ2 (cλ ) lies on γλk−1 ; 3. subsequent iterates decrease through the γλj until, at the k th iterate, the critical orbit lands back on the curve γλ0 . We have shown: Lemma 3.14 When n, d ≥ 5 and for k ≥ 1, the ring S k+1 in parameter space is a simple closed curve that is parameterized by θ → λθ,k and is periodic of period 2π k−1 (n d(n − 1) − nk + 1). m 37 These curves accumulate on the boundary of the McMullen domain. More- over, there are precisely τkn,d = dnk−2 (n−1)−nk−1 +1 parameters along S k+1 that are superstable parameters and the same number of centers of Sierpiński holes. This concludes the proof of our main result Theorem A for n, d ≥ 5. 4 Discussion For the case when n or d are not larger than four, the proof does not work because the function G(λ) does not cover enough of the curves γλk to find the fixed point λθ,k , see [3]. We believe that the result still holds for all these cases, as long as 1/n + 1/d < 1. 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