Identifications in Escape Regions of the Parameter Space of Cubic Polynomials

We investigate a relationship between escape regions in slices of the parameter space of cubic polynomials. The focus of this work is to give a precise description of how to obtain a topological model for the boundary of an escape region in the slice consisting of all cubic polynomials with a marked critical point belonging to a two cycle. To obtain this model, we start with the unique escape region in the slice consisting of all maps with a fixed marked critical point, and make identifications which are described using the identifications which are made in the lamination of the basilica map z 7→ z − 1.


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Illustrations which demonstrate the interactions between the two critical points in the four different types of components. . Polynomials are also widely studied as a special case of the study of rational maps.
The field began in the 1920's with two French mathematicians, Pierre Fatou and Gaston Julia. They showed that given a rational map, the Riemann sphere can be split into two disjoint sets. These sets came to be known as the Fatou set, where the dynamics are tame, and the Julia set, where the dynamics are chaotic.
After this period of work, the subject was largely forgotten until the 1980's when Adrien Douady and John Hubbard proved many important results, see [1]. This resurgence of interest was due at least partially to having computers which could draw the complicated fractal images which arise from the iteration of holomorphic maps.
In studying quadratic polynomials, it is often helpful to consider those of the form p c (z) = z 2 + c. We do not lose any information by doing so, since every quadratic polynomial is equivalent under a conformal change of variables to a unique polynomial of this form. In this way, we may study the parameter space of quadratic polynomials in one complex dimension using the single parameter c ∈ C, see Figure 1. Given a parameter c, and a complex number z, we are interested in understanding the orbit of z under p c , that is, the set {p n c (z)|n ∈ N}, where p n c represents the n-times composition of the map p c .
The study of cubic polynomials is more difficult, and far less is currently known than for quadratics. In polynomial dynamics, the orbits of critical points play an mean that there is an extra critical orbit, but the orbits of the two critical points may interact, which creates new dynamical behavior not present in the quadratic case. Also, the parameter space of cubic polynomials is a two complex, or four real dimensional space. As a result, it is difficult to visualize. Therefore, as a way of studying this space, it is beneficial to slice it into one complex dimensional "cross sections," some of which will be studied in this work.
John Milnor came up with a very helpful way of defining these cross sections, see [2]. He started by considering the parameter space of cubic polynomials using the normal form P a,v (z) = z 3 − 3a 2 z + (2a 3 + v). For a polynomial of this form, the critical points are at a and −a. As with quadratics, every cubic polynomial is conformally conjugate to a unique polynomial of this form. He then defined, for each natural number n, the curve S n as the space consisting of all of the maps P a,v for which the critical point a has period exactly n. Fixing the behavior of the critical point a then simplifies our study, allowing us to look at the orbit of −a, and how it interacts with the known periodic orbit of a.
Each of these curves may be partitioned into two pieces. One of these pieces will be called the connectedness locus, C(S n ), where the orbit of the free critical point remains bounded. This results in a Julia set which is connected. The other is called the escape locus, E(S n ), where the orbit of the free critical point escapes to infinity. Julia sets for maps in the escape locus are disconnected. The escape locus of S n can be divided into connected components, each of which is called an escape region, see [3].  and Pascal Roesch, among others. This work will focus on the escape regions of S 1 and S 2 . As shown in Figure 3, in S 1 there is a unique escape region, denoted E 1 , while in S 2 there are two: E D 2 , where the non-trivial connected components of the filled Julia sets resemble discs, and E B 2 , where they resemble the basilica pictured above in Figure 2, see [3]. We aim to give a precise description of a topological relationship between the boundaries of the escape regions E 1 and E B 2 .
Chapter 2 will present much of the background information necessary to the work, as well as develop a large portion of the notation to be used in the later chapters. A significant amount of time is spent here discussing various types of rays. We will discuss dynamic and parameter rays, and in each of these settings, both internal and external rays. Capital letters will be used in the notation of parameter rays, with lower case used in the case of dynamic rays. Subscripts will then be used to differentiate between internal and external rays. All four types of rays will be used frequently in obtaining the main results.
Chapter 3 will be a discussion of some known results in polynomial dynamics, with a focus on the basilica map which plays an essential role in the main results of this work. This chapter begins with the notion of a Hubbard tree, which is a topological tree associated with a polynomial p from which we can extract a surprising amount of dynamical information for p. Laminations are then discussed.
The lamination of a polynomial p is an equivalence relation on R/Z, viewed as the boundary of a disc, which provides a topological model of the filled Julia set of p by considering the quotient of this closed disc by this equivalence relation. It is established that in the lamination of the basilica, any argument t with nontrivial equivalence class has a unique partnert = t. Such t will be referred to as a basilica angle with partnert, and the pair {t,t} will be referred to as a basilica pair.
The chapter is concluded with a brief discussion of hybrid equivalence, which will require the notion of a polynomial-like mapping. established that an identification will occur on the boundary of H 0 at any landing map F of an internal ray whose argument is a basilica angle t with partnert, except when t ∈ 1 3 , 2 3 . There are two possibilities for the map which is identified with F , namely the landing maps of each of the two internal rays of argumentt.
It is then established in Theorem 4.18 that it is the unique landing map of the ray of argumentt in the same quadrant of H 0 as F which is identified with F . In Theorem 4.19, it is shown that there are no identifications at the landing maps of the four internal rays in H 0 of arguments 1 3 and 2 3 .

Preliminaries
We begin our work on a topological relationship between boundaries of escape regions in slices of the parameter space of cubic polynomials with a discussion of the well known results necessary to the work. We will also set up much of the notation used throughout the work here.

Dynamical Systems and Conjugacy
We begin with the basics of general dynamical systems.
Often the set X will have some type of structure, and the map f will preserve this structure. For example, X may be a topological or a measure space, then f would be a continous or measurable function, respectively. We are interested in studying the repeated application, or iteration, of the map f . We will use the notation for the n-th iterate of f . Definition 2.2. Let (f, X) be a dynamical system, and x ∈ X. The forward orbit of x is the set {f n (x)| n ∈ N}.
One goal of the study of dynamical systems is to understand as much as possible about the eventual tendencies of all forward orbits. Therefore, points with the simplest forward orbits play an important role.
Further, if (f, X) and (g, Y ) are both conformal dynamical systems, by a conformal conjugacy, we will mean a conformal isomorphism ϕ : X → Y which satisfies the same composition property as given above. It can be helpful to view a conjugacy with a commutative diagram.
If such a conjugacy exists, the orbits of the map are necessarily preserved in this diagram, and this gives us a way of comparing the dynamical systems.

Periodic Points and the Multiplier
Consider a polynomial p on C. Given a fixed point z 0 ∈ C of p, we define the multiplier of p at z 0 as We then classify the fixed point according to the size of its multiplier in the following way: In the case of a periodic point z 0 of period n for p, we define the multiplier We then classify the periodic orbit in exactly the same way we classified fixed points above, by the size of this multiplier. For example, when p n (z 0 ) = z 0 with n minimal and |λ z 0 | < 1, we say z 0 belongs to an attracting orbit of period n, or attracting n-cycle, for p. points whose orbits converge to the orbit of z. Further, the immediate basin of attraction of z, denoted A z , is the connected component of A z containing z.

Dynamics of Rational Maps and Polynomials
Definition 2.6. A family F of holomorphic functions on the Riemann sphere is called a normal family if every infinite sequence of functions {f n } n∈N from F contains a subsequence which converges locally uniformly to some limit function f .
Note that this limit function f is necessarily holomorphic, and need not be a member of the family F. is normal is called the Fatou set of p, and its complement is called the Julia set.
We will denote the Julia set of p by J(p), or simply J when the map is clear, and the Fatou set by C \ J. When referring to a Fatou component, we will mean a connected component of the Fatou set.
We will illustrate with the simple example of the polynomial p(z) = z 2 , with one critical point at 0. Let z = re iθ ∈ C. We observe that p(z) = r 2 e 2iθ , and further p n (z) = r 2 n e 2 n iθ . It then follows that when r < 1, lim  In order to simplify our study of the dynamics of polynomial maps we will focus mainly on polynomials which are "centered." We have the following definition.
Definition 2.7. A polynomial will be called centered if the sum of its critical points is zero.
Quadratic polynomials have a single critical point in C, so in order to be centered, this critical point must be at 0. Quadratics of the form p(z) = z 2 + c satisfy this requirement. The centered cubic polynomials we will discuss take the It is easily checked that the critical points of such polynomials are at a and −a, and these polynomials are therefore centered. It is a result of Douady and Hubbard from [1] that the Julia set is the topological boundary of the filled Julia set, i.e. J = ∂K. We will need the following result from the same work. From the earlier example p(z) = z 2 , K(p) is the closed unit disc, which contains the critical point 0, and is connected.
In C, it is clear that ∞ is fixed by any polynomial p. We define the multiplier at ∞ in general for a rational map, and comment on the special case for polynomials.
Definition 2.10. Let R be a rational map for which ∞ is periodic of period n.
The multiplier at ∞ is defined as λ ∞ = lim z→∞ 1 (p n ) (z) . Now, for a polynomial p of degree d ≥ 2, we know that p (z) is a polynomial of degree d − 1 ≥ 1, so that the multiplier λ ∞ is always 0, and hence ∞ is a superattracting fixed point. This agrees with our previous definition of a basin of attraction. Further, due to the maximum modulus principle, the basin of attraction of ∞ corresponds exactly with the immediate basin as defined earlier.
Here we discuss another example, which will be of significant importance to this work. Consider This polynomial is centered, as its one critical point is at zero. The critical point is periodic of period two, as p(0) = −1 and p(−1) = 0. Therefore, 0 ∈ K(p), and K(p) is connected by Theorem 2.9. The filled Julia set of this quadratic polynomial is called the basilica, and is pictured in Figure 5.

Böttcher's Theorem and Rays
We will make use of the following theorem of Böttcher, from [2, Theorem 9.1], to make some constructions which will be essential to our work. The local degree of a holomorphic function f about a point z ∈ C is the number of preimages under f , counted with multiplicity, of any point in a sufficiently small neighborhood of the point z.
Theorem 2.12 (Böttcher). Let n ≥ 2, and suppose f (z) = a n z n + a n+1 z n+1 + ... is a holomorphic function with a superattracting fixed point at 0, around which f has local degree n. Then there is a local holomorphic change of coordinate w = φ(z), with φ(0) = 0, which conjugates f to the nth power map w → w n throughout some neighborhood of zero. Furthermore, φ is unique up to multiplication by an (n -1)st root of unity.
Let p be a polynomial with a fixed simple critical point at z 0 , for which the immediate basin A z 0 does not contain another critical point of p. Then we may apply Böttcher's theorem to get a map φ in some neighborhood of z 0 . We then extend, pulling back by the dynamics of p. This expands the domain and range of φ. Since there are no other critical points in A z 0 , we may extend so that this extension is defined on all of A z 0 , and takes its values in the unit disc D. This Böttcher coordinate conjugates p to the squaring map, since p has local degree two about a simple critical point. It also follows that this map is unique since the local degree is two.
Definition 2.13. Let p and z 0 be as described above. Given t ∈ R/Z, the internal ray of argument t for A z 0 is For such rays, λ i (t) := lim s→1 − φ −1 (se 2πit ) ∈ ∂ A z 0 exists, and is referred to as the landing point of the ray.
When necessary to emphasize the map for any ray or landing point, we will use a superscript, i.e. in this case, r p i (t) and λ p i (t) will be the internal ray of argument t and its landing point for the polynomial p.
Notice that in this local degree two case, p(r i (t)) = r i (2t). We may define internal rays in any other component of A z 0 by taking appropriate preimages of the rays in the immediate basin A z 0 .
Consider a polynomial p of degree d ≥ 2. From the discussion after Definition 2.10, this polynomial has a superattracting fixed point at ∞. From Böttcher's theorem, there is a maximal neighborhood V of ∞ and a corresponding minimal Adding the requirement that lim z→∞ φ(z) z = z, in which case we say φ is tangent to the identity near ∞, there is a unique such map. When p is such that K(p) is connected, it follows that we may choose V = A(∞), and r = 1 since there are no critical points in A(∞). In order to avoid unnecessary complexity, we will hold off on further discussion of the case of disconnected filled Julia set, and discuss later only a special case for certain cubic polynomials. We will refer to the map φ as the Böttcher coordinate, or Böttcher isomorphism of the polynomial p. When necessary, we will use the notation φ p to emphasize the map.
Definition 2.14. Let p be a polynomial with connected filled Julia set. For t ∈ R/Z, the external ray of angle t for p is defined as We say that the ray r e (t) lands at z ∈ J(p) if z = lim s→1 + φ −1 (se 2πit ) (in particular, the limit exists). We will denote the landing point by λ e (t). φ −→ Figure 6. Rational external rays of denominator eight for a quadratic polynomial known as Douady's rabbit.
We have the following result, from [2, Theorem 17.14].
where φ p (z) is the Böttcher coordinate for p.
The Green's function is continuous, harmonic outside of K(p), takes the value 0 on K(p), and increases as the input travels along an external ray from K(p) towards ∞.
Definition 2.17. The level curve of the Green's function {z|G p (z) = c} will be referred to as the equipotential of height c.
The equipotential of height c for a degree d polynomial is mapped under p onto the equipotential of height c · d by a d-fold covering.

Cubic Polynomials
We will now begin to focus our study on cubic polynomials. Every cubic polynomial is affinely conjugate to one which is monic and centered. We are interested in studying the parameter space. We may represent any monic and centered cubic polynomial in the Branner Hubbard normal form Using the coordinates (a, b) we may identify the parameter space of such polynomials, which we will denote by P(3), with C 2 .
Following John Milnor in [4], it will be convenient for us to consider monic and centered cubic polynomials with a marked critical point a having the associated critical value v. Such a polynomial takes the form We will refer to the critical point at −a as the free critical point, and refer often to the associated cocritical point 2a, for which F (−a) = F (2a). We may use the coordinates (a, v) to identify the space of such polynomials with C 2 . This parameter space we will denote by P(3), and we notice that it is a two fold covering of P(3), ramified at a = v = 0.
Definition 2.18. The connectedness locus in P(3) will be defined to be the set of all maps in P(3) whose filled Julia set is connected. We will denote the connectedness locus by C( P(3)).
Recall from Theorem 2.9, it follows that for a map belonging to the connectedness locus, the orbits of both critical points remain bounded. This connectedness locus is difficult to visualize and study since it lives in two complex dimensions.
Therefore, John Milnor came up with a way of slicing it into one complex dimensional pieces which can then be viewed and studied.

Milnor's Curves in Cubic Parameter Space
The following definition is due to Milnor in [4].
Definition 2.19. For n ∈ N, the curve S n in P(3) will be defined to be the set of all maps in P(3) for which the marked critical point a has period exactly n.
Further, C(S n ) will be used to denote the connectedness locus C( P(3)) intersected with the curve S n .
Our focus will be on the curves S 1 and S 2 , or more specifically, the boundaries of C(S 1 ) and C(S 2 ).
Suppose we have a map F ∈ S 1 . Then the marked critical point has period 1, or in other words, is a fixed point of F . Therefore, we have that F (a) = v = a, and Milnor gave a parameterization of this curve using the normal form depending only on the one complex parameter a. This allows us to get a one complex dimensional picture of this curve, and its connectedness locus, see Figure   8.
Similarly for a map F ∈ S 2 , we will now require that the marked critical point belong to the period two orbit a → v → a. We define the displacement δ = v − a ∈ C, and after some algebra we arrive at the normal form which can be used to parameterize the curve S 2 with the single complex parameter δ. We now make the important simple observation that for a map in any curve S n , connectedness of the filled Julia set is equivalent to boundedness of the orbit of the free critical point −a. This is because the only other finite critical point belongs to a periodic orbit which is necessarily bounded.
We now turn our attention to the hyperbolic components of the curves S n . While we may define rays for any hyperbolic component, they are all defined in a different way, and we will only require the definitions for type A and C components. We will begin with the simplest case of a type C component of S 1 . We know that for a map p in such a component, the cocritical point 2a p is mapped eventually into the immediate basin of the marked critical point a p . Let n ∈ N be Definition 2.21. For a map p in a type C component of S 1 , there is a unique argument t such that p n (2a p ) lies on the dynamic ray r i (t) in A ap . We then say that p lies on the internal parameter ray of argument t. We will denote this internal parameter ray by R i (t). When R i (t) lands, the landing map will be denoted Since, by [ The Böttcher coordinate may be extended until the domain reaches −a. There is then an internal ray for which the landing point is −a. will be referred to as an internal parameter ray of argument t, and denoted We observe that as the parameter travels a loop around the center of H 0 , the free critical point makes two loops around the marked critical point in the dynamical plane. The map a → t sending a parameter in H 0 to its internal argument t ∈ R/Z is then a two fold cover, ramified at a = 0. For more information, see [4, Section 3]. We will use the notation Λ i (t) to represent the set of the two landing maps for the internal rays of argument t.

Escape Regions
We now look at the subset of hyperbolic maps in S n called the escape locus, which consists of all maps outside of the connectedness locus, as studied in [7]. The name comes from the fact that for such maps, the orbit of the free critical point "escapes" to ∞, and hence the filled Julia sets are disconnected. We will refer to the connected components of the escape locus as escape regions. There is a unique escape region in S 1 , which we will denote by E 1 . In S 2 , the escape locus consists of two escape regions, which we will distinguish by the connected components of the filled Julia sets, see [7]. We will also provide further details on this in Chapter 3.
There is an escape region of S 2 where each nontrivial connected component of the We now want to consider dynamic rays for the maps in escape regions of S 1 and S 2 , whose filled Julia sets are disconnected. Consider such a polynomial p. As before, there is a Böttcher coordinate φ defined in a neighborhood of ∞, which conjugates p to z → z 3 since a cubic polynomial has local degree three near ∞.
Similarly to our previous discussion, the Böttcher coordinate may be extended until  Given a map F in any escape region of S 1 or S 2 , the cocritical point 2a lies on the equipotential passing through the free critical point −a. Therefore, there is a unique dynamic ray landing at 2a. We use this unique dynamic ray to define parameter rays.
Definition 2.24. The external parameter ray of argument t, denoted R e (t), consists of all maps F in the escape region for which the unique dynamic ray for F landing at the co-critical point 2a has argument t. If this parameter ray lands, we denote its landing map by Λ e (t).
Recall that capital letters will be used for parameter rays, while lower case will be used in the case of dynamic rays. We make the important observation that for a map on the parameter ray of argument t, in the dynamic plane we have the external rays of arguments t + 1 3 and t − 1 3 crashing together at the free critical point −a, as shown in Figure 12.
We will make use of the following result of Roesch in [6, Theorem 6], which was originally proven by Faught in [8]. This then means that any external parameter ray which accumulates on a type A or C component indeed lands at a unique map on the boundary due to Theorem 2.15.
We will have a particular interest in certain external parameter rays. The following definition is adapted from [9]. Definition 2.26. An argument t ∈ R/Z is co-periodic of co-period n if either t− 1 3 or t+ 1 3 is periodic of period n under tripling. A parameter ray whose argument is co-periodic of co-period n will be called co-periodic of co-period n.
An argument t ∈ R/Z is co-periodic of co-period n if and only if it can be written in the form where m is relatively prime to 3, and n is minimal. Now, given n, q ∈ N, we will define the period q decomposition of S n V (q) := S n \ rays of co − period q, and will denote by V 1 , V 2 , ..., V m the connected components of V (q), as in [7]. We will make use of the following result from [7, Theorem 3.1].
Theorem 2.28 (Bonifant and Milnor). For each p ∈ V i and argument t which is periodic of period q under tripling in R/Z, the ray r e (t) for p lands at a repelling periodic point z(p) ∈ J(p) ⊂ C. Furthermore, the map p → z(p) from V i to C is holomorphic. The orbit portrait for dynamic rays of period q is the same for all   Figure 13 with some external rays marked

A Conformal Isomorphism
For a general escape region E in any S n , there is a multiplicity µ E ∈ N, and a canonical isomorphism between E and the µ E -fold cyclic covering of C \ D, see [4,Lemma 5.9]. The escape regions in S 1 and S 2 all have multiplicity one, see [7], so that each is conformally isomorphic to the complement of the closed unit disc.
We will have particular interest in the conformal isomorphisms  which are defined by mapping parameter rays to radial lines of corresponding argument. The composition is then a parameter ray preserving conformal isomorphism between the two escape regions. Let Ψ be the continuous extension of Ψ to the set of all maps on the boundary of a type A or C component of C(S 1 ). Such an extension is well defined by Theorem 2.25. Now, we let Ψ be the restriction of Ψ, mapping the boundary of components of types A and C in S 1 to the boundary of corresponding components of types A, B, and C in S 2 . We are interested in the set of points on the boundary of type A or C components in S 1 where Ψ is injective. This will be explored in detail in Chapter 4. First, we must spend some time building tools and establishing facts about the basilica map.
List of References CHAPTER 3 The basilica map We require some facts about the basilica map z → z 2 − 1. We will introduce the concepts of Hubbard trees and laminations for quadratic polynomials, with a particular focus on the basilica map. We will then discuss the basics of hybrid equivalence, and an important result of Branner and Hubbard.

Hubbard trees
The notion of a Hubbard tree was introduced by Douady and Hubbard in [1].
The following definitions are necessary to the concept of a Hubbard tree.
The polynomial p is said to be post-critically finite if P (p) is a finite set.
It follows that a polynomial is post-critically finite if and only if every critical point is pre-periodic.
Consider a post-critically finite polynomial p. We will follow Milnor and Poirier's appendix on Hubbard trees in [2], which is more general than we will need. Each Fatou component for p is conformally isomorphic to a disc, and is either periodic or pre-periodic due to Sullivan's non-wandering theorem, see [3].
Since we are considering a post-critically finite polynomial, it follows that each component contains a unique point which is either periodic or pre-periodic. In the standard disc model of hyperbolic geometry, a geodesic is an arc of a circle in the disc which meets the boundary of the disc at right angles, see Figure   17.

Laminations
Consider a polynomial p whose Julia set is connected and locally connected.
We will define an associated equivalence relation on R/Z. Definition 3.6. The equivalence relation L which relates arguments t, s ∈ R/Z whenever λ e (t) = λ e (s) ∈ J(p) is called the lamination of p.
The concept of a lamination is more general than what we will require, see [6].
A lamination can be visualized as a closed disc D with non-intersecting hyperbolic geodesics connecting the points on the boundary which are identified. Each arc in this model will be referred to as a leaf, and when necessary, we will use s,t to denote the leaf connecting arguments s and t. The quotient D/L is homeomorphic to the filled Julia set K(p) as explained in Dierk Schleicher's appendix to [6].
The length of the leaf connecting points at arguments t and s will be defined as min{a, b}.
Since, for a quadratic polynomial, forward iteration corresponds to doubling arguments of external rays, we expect that leaves should get longer under such iteration. There are important leaves for which this is not the case, which motivates the following definition, adapted from Thurston in [6, Definition II.6.2]. We would like to describe an inductive process for finding the lamination of the basilica map. The process consists of finding all of the periodic rays which land together in the first step, and for each subsequent step, taking all preimages of all leaves from the previous step. Therefore, we need to describe how to find a preimage of a leaf. We have the following useful result, which also allows us to introduce some terminology.  for which there is n ∈ N such that 2 n t ≡ 1 3 , there is a unique t ∈ R/Z, t = t, such that L relates t and t, i.e. λ e t = λ e (t).
Proof. Let p(z) = z 2 − 1 be the basilica map, and z 0 ∈ J(p) be the landing point of more than one external ray. From Theorem 3.5, we have that z 0 eventually lands on the Hubbard tree H p under iteration. The intersection H p ∩ J(p) consists of a unique point, namely the common landing point of the external rays of arguments 1 3 and 2 3 . Define α := λ e ( 1 3 ) = λ e ( 2 3 ), and let n be the minimal natural number such that p n (z 0 ) = α, and δ > 0 be the distance between z 0 and the nearest center of a Fatou component. Choose ε with 0 < ε < δ. It then follows that p n | D(z 0 ,ε) is a homeomorphism onto a neighborhood of α. Therefore, we may conclude that there is an argument t for which λ e (t) = z 0 and 2 n t ≡ 1 3 , as well as a unique corresponding t with λ e t = z 0 = λ e (t).  10. An argument t ∈ R/Z will be called a basilica angle if there is n ∈ N such that 2 n t ≡ 1 3 . If t is a basilica angle and s = t is the unique argument in R/Z such that s and t are related in the lamination of the basilica, then (s, t) will be referred to as a basilica pair.
Considering Lemma 3.9, to find a preimage of a leaf s,t of the lamination of the basilica, we first choose a preimage under doubling of one of the arguments, say t. We then identify this preimage with its unique partner, which is the preimage of s which is accessible from our preimage of t, in the sense that this new leaf created doesn't cross any existing leaves of the lamination. The lamination of the basilica is then obtained by starting with the major leaf

Hybrid equivalence
We will require the concept of hybrid equivalence, introduced in [7], which will necessitate the ideas of both quasiconformal and polynomial-like mappings.
We begin by considering an orientation preserving C 1 homeomorphism from some region of C to another. We then have the standard partial derivatives ∂f ∂x and ∂f ∂y , which we use to define the z and z-derivatives Definition 3.11. The quantity is called the dilatation of f at the point z. For more information, see [8]. Let p be a polynomial in any escape region of S n , lying on the parameter ray of argument t. Then the dynamical plane is separated into two components by the rays r e (t+ 1 3 ) and r e (t− 1 3 ) together with the free critical point −a, and the point at infinity. We will label U 0 the component containing the marked critical point a, and the other we will label U 1 . We then form a binary sequence σ = (σ 1 , σ 2 , ...) from the orbit of the marked critical point a, by setting σ m = i whenever p m (a) ∈ U i .
Since a is periodic of period n under p, this sequence is also periodic, having some period which divides n. This sequence σ will be referred to as the kneading sequence for the polynomial p. Recall that K a refers to the component of the filled Julia set of p containing the marked critical point a.
Theorem 3.14 (Milnor). For a polynomial p in an escape region of S n for n = 1, 2, the restriction of p to a neighborhood of K a is hybrid equivalent to a unique quadratic polynomial, which has periodic critical orbit of period n where is the period of the kneading sequence for p.
Remark. It is noted in [10] that [2,Theorem 5.15] is not true in its full generality.
However, when considering small values of n (up to and including n = 3), the result holds as stated.
From [10], the kneading sequence is an invariant of each escape region, and furthermore each escape region of S 1 and S 2 is uniquely determined by its kneading sequence. Observing that a quasiconformal conjugacy necessarily preserves topology, this justifies our earlier terminology of referring to the escape regions in

CHAPTER 4 Identifications in Escape Regions
We begin with a way of representing maps in escape regions based on combinatorial data. We then obtain some results using this representation which will help us to describe the map Ψ. This will be helpful in obtaining our main results.

Preliminary Results
Lemma 4.1. Let E be an escape region of multiplicity 1 in S n for some n ∈ N, and let F ∈ E. Then F is uniquely determined by the ordered pair Proof. Since the multiplicity of the escape region E is 1, from a result of Milnor, [1, Lemma 5.9], we have that the covering map Φ : E → C \ D defined by is in fact a conformal isomorphism. For existence, let F ∈ E. Then represents the map F .
We will use (ρ, t) 1 and (ρ, t) B 2 to denote the unique maps in E 1 and E B 2 , respectively, determined by (ρ, t). If the parameter ray of angle t lands, the landing map will be denoted (1, t). The following gives a useful description of the map Ψ in terms of this combinatorial description of the maps.
Proof. We have the commutative diagram By definition, both Φ 1 and Φ 2 map parameter rays to radial lines of corresponding argument. Therefore, we may conclude that Ψ preserves parameter rays, and that t = t * . Further, by the definitions of ρ, ρ * , and the maps Φ 1 , Φ 2 , and Ψ, we have At this point, we make a conjecture which will need to be assumed for the remainder of the work. This is an analogous result to Theorem 2.25 for the basilica escape region of S 2 .
Conjecture. The boundary of E B 2 is locally connected at every point which is not in the boundary of a type D component.
thus the result is established.

Behavior of the Conformal Isomorphism
We would like to now describe the identifications that are made by Ψ in sending a map (ρ, t) 1 to its image (ρ, t) B 2 .
Recall the unique fixed point α on the boundaries of the Fatou components containing a and its associated critical value v from Definition 3.16. Since the components containing a and v form a two-cycle, the arguments of the rays landing at α must be periodic under tripling with period two. The possible arguments of pairs of rays landing at α are then 1 8 , 3 8 , 2 8 , 6 8 , and 5 8 , 7 8 . We will narrow down as much as possible for each map in E B 2 which pairs of period two rays could possibly land at α. Proof. As we saw in defining a kneading sequence, the rays r e (t − 1 3 ) and r e (t + 1 3 ), together with the free critical point −a and the point at ∞, cut the Riemann sphere into two components. We will again label these components U 0 and U 1 as before, where U 0 is the component containing the marked critical point a, see  Proof. Since α is on the main basilica component K a of K((ρ, t) B 2 ), as shown in Figure 24, this follows directly from Lemma 4.4.
Lemma 4.6. At most two pairs of period two rays could possibly land at α for any Proof. Any arc containing all six period two arguments will necessarily have length greater than 2 3 . Therefore, by Corollary 4.5, at least one pair must always be excluded.
We will now decompose E B 2 into regions depending on how many pairs of rays can possibly land at α. In regions where only one pair of rays is possible, it follows Figure 24. An updated version of Figure 12 for a map in E B 2 . The same applies to a map in E 1 with a disc in place of the main basilica component illustrated. that those rays must land at α. We will further show that in regions where two pairs are possible, all four of the rays from these two possible pairs do indeed land at α. which are co-periodic of co-period two.
Then all period two dynamic rays land on J(F ) if and only if t ∈ R/Z is not co-periodic of co-period two.
Proof. Let F ∈ R e (t). Suppose first that t is co-periodic of co-period two. Then one of the dynamic rays r e (t+ 1 3 ) and r e (t− 1 3 ) is a period two ray which crashes into the free critical point −a and therefore does not land on J(F ). Conversely, suppose that there is a period two dynamic ray r e (t) which does not land. Then there is a minimal n ∈ N such that F −n (−a) ∈ r e (t). Taking forward iterates, it then follows that one of the rays r e (t) or r e (3t), depending on the parity of n, crashes into the free critical point −a. This implies thatt ∈ {t − 1 3 , t + 1 3 } or 3t ∈ {t − 1 3 , t + 1 3 }, and therefore that t is co-periodic of co-period two by definition.
We finish off the decomposition of the escape region E B 2 into regions depending on the number of possible rays landing at α with the following result. Proof. First assume that t ∈ ( 23 24 , 13 24 ), refer to figure 25 to visualize which maps we are considering. For such angles we find that t + 1 3 ∈ ( 7 24 , 7 8 ) and t − 1 3 ∈ ( 5 8 , 5 24 ). Therefore, at least one of the angles from the pair { 5 8 , 7 8 } is in the interval (t+ 1 3 , t− 1 3 ), which excludes this pair from landing at α by Corollary 4.5. Similarly, we may exclude the pair { 1 8 , 3 8 } exactly when t ∈ ( 11 24 , 1 24 ), and the pair { 2 8 , 6 8 } exactly when t ∈ ( 2 24 , 10 24 ) or t ∈ ( 14 24 , 22 24 ). On the co-period 2 rays which bound W , we know from Lemma 4.8 that one of the two possible pairs remaining is excluded by means of one of the rays in the pair crashing into −a. Considering all of these exclusions at once, the result is proven, and we have Figure 26 showing which pairs land at We use the previous result to get an analogous result for the unique escape region E 1 in S 1 , which will be important to our main results. Recall that for a map F ∈ S n , A a is the immediate basin of attraction of the marked critical point a, that is, the connected component containing a of the set of all points whose orbits converge to a under iteration of F . Figure 26. The pairs of rays landing at α in each piece of the basilica escape region.
, it follows that these points must be the landing points of a pair of period two external rays, and therefore we must have the set equality For the second case, suppose Ψ(F ) ∈ W , and first assume that Ψ(F ) ∈ W 1 .
The remaining cases then follow similarly. In this case, for the two pairs,  Figure 29. Therefore, we have the two cycle The proof follows similarly assuming Ψ(F ) is in any other W i .
We are now prepared to obtain a very important result. The purpose of this result is to take a step towards describing the behavior of Ψ, by first describing this desired behavior for Ψ. This behavior of Ψ can be pictured as taking each of the disc components of the filled Julia set K(F ) for a map F ∈ E 1 , and replacing them with homeomorphic copies of the basilica in order to obtain a topological object which is homeomorphic to K(Ψ(F )). The intention of the main results is then to extend this result on the behavior of Ψ to the boundary in order to describe the behavior of Ψ. In order to do this, we will have to split up into separate cases based on whether we are on the boundary of a type A or C component of S 1 . The remainder of our main results will apply the dynamical information obtained on the boundary in order to describe what is happening topologically in parameter space. Here we are following the words of Adrien Douady, who said "we sow the seeds in the dynamical plane, and reap the harvest in the parameter plane."  The three preimages of r e (t) must land at preimages of λ i ( 1 3 ). These preimages are λ i ( 2 3 ), λ i ( 1 6 ), and a point off of the main component which we will disregard for now. Each of these landing points of internal rays is also the landing point of a corresponding external ray. We have λ e (t) = λ i ( 2 3 ), and we denote by t 1 the argument for which λ e (t 1 ) = λ i ( 1 6 ). We similarly find an argumentt 1 such that λ e (t 1 ) = λ i ( 5 6 ), which is a preimage of the point λ i ( 2 3 ). Now, in K(Ψ(F )), we know that there is exactly one preimage of α under Ψ(F ) in the main basilica component other than α itself, and it follows that this point is λ e (t 1 ) = λ e (t 1 ).
This identification of external rays corresponds to the leaf 1 6 , 5 6 of the lamination of the basilica along the internal rays of the immediate basin of a. We find further identifications by taking preimages. We choose an internal argument where the previous identification occurred, say 1 6 , and further choose one of its preimages, say 1 12 . At this preimage, there is a corresponding external argument t 2 with λ i ( 1 12 ) = λ e (t 2 ) in K(F ), for which, similarly to above, we can find another external argumentt 2 for which λ e (t 2 ) = λ e (t 2 ) in K(Ψ(F )), witht 2 landing at the same point as an appropriate preimage of λ i 5 6 in K(F ). By Lemma 3.9 there is exactly one possible internal argument to pair up with, in this case 11 12 . Continuing to find all preimages, this corresponds exactly with the inductive process for putting the lamination of the basilica in this component of K(F ). Now, we consider the preimage of λ i ( 1 3 ) ∈ J(F ) off the main component, which was disregarded earlier. We similarly find an identification of external rays which produces the leaf

Main results
We begin with our main results by extending the result of Lemma 4.13 to the boundary to obtain a similar result describing how to obtain K( Ψ(F )) topologically from K(F ).
We have the following important corollary to Theorem 2.28. It essentially says that the fixed point α for maps in E B 2 persists in the landing map of any ray which is not co-periodic of co-period two. This is helpful for extending our results on identifications to the boundary of the escape region E B 2 . The co-period two case will be handled separately later.
Lemma 4.14. Suppose t ∈ R/Z is not co-periodic of co-period two. The map (1, t) B 2 ∈ ∂E B 2 has a unique fixed point in J on the boundary of the Fatou components containing the marked critical point a, and its associated critical value v.
Furthermore, the rays which land at this fixed point are exactly those which land at the fixed point α for any map (ρ, t) B 2 ∈ Λ e (t) ⊂ E B 2 .
Proof. Suppose t is not co-periodic of co-period two. The landing point (1, t) B 2 is in the same component of the period two decomposition as the rest of the ray R e (t). Therefore, by Theorem 2.28, the dynamic rays which land together at α for any map along the parameter ray R e (t) still land together at a fixed point in the We will continue to refer to this unique fixed point as α for maps on the boundary just as we did for maps in the escape region.  1. Assume that R e (t) is not co-periodic. By Lemma 4.14 we have the fixed point α ∈ K( Ψ(F )). Similarly to Lemma 4.13, there are one or two pairs of period two dynamic external rays landing together at α, for which the corresponding dynamic rays for F do not land together. We describe this as an identification made by Ψ, and we find other identifications by taking preimages as we did previously in Lemma 4.13. Thus we conclude that the identifications from the lamination of the basilica along the internal rays of every Fatou component of K(F ) exist in K( Ψ(F )). We want to show that these are the only identifications made by Ψ.
Since R e (t) is not co-periodic, neither map F, Ψ(F ) is a parabolic map, see [2].
Therefore, the free critical point is in the Julia set for both of these maps. Since 2. Now, suppose that R e (t) is co-periodic. It follows that F and Ψ(F ) are parabolic maps, see [2]. We again have the identifications arising from preimages of We now look to apply this dynamical information to prove our main results in parameter space. We will begin with the easier of our two cases, where we consider a map on the boundary of a type C component of S 1 . The following result describes where to find pairs of distinct maps on the boundary of a type C component of S 1 which will be mapped by Ψ to the same image in S 2 .  If t is not a basilica angle, then it follows from Theorem 4.15 that these are the only external arguments for rays landing at 2a G . Therefore, we conclude that these are the only arguments for parameter rays landing at G, and Ψ −1 (G) = {F }. However, if t is a basilica angle with partner t, then by Theorem 4.15 the arguments of the external rays for F landing on the component whose boundary contains 2a F at internal argumentt will be arguments of external rays for G which also land at 2a G . This means that G is the landing point of parameter rays of arguments t andt in S 2 . We may then conclude that Ψ −1 ( Ψ(F )) = {F,F }.
We will now focus on the principal hyperbolic component H 0 in S 1 , or in other words, the unique type A component. This case is more difficult due to the double cover of internal arguments in this component. It will be convenient to split this component up into four quadrants, which will be separated by the four internal rays of arguments 1 3 and 2 3 . We will label them so that the first quadrant contains on its boundary the landing map of the external ray of argument zero, and the rest will be labeled in the counterclockwise direction as usual, see Figure 30. We will refer to Quadrants I and III, and similarly II and IV, as opposite quadrants. We will use these quadrants to give a useful notation for internal rays in H 0 . By R I i (t), we will mean the internal ray of argument t which lies in quadrant I. Note that we need to be sure that t ∈ ( 1 3 , 2 3 ) for R I i (t) to exist. We will take care of the internal rays which bound the quadrants by choosing the odd numbered quadrant which it bounds, i.e. R I i ( 1 3 ), R I i ( 2 3 ), R III i ( 1 3 ), and R III i ( 2 3 ) will denote these four internal rays in H 0 .  First, assume that F is on the boundary of quadrant I in S 1 . It then follows that the argument s of the external parameter ray landing at F is in 23 24 , 1 24 , and the arguments of the external parameter ray landing atF o is in 11 24 , 13 24 . Now, define G := Ψ(F ) = Ψ(F o ) ∈ S 2 . Then we know that λ e ( 2 8 ) = λ e ( 6 8 ) = α G ∈ K(G), see Figure 26. However, we must also have λ e (s) = λ e (s) = 2a G ∈ K(G). This is a contradiction, since the crossing of rays would violate the injectivity of the Böttcher coordinate. Therefore, we have Ψ −1 ( Ψ(F )) = {F,F s }, as desired. The remaining cases follow similarly, with λ e ( 5 8 ) = λ e ( 7 8 ) = α G ∈ K(G) when F is assumed in quadrant II, and λ e ( 1 8 ) = λ e ( 3 8 ) = α G ∈ K(G) when F is assumed in quadrant IV.  Proof. The four landing points of internal rays of H 0 of arguments 1 3 and 2 3 are landing points of co-periodic external rays of co-period two. From [2], we know that such landing points are necessarily parabolic maps. Since Ψ sends the landing point of the parameter ray of argument t in S 1 to the landing point of the parameter ray of argument t in S 2 by Lemma 4.3, it follows that these four distinct parabolic maps correspond bijectively under Ψ with the four distinct parabolic maps in S 2 on the boundaries of both escape regions which are the landing points of the corresponding co-periodic rays of co-period two.
S n The set of all cubic polynomials with a marked critical point having period exactly n.
C(S n ) The connectedness locus in S n .

E(S n )
The escape locus in S n .

E 1
The unique escape region in S 1 .
The basilica escape region in S 2 .
H 0 The unique hyperbolic component of type A in S 1 .
A z The basin of attraction of the point z.
A z The immediate basin of attraction of the point z.

K(p)
The filled Julia set of the polynomial p.

J(p)
The Julia set of the polynomial p.
The internal dynamic ray of argument t.
λ i (t) The landing point of the ray r i (t).
r e (t) The external dynamic ray of argument t.
λ e (t) The landing point of the ray r e (t).
The internal parameter ray of argument t.
Λ i (t) The landing point of the ray R i (t).
Λ i (t) The set of landing points of the two rays R i (t) ∈ H 0 .
R e (t) The external parameter ray of argument t.
Λ e (t) The landing point of the ray R e (t).
Ψ A conformal isomorphism mapping E 1 to E B 2 .

Ψ
The continuous extension of Ψ to part of ∂E 1 .

Ψ
The restriction of Ψ to part of ∂E 1 .
Thurston, W. P., "On the geometry and dynamics of iterated rational maps, with appendix by D. Schleicher," in Complex Dynamics Families and Friends, 2009, pp. 3-137.