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The Dynamics of Twisted Tent Maps

This paper is a study of the dynamics of a new family of maps from the complex plane to itself, which we call twisted tent maps. A twisted tent map is a complex generalization of a real tent map. The action of this map can be visualized as the complex scaling of the plane followed by folding the plane once. Most of the time, scaling by a complex number will"twist"the plane, hence the name. The"folding"both breaks analyticity (and even smoothness) and leads to interesting dynamics ranging from easily understood and highly geometric behavior to chaotic behavior and fractals.


Introduction
Real tent maps are piecewise-linear maps from the real line R to itself and have been extensively studied. Real tent maps can be visualized as a real scaling followed by a folding. Working on the complex plane C instead of R and replacing the real scaling by multiplication by a complex parameter c, we get a complex generalization of real tent maps. Because multiplication by a complex number usually rotates (or twists) the complex plane, we call these new maps twisted tent maps or TTM's. This work attempts to both study and lay a foundation for the future study of TTM's. This subject has several inherent advantages for those who study TTM's. The first is the simplicity of the map. A twisted tent map sends a line segment to either a line segment or to a bent line segment. This often gives rise to structures that can be studied using geometry, although quite often these structures are very complicated. The second advantage is the ability to make pictures, which aid in intuition, understanding, and interest, since these pictures are often beautiful and pleasing on their own (see Figure 1). The primary disadvantages of the subject are the lack of analyticity and smoothness due to folding. For us, the natural object of study is the filled-in Julia set K = K(c), which is the set of all points whose trajectory stays bounded under forward iteration. Frequently, a TTM restricted to a forward invariant subset of K is conjugate (in the dynamical sense) to a real tent map. Depending on the choice of c, K(c) can be a line segment, a double spiral of line segments, a polygon, a fractal, or even a Cantor set. In this work, necessary and sufficient conditions will be given for K to be a polygon. Figure 2 shows a few examples of K.
For the most interesting TTM's there are no attracting periodic points in C. Instead, there can be sets of points we call hungry sets that attract neighboring points, "eat" them, and retain them. These hungry sets can be simply connected, a topological annulus, or be comprised of periodic components. There can be several distinct hungry sets in a filled-in Julia sets and some hungry sets can contain smaller hungry sets.
A very standard way to make pictures of filled-in Julia sets is to color pixels based on the escape-time algorithm (such as in Figure 2). However, when K has nonempty interior, the escape-time algorithm shows only black throughout the interior of K. We introduce a new coloring algorithm called the coded-coloring algorithm which makes it possible to see periodic structures and their preimages in the interior of K. Some of these structures behave like periodic copies of filled-in Julia sets for a different choice of parameter (see Figure 3).
The coded-coloring can also be used in making pictures of the parameter plane. Any structures that appear near c in the coded-coloring of the parameter plane are approximate previews of the structures found in the coded-coloring of K(c) (see Figures 4 and 5).
There have been several generalizations of real tent maps, such as the ctent maps which are defined in [2]. We believe that TTM's are also suitable subjects to study, due to the simplicity of the formula, the complexity of the dynamical behavior, and to the beauty of the pictures. Further motivation for the study of TTM's include the following:   as designs for frequency independent antennae. Koch curves appear as filled-in Julia sets for certain TTM's. Thus, it is possible that TTM's could be useful in finding fractals that have not yet been considered for this purpose. This present work is organized in the following way: Chapter 2 provides the formal definition of a TTM, some of the basic terms, and shows that due to conjugacy (both topological conjugacy and complex conjugation) there is a canonical subset of the family of twisted tent maps that are sufficient representatives of the entire family. This subset consists of TTM's whose folding line is the set of points with imaginary part equal to −1 and whose parameter has non-negative imaginary part. For a canonical TTM we show that if |c| < 1 then the dynamics are trivial. We then study the dynamics that can occur if |c| = 1 and finish the chapter with a list of assumptions that will be held throughout the rest of the text.
Chapter 3 describes K when c ∈ R. In this chapter we also give some partial results about the connectivity of K as well as several conjectures. We also give a several different sufficient criteria for K to be a Cantor set.
In Chapter 4 we define and use the perimeter set P , which is the primary tool we use to categorize K. The definition of P was chosen such that the following are true.
• P is compact, • P = K for some cases, • P always contains K, • P is easily calculated, • P gives us significant information about K.
In the process of defining P , we show that when K is a polygon, then its boundary can be explicitly calculated.
Chapter 5 lays down a framework for the study of hungry sets and gives the proofs for a few results. The rest of this chapter is broken up into sections primarily consisting of experimental results. These experimental results include examples where hungry sets swell in size as the modulus of the parameter is increased as well as examples illustrating that the tools of renormalization might be useful in studying TTM's.
In Chapter 6 describes what we know about different pictures of the parameter plane. These pictures include the polygonal locus, which is the set of parameters c such that K(c) is a polygon. The coded-coloring of the parameter plane is discusses and several open problems and conjectures can be found in this discussion.
Chapter 7 proves several results about the entropy of a TTM, such as Theorem 7.10 which states that h(f | K ) ≤ log min(2, |c| 2 ) and that this inequality is sometimes strict.
Lastly, scattered throughout this text are conjectures and open questions to aid those who wish to continue to develop this theory.

Aknowledgements
This is my doctoral thesis for my Ph.D. in mathematics from IUPUI. I would like to thank Micha l Misiurewicz, my advisor, for patiently working with me for many hours to develop this theory, while sharing his love of mathematics with me. I would also like to thank the dynamics group at IUPUI for their many helpful suggestions. I am grateful to the U.S. Department of Education because this work was supported by the GAANN grant 42-945-00.

Basic Cases
Let FL denote a line in the complex plane. (FL will later be called the folding line.) Then FL divides the plane into two closed half planes H − and H + , where 0 ∈ H + . (If 0 ∈ FL then the choice of H + is arbitrary.) Let Reflect(z, FL) be the reflection of the point z about the line FL. Let c ∈ C and fix FL. We wish to study the dynamics of a family of functions {f c } where f c : C → C is of the form: If 0 ∈ FL then the dynamics are trivial. Let r = |c|.
Then every point is eventually mapped into the invariant sector S with the negative real axis as its bottom edge, vertex 0, and with angle θ measured clockwise from the negative real axis.
1. If r < 1 then 0 is an attracting fixed point whose basin of attraction is C.
3. If r > 1, then the orbit of every nonzero point diverges to infinity.
Proof. Each of these follow easily after the observation that since 0 ∈ FL, then reflection about FL does not affect the modulus of a point.
From now on, we will assume that 0 / ∈ FL. Theorem 2.2 shows that all choices of FL that do not pass through the origin are equivalent. This allows us to make a canonical choice for FL which simplifies things greatly.
Theorem 2.2. Let FL F and FL G be two different choices for FL. Let F = {f c : c ∈ C} and G = {g c : c ∈ C} be the resulting families of functions with those choices of folding lines, defined as in (1). Then for every c ∈ C, f c is conjugate to g c .
Proof. There exists a unique circle centered at the origin such that the line FL F is tangent to it at a point z F . Similarly, we define z G to be the point where FL G is tangent to some circle centered at the origin. Note that the points z F and z G depend only on the choice of FL and not on c. We denote the closed half planes on either side of FL F by H − F and H + F . Likewise, the closed half planes that meet along FL G are denoted by H − G and H + G . Now let ϕ : C → C be define by ϕ(z) = z G z F z. Since neither FL F nor FL G pass through the origin, then z F , z G = 0. Thus ϕ and its inverse are well defined. The following hold trivially: Thus ϕ(f c (z)) = g c (ϕ(z)) as desired.
Definition 2.3. We will denote the real and imaginary parts of z by Re(z) and Im(z) respectively.
Theorem 2.2 allows us to define FL = {z : Im(z) = −1} as our canonical choice for FL. This will be our choice for FL throughout the rest of this paper and we will denote the family of functions induced by this choice of FL by F = {f c : c ∈ C}. Now that a specific family of functions has been chosen, we will write FL instead of FL F and will also drop the family subscript for the half-planes and write H + and H − . Throughout this paper we will write z for the complex conjugate of z. Now, for every z ∈ C we have Reflect(z, Thus, for every f c ∈ F we have: ( Each f c ∈ F scales and rotates the plane and then reflects H − across FL. This family of maps is a generalization of the family of real tent maps. Recall that the tent map acting on R scales the real line by a real constant and then folds it at a particular point on R. A twisted tent map scales C by a complex constant and then folds it across the FL. More specifically, a tent map scales and folds; a twisted tent map scales, rotates (twists), and folds. This is why we call each f c a twisted tent map.
If f is a twisted tent map, then it is fairy common for a one dimensional forward invariant subset of the complex plane to have the dynamics of a tent map. For example, when c = 2 the line segment joining the origin and −i has the same dynamics as the tent map G : [0, 1] → [0, 1] defined by: The following proposition describes the dynamics of the simplest TTM's with canonical FL. Proposition 2.4. If f c is a twisted tent map with |c| < 1 then every point in the complex plane is attracted to the origin.
The next proposition shows that f c is conjugate (in the dynamical sense) to f c . Thus, without loss of generality we can focus our study on parameters c = α + βi where β ≥ 0. Written another way, if c = re iθ then we may always assume that θ ∈ [0, π].
Proof. Let ϕ(z) = −z so that ϕ is reflection about the imaginary axis. We will show that ϕ(f c (z)) = f c (ϕ(z)). That is, we wish to show that the following diagram commutes.
This shows that ϕ semi-conjugates f c with f c . Lastly, since ϕ is an isometry, then it conjugates f c and f c .
The dynamics of a TTM become more interesting when |c| = 1. We need the following.
Definition 2.6. We will use PFL to denote the pre-folding line, which is the preimage of FL under f . This is also equal to the set of points {z/c : z ∈ FL}.
Definition 2.7. The PFL partitions the plane into two half planes. We define the pre-upper half plane as P H + = {z : f (z) = cz} and define the pre-lower half plane as P H − = {z : f (z) = cz − 2i}. Note that both of the pre-half planes contain PFL.
The PFL plays an important role since it is an axis of symmetry. That is, if z = Reflect(z, PFL), then f (z ) = f (z). Also note that an alternative and equivalent way to define f would be to reflect every point in P H − about the PFL and then multiply the result by c. That is, fold then multiply. This way of visualizing the action of f is sometimes helpful. Definition 2.8. If c / ∈ R then the intersection between the FL and the PFL is unique. In this case we define {γ 0 } = FL ∩ PFL.

2.
If θ = π then every point is eventually mapped into a horizontal strip of height 2. For every point z in this strip, f 2 (z) = z.
• j k is written in lowest terms, • j, k ∈ {1, 2, 3, ...}, then there is a regular periodic polygon, centered at the origin, with period k, such that the orbit of every point eventually intersects the polygon.
4. If θ is an irrational multiple of π, then for every point z there exists a point q ∈ D such that |f n (z) − f n (q)| goes to 0 as n goes to infinity.
Proof. Case 1: Let θ = 0 so that c = 1. The first image of the complex plane is always H + and since c = 1 then (f c ) restricted to H + is the identity map. Case 2: Let θ = π so that c = −1 and let z = a + bi ∈ C be given. Since f (C) = H + then it is enough to prove the claim only for points in H + . Let S = {z : −1 ≤ Im(z) ≤ 1}. If z ∈ S ⊂ H + then f (z) = cz = −z ∈ S and f 2 (z) = −(−z) = z. Thus, every point in S is a periodic point of period 2. Now for every z ∈ (H + \ S) we have that z = a + bi, b > 1 and so f (z) = cz −2i = −a+bi−2i. Thus if z ∈ (H + \S) then f negates the real part of z subtracts 2i. This means if the first n − 1 iterates of z are in H + \ S, then f n (z) = (−1) n a+bi−2ni. We now write b = 2m+r where m is a non-negative integer and 0 ≤ r < 2. Then Im(f m (z)) = b − 2m = 2m + r − 2m = r < 2. Now if r ≤ 1 then f m (z) ∈ S and we are done. Otherwise, f m+1 (z) ∈ S.
Case 3: Let θ = j k 2π, where • j k is written in lowest terms, • j, k ∈ {1, 2, 3, ...}, • and j < k. .., [c k−1 γ 0 , γ 0 ] form a k-pointed star which partitions the plane and has the origin as the center of the star. By symmetry, the piece of the partition containing the origin is a regular polygon P with k sides. By the definition of γ 0 ∈ FL, we have f (γ 0 ) = cγ 0 ∈ FL, and so the bottom-most edge of P lies on FL. In both cases, it is evident that P = {z : cf n (z) ∈ H + , n = 0, 1, 2, ...} and so f (P ) = P . Note that f k (z) = z for all z ∈ P .
We now show that every point in the plane is eventually mapped into P . Choose z ∈ C and let D be a closed disk, centered at the origin, such that z ∈ D. If cz / ∈ H + , then by Lemma 2.11 |f (z)| < |z|. It is easily seen that for every z / ∈ P we have cf n (z) / ∈ H + at least once as n takes on the values 1, 2, 3, ..., k. Thus, every k iterates, a point either lands in P or gets closer to the origin. Consider the sequence of iterates of z, (z n ), n = 1, 2, 3, .... Then since D is compact, and since the iterates of z are contained in D, then this sequence has an accumulation point q inside of D. If q is in the interior of P then we are done. If q / ∈ P then by Lemma 2.11 |f k (q)| < |q| which contradicts the assumption that q was an accumulation point. However, we still have the possibility that q is on the boundary of P .
We now show that every point sufficiently close to P gets mapped into P . First consider the special case when k = 3. Then j is either 1 or 2. But Proposition 2.5 implies that it is sufficient to assume j = 1. (If j = 2 then c = e iθ where θ / ∈ [0, π].) When j k = 1 3 , P is an equilateral triangle with the bottom edge on FL. Figure 8 shows a neighborhood U P of P that has been partitioned by triangles congruent to P and shows that f 5 (U P ) = P . Figure 8: Every point sufficiently close to P is eventually mapped into P . These diagrams show the partitions of U P and show the first few iterates of U P in the case that j = 1. P is shaded. Now assume that k ≥ 4. (See Figure 9) Let S be the union of lines collinear to the segments forming the k−pointed star in the construction of P . We define U P = {(1 + ε)z : z ∈ P } for ε > 0 where ε is small enough that for every z ∈ U P the line segment [z, 0] intersects at most 2 lines in S. Note that S partitions U P into the sets A, B, and P as shown in Figure 9. During each iteration, multiplication by c rotates one connected component of B and two connected components of A below FL. After this, folding maps the component of B into P and maps the two components of A into P ∪ B. It is now easily seen that f k (B) ⊂ P , f k (U P ) ⊂ (P ∪ B), and thus f 2k (U P ) = P .
Case 4: Let θ be an irrational multiple of π. Fix z 0 ∈ C and let D be a closed disk containing D ∪ {z} and centered at the origin. It is easily seen that since |c| = 1 then |f (z) − f (w)| ≤ |z − w| for all z, w. In particular, when w = 0 this means that f cannot increase the modulus of any point. For this reason and since D is centered at the origin, then the iterates of z are contained in D. Since D is compact, then the sequence of iterates (z n ) = f n (z 0 ) c n , n = 1, 2, ... contains a subsequence (z n j ) that converges to an accumulation point q ∈ D. Then |z n j − q| goes to 0 as j goes to infinity. If q / ∈ D then for some k > 0 f k (q) ∈ P H − and by Lemma 2.11 |f k+1 (q)| < |q|, which contradicts the assumption that q is an accumulation point. Thus, q ∈ D. Now consider the sequence (|z n − q|) for n = 1, 2, .... Note that since f cannot increase the distance between two points, then |f n+1 Thus, the sequence (|z n − q|) is a decreasing sequence of nonnegative numbers and (by the Monotone Convergence Theorem of real numbers) has a limit. Since the subsequence (|z n j − q|) goes to 0 as j goes to infinity, then this limit must be zero. Lastly, which we just showed goes to zero as n goes to infinity.
As we will see, much more interesting dynamics occur when |c| > 1. Thus, unless explicitly mentioned, for the rest of this paper we will assume that |c| > 1. The following list of assumptions is for the reader's reference.
Standing Assumptions: We will use the following definitions and assumptions throughout this text. They are listed here for the reader's convenience.
1. |c| > 1. 3. c = α + βi = |c|e iθ , 0 ≤ θ < 2π. 4. f c = f is a twisted tent map with folding line consisting of points with imaginary part equal to −1. We will only write f c if the choice of c needs to be explicitly stated.
7. We will use PFL to denote the pre-folding line, which is the preimage of FL under f .
8. We will denote the line segment between z, w ∈ C by [z, w] = [w, z].

The Filled-in Julia Set
Some of the most notable objects of study in quadratic complex dynamics on the complex plane (or Riemann sphere) are the Julia set J, filled-in Julia set K, and the Fatou set F . We now define these notions for Twisted Tent Maps, give examples, and note some of their key differences with their counterparts in quadratic dynamics. Note that each TTM can be extended to a map from the Riemann sphere to the Riemann sphere by defining f (∞) = ∞. These maps will also be called TTM's and it should be clear from context which definition of TTM's we are using.  The following is a short list of dynamical properties of TTM's that are distinct from their rational counterparts. Definition 3.3. We will call a point preperiodic if it is strictly preperiodic. We will call a point eventually periodic if it is either periodic or preperiodic.
Definition 3.4. The diameter of a set X will be denoted Diam(X).

Proof. Let lim
m→∞ Diam(f m (U )) = 0. If U contains an eventually periodic point z 0 , then let z be a periodic point of period n > 0 in the orbit of z 0 . If each of the n points in the periodic orbit of z is on FL, then every short line segment starting from from a point in this orbit increases in length under iteration, a contradiction. Thus, at least one point of this orbit is not in FL. Let δ = min{| Im(f n (z)) − (−i)| = 0 : n = 1, 2, · · · , n}. Then δ is the shortest non zero distance from a point in the orbit of z to FL. Clearly, every line segment of length δ/2 starting from a point in this orbit will increase in length under one application of f . This also is a contradiction and we are done.
Proposition 3.6. The point at infinity is the only attracting periodic point.
Proof. Since |c| > 1 then multiplication by c increases the modulus of a point. Folding can only decrease the imaginary part of a point, and can decrease the modulus of a point by at most 2. Thus, |f (z)| ≥ |cz| − 2. Let k > 2 where k = |c|. Then solving |cz| − 2 > k|z| for |z|, we get |z| > 2 |c|−k . Thus, for all z such that |z| > 2 |c|−k , we have |f (z)| > k|z| > 2|z|. It follows easily by induction that |f n (z)| > 2 n |z| which goes to infinity exponentially as n increases. It then follows easily that infinity is an attracting fixed point. A likely consequence of Propositions 3.5 and 3.6 is that the basin of attraction of infinity is the only Fatou domain. However, there is still the possibility of the existence of wandering domains within K. We make the following conjectures.  If these conjectures are true, then the study of K, J, and F are in many ways equivalent. Since K is the easiest to define, it is the one we choose to study.
We now wish to describe the simplest K when |c| > 1. We begin by characterizing K(c) for c ∈ R; these K are contained in the imaginary axis, and are line segments or Cantor sets. We then describe K(c) where Im(c) is small; these K are double spirals made from line segments. We conclude this chapter with a discussion on how continuously changing c can lead to the discontinuous creation or deletion of isolated points or components of K.
For this chapter, let G(x) be the real tent map defined by Lemma 3.9. If c ∈ R then K(f ) is a subset of the imaginary axis.
Lemma 3.10. If c ∈ R then f restricted to the imaginary axis is conjugate to the real tent map G.
We first note that the intersection of PFL with the imaginary axis is the point −i c , and that ϕ( −i c ) = 1 2 . It then follows easily that z is a point in the intersection of P H + and the imaginary axis if and only if ϕ(z) ≤ 1 2 . Thus, we have two cases.
Lemma 3.11. Let G(x) be the real tent map. Then if |c| > 2 then the set of points on the real line whose trajectories do not diverge to infinity is a Cantor set.
Proof. This is a well known result.
Proof. This follows immediately from Lemmas 3.10 and 3.11.
Proof. This follows from Lemma 3.10 and well known results in the study of the real tent map.
For ease of reference, we state as a theorem the collection of the above results to describe K for all c ∈ R.
2. If c = 1 then f (C) = H + and f restricted to H + is the identity map.
Proof. The cases when |c| < 1, c = 1, and c = −1 follow from Proposition 2.4 and Theorem 2.12. The rest of the cases follow from Lemmas 3.13 and 3.12.
Things start to get more exciting when θ = 0, π. When Im(c) is small and nonzero, then a spiral made of line segments emanates from the origin. The PFL acts as an axis of symmetry and so K is a double spiral. As the parameter changes, the bottom spiral may collide with FL. When the intersection of the bottom spiral with H + is disconnected, then K will have isolated point and/or components which are preimages of the portion of the lower spiral that was introduced into H + . An illustration of this process is given in Figure 10. The creation of these new components of K can be more fully understood when considering the inverse map. The preimage under f is found by removing the open lower half-plane, unfolding a copy of H + onto H − and then dividing by c. Thus, if locally the intersection K ∩ H + consists of one point, then there will be infinitely many preimages of that point which are all isolated points of K.
As an example, there exists a K(c) that has both isolated points and disjoint connected components where c = 0.971 + (0.851 + ε)i and 0 < ε < 0.001. (See Figure 11) We formalize these ideas in the following results.
Proof. K is completely invariant trivially. Thus, f −1 (K) = K. Since the first step in finding the preimage of K is to remove the open lower half plane, then Proposition 3.16. If every point z ∈ K has the property that Im(z) > −1, then K has uncountably many components and is a Cantor set.
Proof. We will closely follow the proof in [7, pg 99]. Assume that every point z ∈ K has the property that Im(z) > −1. Then PFL cuts the plane into two open half planes with f (K n 0 ···np ) = K n 1 ···np . Similarly, for any infinite sequence n 0 n 1 n 2 · · · of zeros and ones, let K n 0 n 1 n 2 ··· be the intersection of the nested sequence Each such intersection is compact and nonvacuous. In this way, we obtain uncountably many disjoint nonvacuous subset with union K. Every Figure 12: On the left is the set of parameters c in the parameter plane such that K(c) has at least one point in H − . On the right is the same set along with 3 disks centered at the origin with labeled radii for reference.
connected component of K must be contained in exactly one of these, so K has uncountably many components. Each of the sets K n 0 n 1 ···np is disjoint from FL and since |c| > 1 then by construction Diam(K n 0 n 1 ···np ) = |c| Diam(K n 0 n 1 ···npn p+1 and thus the diameter of these sets go to zero. Thus, K is a Cantor set. It is worth noting that when K is a Cantor set, then the dynamics on K are conjugate to the one-sided 2-shift. Figure 12 shows the region in the parameter plane where K(c) has at least one point in H − . (This is equivalent to the existence of j ∈ H − , j ∈ Z.) Outside of this region, K(c) is contained in the open upper half plane and by Proposition 3.16, K(c) is a Cantor set. We conjecture that K(c) is a Cantor set if and only if j / ∈ H − for all j ∈ Z. (We define j on page 26.) In Chapter 4 we prove that f has two fixed points Since the image of every point is in H + then these fixed points are also in H + .
Proof. If K ∩ H + is connected then both fixed points 0 , 0 are in the same Proposition 3.20. If K has no trivial components and if K ∩ FL is connected, then K is connected.
Proof. Assume K has no trivial components. Assume K ∩ FL is connected. Let K 1 be the unique component containing K ∩ FL. Take any other component K 2 ⊂ K. If for every n > 0, f n (K 2 ) ⊂ K 1 , then Diam(f n (K 2 )) goes to infinity as n increases, since f n (K 2 ) ∩ FL = ∅. This is a contradiction. Thus, for all K 2 there exists an n such that f n (K 2 ) ⊂ K 1 . Let K 3 be the component containing 0. Since f (0) = 0, then for all Figure 13 shows an example when K can be connected even if K ∩ FL is disconnected. For this example c ≈ 1.191487884 + 1.191487884i. The exact value of c has α = β = (1/6)(54 + 6 √ 33) 1/3 + 2/(54 + 6 √ 33) 1/3 . For this choice of c, the bottom of K is both a subset of FL and a Cantor set. In particular −1 , −2 ∈ FL (see Chapter 4). The set K can be seen to be connected by the following argument. Cover K with closed balls of radius ε centered at every point of K where ε > 0 is chosen such that the union of these balls, B, is connected. Since every point in K is in H + , then removing the open lower half plane does not disconnect B (this is the crux of the argument). Since −2 ∈ FL ∩B then B ∪ Reflect(B, FL) is a connected set. Dividing by c reduces the diameters of the balls and rotates, but has no May 10, 2014 It is now easily seen that K is the intersection of a nested sequence of nonempty, compact, connected sets of the form f −n (B). Figure 13: An example where K is connected but FL ∩K is disconnected.
Proof. Let |c| > 2 and assume that K has nontrivial connected components. Let K 1 be one of these components with maximal diameter. Then Diam(K 1 ) = d > 0 and Diam(f (K 1 )) ≥ d|c| 2 > d. Since the continuous image of a connected set is connected, then f (K 1 ) ⊂ K is connected with diameter strictly larger than d, a contradiction. Thus, K has only trivial components.  Proof. Let µ(S) denote the Lebesgue measure of a set S and for all w ∈ C, let wS = {ws : s ∈ S}. By definition K is forward invariant. Noting that multiplication by c scales both dimensions by a factor of |c|, we see that if |c| > √ 2 then µ(cK) = |c| 2 µ(K) ≥ 2µ(K) with equality exactly when µ(K) = 0. Since folding can reduce the measure of a set by at most half, then Since |c| > 1, then by induction we have that every point outside of the open unit disk has a trajectory that diverges.

The Perimeter Set P
In this chapter we introduce the perimeter set, P c , which will be of primary importance throughout this paper. We will write P in stead of P c whenever possible. Note that Reflect(z, PFL) = 1 c (cz − 2i). We begin with some definitions.  Figure 14) We define, 0 = 2i(1−c) |c| 2 −1 . We also define k , where k = 0, as We now show that the point 0 is a fixed point of f . First we need a lemma. Proof. Since |c| > 1 then |c| 2 − 1 > 0 and so the inequality Im(c 0 ) < −1 is true precisely when 2α − 2|c| 2 < −|c| 2 + 1, or equivalently when where the inequality is strict. Now since (with equality only when c is real) then with equality only when α = 1. Now if c is real and α = 1 then c = 1. Since we assume that |c| > 1 then at least one of the inequalities in (17)   Proof. Suppose that Im(cz) ≥ −1. Then since |c| > 1 we see that f (z) = cz = z has a unique solution z = 0. Now suppose that Im(cz) < −1. Then we want to solve f (z) = cz − 2i = z for z. We will show that z = 0 is the desired solution. This solution is unique since h(z) = cz − 2i multiplies all distances by |c|. By Lemma 4.2 we have: Case 3: If k > 0 then In quadratic complex dynamics, every connected filled-in Julia set has diameter less than or equal to 4. This is useful since it allows one to choose a fixed bailout value for all computer pictures of connected filled-in Julia sets. We now show that for TTM's, the diameter of K can be arbitrarily large when |c| is close to 1. Thus, the choice of our bailout value will need to depend on c.   Proof. Lemma 4.8 gives us that the PFL is perpendicular to L 0 . Also, f restricted to P H + maps PFL onto FL, maps L 0 to L −1 , and preserves angles. The next proposition states that if L self-intersects, then L intersects FL before it intersects itself. Proof. Suppose L n ∩ L m = ∅ for integers 0 < m < n. Since |c| > 1 then L n ∩ L m consists of a single point which we will call x m , since x m ∈ L m . It follows that f m (L n ) ∩ f m (L m ) = L n−m ∩ L 0 = f m (x m ) and we call this point x 0 . Then x 0 is seen to be the first intersection of L ∪ L 0 with itself.
Assume that α ≥ −1. Later in Proposition 4.29 we will show that Re(γ 0 ) < Re( 0 ) if and only if α < −1. Thus, if α ≥ −1, then PFL either does not intersect cL 0 or PFL intersects cL 0 on/below FL. That and is the intersection between cL 0 and L n−m−1 . Since f (x 0 ) ∈ P H − then L n−m−1 intersects PFL and we conclude that L n−m−2 ∩ FL = ∅ and we are done. Now assume that α < −1. Let ω = π − θ. Then ω is the smallest angle between any two consecutive L k and since α < −1 then π/2 < θ < π. Since 4θ > 2π and from the self-similarity of the L k , it is easily seen that if L 0 ∩ L n = ∅, n = 2, 3 then L 0 ∩ L n = ∅, n > 3. Thus, we have only two more cases.
Case 1: (See Figure 16) Assume L 0 ∩ L 2 = ∅. Then L 0 ∪ L 1 ∪ L 2 bound an isosceles triangle T where at least two of the interior angles are ω. By Lemma 4.8, PFL is the perpendicular bisector of L 0 , which implies that PFL /c is the perpendicular bisector of L 1 . Thus, PFL /c bisects T and so {x 0 } = (L 0 ∩ L 2 ) ∈ PFL /c. But this implies that f 2 (x 0 ) ∈ FL ∩L 0 and we are done. Case 2: Lastly, assume L 0 ∩L 3 = ∅ but L 0 ∩L 2 = ∅. A necessary condition for this is π 2 < θ < 2π 3 . The parameter with the smallest modulus satisfying c = re iθ and Re(c) ≤ −1 is −1 + i √ 3 which has modulus 2. Thus, we may assume that |c| ≥ 2. To increase the chances of L 0 intersecting L 3 we clearly want to minimize both |c| and the angle between L 0 and L 1 . Thus, we assume that |c| = 2 and ω = π 3 . Denote the length of L 0 by d. Now, plotting L n , n = 0, 1, 2, 3 under these conditions, treating vertices 2 , 3 as flexible joints by straightening out L 2 , L 3 towards L 0 , we obtain the simplified diagram shown in Figure 17. Since 1.5d < √ 3d, then L 3 ∩ L 0 = ∅. Since everything that was done increased the chance of an intersection between L 0 and L 3 , then if α ≤ −1 and (L 1 ∪ L 2 ∪ L 3 ) ∩ FL = ∅, then L 0 ∩ L 3 = ∅.  • Let d(z, w) = |z − w| and let d(z, S) = min{d(z, w) : w ∈ S} for any compact set S.
• We will denote the complement of a set S by S c .
• We will denote Reflect(z, PFL) by z and will write X = {z : z ∈ X} = Reflect(X, PFL) for any set X ⊂ C.
• We will denote the midpoint of L k by m k = k + k+1 2 for k ≥ 0. We will also write m = f (m 0 ).
• We will denote the boundary of a set X by Bd(X).        Proof. This follows immediately from symmetry and Corollary 4.9, which says that f (L 0 ) is a vertical line segment. It is easy to see that f (L 0 ) is collinear with L −1 . Proof. Since f restricted to P H + is multiplication by c, then for any set U ⊂ P H + we have f (Int(U )) = c Int(U ) = Int(cU ). Since P c is symmetric with re- Proof. Let f (z) ∈ P ∩ H + . Then z ∈ P since otherwise z ∈ P c and then by Proposition 4.18 we would have f (z) ∈ P c . Lemma 4.20. P is compact.
Proof. P c = Int(S ∪ S ) and is so it is open. Thus, P is closed. To show that P is bounded we begin with a disk D centered at the origin and containing the line segment L 0 = [ 0 , 1 ]. Now for all k > 1 we have that | k | ≤ | 0 | |c| . Thus L k ⊂ D, k ≥ 0. Similarly, L ⊂ D . Let γ 1 = γ 0 /c. Then by construction Conv(P ) = Conv((L ∪ L ∪ {γ 1 }). Therefore, any disk centered at the origin containing D ∪ D ∪ {γ 1 } will contain Conv(P ) and necessarily also P . Thus, P is a closed and bounded subset of the complex plane, and is therefore compact. Proof. Let z ∈ P c be given. Since P is symmetric with respect to the PFL then d(z, P ) = d(z , P ). Also, since f (z) = f (z ) then d(f (z), P ) = d(f (z ), P ). Thus, d(f (z), P ) ≥ |c|d(z, P ) if and only if d(f (z ), P ) ≥ |c|d(z , P ), and so we may assume that z ∈ P H + .
Since P is compact, then there exists w ∈ P such that d(z, w) = d(z, P ). Clearly, w ∈ P H + , for otherwise d(z, w ) < d(z, w) which would contradict d(z, w) = d(z, P ). Noting that f restricted to P H + is multiplication by c, we have that d(f (z), f (P )) = d(f (z), f (w)) = d(cz, cw) = |cz − cw| = |c||z − w| = |c|d(z, P ). Now by Corollary 4.19 P ∩ H + ⊂ f (P ), and so for every point q ∈ P ∩ H + we have d(q, P ) ≥ d(q, f (P )). Setting q = f (z) this becomes d(f (z), P ) ≥ d(f (z), f (P )). Thus if z ∈ P c then d(f (z), P ) ≥ d(f (z), f (P )) = |c|d(z, P ). Proof. Let z ∈ P c . Then by Lemma 4.21, d(f (z), P ) ≥ |c|d(z, P ). By Proposition 4.18 we have that f (z) ∈ P c . Thus, d(f n (z), P ) ≥ |c| n d(z, P ) ans so d(f n (z), P ) grows exponentially as n increases. This shows that z / ∈ K. Since z ∈ P c was arbitrary, then K ∩ P c = ∅ and thus K ⊂ P . We now define a curve Γ which is a subset of the preimages of PFL. Γ is constructed in the same way as L. Recall that {γ 0 } = { α−1 β −i} = FL ∩ PFL. Definition 4.24. We make the following definitions.

We also define
Note that Γ 0 ∩ Γ = {γ 1 } in the same way as L 0 ∩ L = { 1 }. Definition 4.25. If ζ exists and n is the smallest positive integer such that ζ ∈ L n , then we define the outer-most boundary of P to be If ζ does not exist then the outer-most boundary of P is L ∪ L 0 ∪ L and we additionally define the inner-most boundary of P to be Γ ∪ Γ (see Figure 21). For this choice of c, the outer-most boundary is a triangle. However, K is a polygon with infinitely many sides.
When P is a polygon then the outer-most boundary of P is just the boundary of P . This is not true when P is not a polygon. Figure 22 shows an example where K = P , K has nonempty interior, and yet the outer-most boundary of P is not the boundary of P . Proof. It is easily seen that the points of P with the smallest real part belong to the outer-most boundary of P . Due to this and the spiraling of L , for there to be a point z ∈ P with Re(z) < Re( 0 ), then L must have a selfintersection. Then by Proposition 4.10, ζ must exist and by definition, P is contained in the polygon bounded by the outer-most boundary of P . It is clear that every point in this polygon has real part greater than or equal to Re( 0 ).   Proof. Re(γ 0 ) = α−1 β and Re( 0 ) = −2β α 2 +β 2 −1 . Since β > 0 and |c| = α 2 + β 2 > 1, then both denominators are positive. We have is equivalent to α < −1. Thus α < −1 if and only if Re(γ 0 ) < Re( 0 ). As a result, if α < −1 then Re(γ 0 ) < Re( 0 ) and so γ 0 / ∈ P . Theorem 4.22 then gives us that γ 0 / ∈ K.  Proof. We will show that if α < −1 then L 0 ⊂ K. Now assume that α < −1. By Proposition 4.29 Re(γ 0 ) < Re( 0 ). By Lemma 4.28 the point m / ∈ P . Thus by Theorem 4.22 we have that m / ∈ K. This is a contradiction since m ∈ f (L 0 ) ⊂ K. Thus α ≥ −1.
It is important to notice that there are times when the full line segment [ −1 , 0 ] = L −1 extends past a side of P . One such case is illustrated in Figure 23.  Proof. We first note that 0 = 2i(1−c) So where δ = |c| 2 = α 2 + β 2 . Now the claim that Im( −1 ) > −1 is equivalent to the claim that Im( −1 ) + 1 > 0. We have: which is greater than 0 exactly when δ = |c| 2 > 2.    Proof. Let B(ε, z) ⊂ U be a closed ball of radius ε > 0 centered at z ∈ K. If the images of U do not intersect FL, then f n (B(ε, z)) = B(c n ε, f n (z)). Since K is bounded, this is impossible, since for a sufficiently large n the largest distance from a point of K to FL is smaller than c n ε. Proof. If ζ exists, then the outer-most boundary of P is a polygon. Due to symmetry and the fact that L 0 is shared by by both P H + and P H − , it is clear that the outer-most boundary of P will have an odd number of sides. Note that when c is purely imaginary and 1 < |c| ≤ √ 2, then P looks like a rectangle, but the sides L 2 and L −2 are counted separately. In this case, the outer-most boundary of P has 5 sides.
Lemma 4.40. If ζ exists and α ≤ 0, then the outer-most boundary of P has either 3 or 5 sides.
Proof. Suppose ζ ∈ L n , for some n > 0. If n = 1, 2, then the outer-most boundary of P has 3 or 5 sides respectively, and we are done. Thus we may assume that n ≥ 3 and that (L 1 ∪ L 2 ) ∩ PFL = ∅. In particular this means that 3 ∈ P H + . Then α ≤ 0 implies θ ≥ π 2 and so L 3 starts at 3 and slopes away from PFL. Thus, L 3 ∩ PFL = ∅. It is now easy to see that for ζ ∈ L n to be true, it is neccessary for L to have a self-intersection along at least one of L 1 , L 2 , L 3 . However, Proposition 4.10 would then imply that (L 1 ∪ L 2 ) ∩ FL = ∅. But this means that (L 1 ∪ L 2 ∪ L 3 ) ∩ PFL = ∅, a contradiction. We now assume that Im(ζ) ≤ −1 and that α ≥ −1 and show that for every possible number of sides that the outer-most boundary of P can have, that |c| ≤ √ 2. Case 1: Assume that the outer-most boundary of P has 3 sides as shown in Figure 25. Then {ζ} = L −1 ∩PFL which implies that Im( −1 ) ≤ Im(ζ) ≤ −1. By Proposition 4.33 we have that |c| ≤ √ 2. Figure 25: An example where the outer-most boundary of P has 3 sides.
Case 2: Assume that the outer-most boundary of P has 5 sides. We have 3 subcases.
In each of these cases, Proposition 4.33 implies that |c| ≤ √ 2. Figure 27: An example where P is a rectangle but still has 5 sides.
Case 3: Assume that the outer-most boundary of P has 7 sides. This implies that ζ ∈ L 3 ∩ PFL and so L 2 ∩ FL = ∅. The assumption that Im(ζ) ≤ −1 then implies that L 1 ∩ FL = ∅. We now look at three subcases.
1. By Lemma 4.40 α can not be less than 0.
We will need the following estimate: (33) Applying the estimate given in (33) and using the assumption that |c| ≥ √ 2, we now show that | 3 | 2 < 1.
Case 4: Assume the outer-most boundary of P has 9 or more sides. Lemma 4.40 implies this cannot happen if α ≤ 0. Thus we may assume 0 < α. Assume also that Im(ζ) ≤ −1. Then if ζ ∈ L k then L k−1 is the first segment of L to intersect FL. Thus Im( k ) ≤ −1 and so | k | ≥ 1. We will now show this is impossible when |c| ≥ √ 2, proceeding very much as before. We will need the following estimate, which relies on α > 0: Now we show that if |c| ≥ √ 2 then | 4 | 2 < 1.
This completes the proof.
Proof. (See Figure 30) By definition S 0 = cS 1 . Thus S 2 = S 1 c ∩ P H + = S 0 c 2 ∩ P H + . Clearly, S 2 = ∅ since M 1 ⊂ S 2 and 0 < θ < π. If S k−1 = ∅, then S k = ∅ trivially. Now fix an integer k ≥ 3 and assume that S k−1 = S 0 c k−1 ∩ P H + = ∅ and that γ 0 / ∈ k−1 n=2 S n . Let A = PFL ∩S k−1 . Since γ 0 / ∈ S k−1 then A ⊂ H − \ FL . (Here, A may be the empty set.) Thus A c ⊂ P H − \ PFL and so A c ∩ S k = ∅. Thus S k = S k−1 c ∩ P H + = S 0 c k ∩ P H + . We now wish to define the ray M 0 which intuitively is the ray perpendicular to L 0 that starts at m 0 and "radiates outward" from P .  Figure 31: An example where P resembles a ram's head whose horns spiral and never self-intersect.
Proof. Since |c| > 1, then L spirals in toward the origin. Furthermore, since ζ does not exist, then L ⊂ P H + . Since L n intersects S n non-trivially, then S n = ∅, for n = 1, 2, .... Now the collection of sets S 0 c n , n = 1, 2, ... cover C \ {0}. Thus γ 0 ∈ S 0 c n for some integer n > 0. In fact, there may be infinitely many such positive values for n and we will denote the smallest of these values by k. By the choice of k, γ 0 / ∈ k−1 n=2 S 0 c n and so γ 0 / ∈ k−1 n=2 S n . As previously argued, S k−1 = ∅. Then by Lemma 4.42 γ 0 ∈ S 0 c k ∩ P H + = S k . Since γ 0 ∈ S k then Im(M k ∩ PFL) ≥ −1. Then M k−1 ∩ FL = ∅. Thus S k ∩ H + is nonempty and bounded. It follows immediately that S k+n = S k ∩H + c n for n > 0 and so ∞ n=1 S n spirals in toward the origin. It is now easy to see that P resembles a ram's head as shown if Figure 31. Now if S k ∩ FL = {γ 0 } then γ 0 ∈ S k−1 contradicting our choice of k. Letting A = (S k ∪ S k+1 ) ∩ PFL we see that γ 0 ∈ A and that γ 0 is not an endpoint of the line segment A. Then γ 0 ∈ Int(S k ∪ S k ∪ S k+1 ∪ S k+1 ). Thus, γ 0 / ∈ P . Since K ⊂ P , then γ 0 / ∈ K. Lastly, since γ 0 / ∈ P and since P is a closed set, then there is a neighborhood U γ 0 such that U ∩ P = ∅. Now γ 1 ∈ P and every neighborhood of γ 1 gets mapped into a neighborhood of γ 0 . Thus, there is a small enough neighborhood V γ 1 such that f (V ) ⊂ U implying that V ⊂ K and so K P. Theorem 4.45. If ζ exists and Im(ζ) ≤ −1, then: 1. P is a polygon bounded by the outer-most boundary of P , Proof. Since ζ exists, then ζ ∈ L ∩ P H + ⊂ ∞ n=1 S k . Therefore ζ ∈ S k for some k > 0. Since Im(ζ) ≤ −1, then S k ⊂ H − , for if not, then m k−1 ∈ (S k ∩ S k−1 ) would be in H + \ FL. Then m k ∈ (P H + \ PFL) contradicting the assumption that ζ ∈ S k . Thus, S k ⊂ H − and so either S k+1 = ∅ or S k+1 ⊂ PFL. In both cases it is easy to see that S k+n ⊂ P for n > 0. This means that P is simply connected. Thus, P is a polygon bounded by the outer-most boundary of P which must contain γ 0 since Im(ζ) ≤ −1.
2. P is not the polygon bounded by the outer-most boundary of P .

K P
Proof. Since |c| > 1, then L spirals in toward the origin. Since ζ exists, then ζ ∈ L k for some k > 0. We may assume that k is the smallest positive integer such that ζ ∈ L k . Since Im(ζ) > −1, (and by our choice of k) then γ 0 ∈ Int(S k ∪ S k ). Thus, γ 0 / ∈ P and so γ 0 / ∈ K. Since L k ∩ PFL = ∅ then L k−1 ∩ FL = ∅. By our choice of k we see also that L k ∩ FL = ∅. Since L k intersects both FL and PFL, then we can let A be the closed triangle bounded by FL, PFL, and L k . It is clear that A has nonempty interior since Im(ζ) > −1 (see Figure 35).
It follows immediately that Int(S k+1 ) = Int( A c ) = ∅. Now let X be the polygon bounded by outer-most boundary of P . Then P ⊂ X and A ⊂ X. By definition P = X \  Another example of P when Im(ζ) > −1 is shown in Figure 36. In this example, P is a polygon with countably many open sets removed. Compare this to Figure 35 which shows on the far right a P which is the polygon bounded by the outer-most boundary of P with finitely many open sets removed. Figure 36: An example of P and K when ζ is above FL.

Hungry Sets and Crises
Proposition 3.6 states that there are no attracting periodic orbits (other than the point at infinity if we are on the Riemann sphere.) However, we can still have sets that attract in some sense.
Definition 5.1. A compact set A with positive 2-dimensional Lebesgue measure, and with the property that f (A) = A will be called a hungry set if every point whose orbit contains a subsequence that converges to A eventually lands inside A (gets eaten).
Hungry sets often "attract" sets of positive Lebesgue measure. For this reason, we will borrow (and possibly modify) some familiar terminology. Proof. Since C is a nontrivial component, then it is connected and Diam(C) = d > 0. By Lemma 5.4 C ⊂ K and so the sequence (Diam(f n (C))) is bounded. But |c| > 1 and if f (C) ∩ FL = ∅ then Diam(f (C)) = |c| Diam(C) > Diam(C). The result follows easily.
Note that the union of two hungry sets is a hungry set. For this reason, we need the following definition.
Definition 5.6. We say that a hungry set is reducible if it contains a proper subset which is a hungry set. Otherwise, a hungry set is said to be irreducible. Definition 5.10. We will denote the closure of a set X by Cl(X).
Definition 5.11. Recall that the omega-limit set of z is Definition 5.12. We will call the boundary of any set that is significant dynamically a dynamical boundary. A continuous change in parameter can cause dynamical boundaries to come together, meet, and then cross. We will adapt the term boundary crisis to mean the meeting of dynamical boundaries.
Some examples of dynamical boundaries include the outer-most boundary of P , the inner-most boundary of P , the boundary of hungry sets, FL, and the boundary of any periodic sets.
We now discuss some experimental results giving each type of result its own subsection. Let A be a hungry set. We will give examples where the following seem to occur.
1. An example where the coded-coloring of K shows periodic structures in K.
2. An example where A is a topological annulus.

Renormalization.
4. Examples where boundary crises cause sudden changes in dynamics.
5. An example where A has multiple components bounded by a topological annulus.
6. Examples where an increase in the modulus of c causes the components of A to swell. This swelling leads to boundary crises causing sudden changes in dynamics.
7. An example where K contains multiple disjoint hungry sets.

The Coded-Coloring of K Shows Periodic Structures in K
A traditional way to make pictures when studying the dynamics in one complex variable, is to color a pixel based on how quickly the trajectory of the corresponding point in the plane leaves a region known to contain K. This region is usually a ball whose radius is called the bailout value. This coloring method is known as the escape time algorithm. If K has nonempty interior, then using this escape time algorithm produces colorful pictures like the first one in Figure 37, where there is a large black region (which is K) surrounded by color. But the escape time algorithm alone does not give any information as to what is happening on the inside of K. In Figure 37 the last three pictures use the escape time algorithm but also use a new coloring method we call the coded-coloring. Informally, for TTM's, the method of coded-coloring is an escape time algorithm for how long it takes for the orbit of a point to be in P H + N times, where N is the bailout value. We now give a more formal and general definition of coded-colorings.
Definition 5.13. Let X be a topological space and let f be a map from X to X. Let ϕ be a map from X to R and for every x ∈ X we call the sequence S(x) = (ϕ(f n (x))) = (x n ), n = 0, 1, 2, ... the itinerary of x. We will abuse notation and will let η n (x) = n Σ j=0 x j . Then a coded-coloring of X is any escape-time coloring under iteration of η.
Most of our pictures use the fastest coloring between the traditional escape time coloring and the coded-colorings. For our coded-colorings we typically use a bailout value between 80 and 400 and have ϕ defined by The common refinement of a partition is a new partition defined by intersection of preimages of a partition. The coded-coloring is a way to color Figure 37: This shows first the traditional escape time picture of the filled-in Julia set followed by the three progressively closer looks inside K when using the coded-coloring. A close look at the coded-coloring reveals that there is structure inside of K and gives clues as to what that structure is. In each picture c = 0.5567 + 0.8471i. the common refinement of a partition in a way that is sometimes useful. However, the bailout value used in the coded-coloring needs to be chosen appropriately based on the size of the structures for which one is looking. Larger structures most easily seen with a coded-coloring bailout value that is low, but can become very hard to see with much larger values. Obviously, to see very small structures, a very fine partition is needed and this is achieved by using a large bailout value.
Their are two known reasons why the coded-coloring shows the internal structure of K. First, points that stay close together for a long time will tend to be colored similarly. Second, the boundaries of the structures that appear in the coded-coloring are repelling periodic structures. Thus, patterns of color accumulate on the repelling structures. This is comparable to the Theorem in Rational Complex Dynamics that the preimages of almost every point will accumulate on the Julia set.
It is worth noting that a coded-coloring can be used to see structure in the filled-in Julia sets of quadratic complex polynomials. Attracting periodic points cause large groups of points to be colored the same way. Also, since the Julia set is the closure of the set of periodic repelling points, then patterns of color accumulate only on the Julia set. Figure 38 shows the filled-in Julia set for the map f (z) = z 2 − 0.6. Because the parameter is real, then K is symmetric about the real axis. To have a changes in color occur when crossing a preimage of the real axis, we used: Figure 39 shows an example of where A is expected to be a topological annulus bounded inside and out by the images of FL ∩A. This figure shows the trajectory of a point in A. It seems likely then that A = ω(z, f ) for some z ∈ A.

Renormalization
Renormalization is an important topic that is studied in papers like [8] and [6].
The following examples seem to suggest that the tools of renormalization could be used to study TTM's.  Letting c = 1.05i we have c 4 = 1.21550625 ∈ R. Then by Theorem 3.14 K(c 4 ) = [−2i/c, 0] ≈ [−1.6454i, 0]. It is easily seen that for all z ∈ K(c 4 ), f 4 c (z) = f c 4 (z). In particular, this means that K(c 4 ) is embedded by the identity map into K(c). The embedded structure of K(c 4 ) and its preimages under f c can be seen in the coded-coloring of K(c) given in Figure 40. A separate example is given in Figure 41, which shows an overlay of K(c 3 ) and K(c) for c = −0.6 + 0.9i. Proposition 5.14. Let c = −0.65 + 0.88i. There is an affine transformation ϕ such that for every z ∈ K(|c| 2 c), we have ϕ(f |c| 2 c (z)) = f 3 c (ϕ(z)). Furthermore, ϕ(K(|c| 2 c)) is a period 3 component of a hungry set in K(c).
Proof. Figure 43 shows an example where the embedding map is an affine transformation. Let c = −0.65 + 0.88i, g(z) = cz, and h(z) = cz − 2i. It is easily checked that if z 0 = 2i(1−c) |c| 2 c−1 , then f 3 c (z 0 ) = h(h(g(z 0 ))) = z 0 . Let ϕ(z) = (Im(z 0 ) + 1)z + z 0 . Let B be a ball of radius 5.85 centered at 0 + i. The leftmost picture in Figure 42 shows that K(|c| 2 c) ⊂ B. The right-most picture in Figure 42 shows that ϕ(B) is contained in a non-shaded region, which is the set of points where f 2 c (z) = h(g(z)). It is then a straightforward computation to show that for every z ∈ K(|c| 2 c), we have ϕ(f |c| 2 c (z)) = f 3 c (ϕ(z)). This conjugacy also implies that ϕ(K(|c| 2 c)) is a period 3 component of a hungry set in K(c).

Boundary Crises and Sudden Changes in Dynamics
By perturbing the parameter so that c = −0.04 + 1.05i we find that c 4 = 1.20492481 + 0.18495120i / ∈ R, and yet K(c 4 ) is still embedded into K(c) by the identity map. This is seen in the coded-coloring of K(c) given in Figure 46. In both of the cases given by Figures 40 and 46 applying f c four times was equivalent to applying f c 4 once. This is due to no point getting folded more than once every four iterations.
The coded-coloring becomes more interesting when a continuous change in parameter bringsFL and a preimage of K(c 4 ) together causing a boundary crisis. A small perturbation of parameter can cause these dynamical boundaries to cross. This leads to the creation of islands as shown in Figure 47 and is the same mechanism as was discussed in reference to Figure 10.
Another boundary crisis occurs when c is perturbed until K(c 4 ) touches PFL. If K(c 4 ) crosses PFL, then there is a loss of the fixed point 0 of f c 4 . This in turn leads to a loss of structure. An example of this is given in Figure 48.  Figure 46, we see that islands are created when a structure is introduced into H + from below. The source of the islands is indicated.

Hungry Set with Multiple Components in an Annulus
In Figure 49 shows the coded-colorings of Figures 47 and 48 with 10,000 points in the orbit of 0.1 − i overlayed. This suggests that the boundary crisis can cause a hungry set with multiple components to diffuse into a larger forward invariant set. Often, the larger forward invariant set is a topological annulus. Figure 49: A hungry set with periodic components becoming unstable and filling out a topological annulus bounded inside and out by images of FL. repeated. Once the orbit of points begin to hop components, then A is a topological annulus. Further increasing the modulus of c will eventually lead to the annulus swelling until it contains the origin, at which point A is a topological disk. Figure 51: Each hungry set is a subset of K and therefore is bounded. Thus, each hungry set must intersect FL. Also, as the modulus of c increases, then |cz| increases for every z ∈ FL. This is why the components of the hungry set swell as the modulus of c increases.

Coexisting Disjoint Hungry Sets
By Lemma 5.3, any sub-K with nonempty interior is a hungry set. Figure 52 shows an example of a K where there is a hungry set near the origin and also a period 4 sub-K near the boundary.

Partitioning the Parameter Space
There are several different ways to draw pictures of the parameter space. Each way has advantages and disadvantages. We begin with the definition and picture of the polygonal locus.
Definition 6.1. The polygonal locus is the set of points in the parameter space for which K is a polygon.
By Theorems 4.44, 4.45, and 4.46, K is a polygon exactly when γ 0 ∈ K. Thus, we can make an escape-time picture of the polygonal locus by using γ 0 as our test point. Figure 53 is the result. Notice the vertical line at α = −1.
The boundary of the polygonal locus consists of the union of many curves. Every parameter on a curve has an associated K(c) and for all c on that curve, K(c) will have the same number of sides. The equations of these curves are implicitly defined by c n γ 0 = (1 − t) 0 + t 0 /c, that is, f n (γ 0 ) ∈ L 1 . Figure 54 shows the unit disk centered at the origin and parameters where there is a smallest positive integer n with n ∈ H − . If n is any integer, then the corresponding picture is Figure 57.
Another useful picture of the parameter plane is given in black and white in Figure 55 and in color in Figure 56. Each of these pictures mark the circles of radii √ 2 and 1, the polygonal locus, and the regions marked in Figure 54. The escape-time method of coloring the parameter plane provides visual representations of the external structures of K(c) near a test point when      K(c) has empty interior. Close inspection of Figure 4 leads to Conjecture 6.2. Experiments suggest that the closer the test point is to the origin, the more "little filled-in Julia sets" appear along the outside of the polygonal locus. Conjecture 6.2. Drawing an escape-time picture of the parameter plane using a test point very close to 0 produces an atlas of K's with empty interior all around the outside of the polygonal locus.
Using the coded-coloring instead of just the escape-time algorithm we can get a preview of the coded-coloring inside of K(c) near the test point. Figure 5 shows that the coded-coloring of the parameter plane can help to find parameters c such that K(c) contains sub-K's. Conjecture 6.3. In a neighborhood of c in the parameter plane, the codedcoloring using a test point very close to 0 produces an atlas of sub-K's which can be found in the coded-coloring of K(c) near the test point. Open Question 6.4. For which parameters are all the internal structures of K near 0 the same? Conjecture 6.5. When making coded-colorings of the parameter plane, the smaller the modulus of the non-zero test point, the closer the similarity between the structure in K(c) near the test point and the structure seen in the coded-coloring of the parameter plane near c. Any sub-K's found in this way have sides that approach uniform length as |c| goes to 1.
The coded-coloring of the parameter plane in Figure 59 reveals what resemble sun flares coming out of the unit disk. We will refer to these as flares. In Figure 59 one of the flares is circled and Figure 60 shows a close-up of this flare along with the coded-colorings of various K(c) for six different choices of c near this flare. Experiments suggest that a flare is a set of parameters where there is an n such that K(c n ) is embedded in the dynamical structure of K(c). The larger the flare, and the sharper the image in the flare, the more "stable" the embedding.
Now each flare has a central curve. Figure 60 shows that the closer a parameter is to this central curve (near location 3), the straighter K(c n ) tends to be. This suggest that the central curves of flares meet the unit circle at points whose argument is a rational multiple of π. Conjecture 6.6. Flares in the parameter plane meet the centered unit disk at points whose argument is a rational multiple of π.
Lastly, if z ∈ Bd(K) then at most two A ∈ A n have a common boundary with K in the small neighborhood U of z. Since the sides of the polygon K do not need to be parallel to a preimage of FL, then this may add at most two more angles to consider. Thus, for all z ∈ K we have N (z, n) ≤ 2(2n + 3) = 4n + 6.
Lemma 7.7. There exists an open cover B of A n such that each element of B intersects at most k(n) elements of A n where k(n) grows linearly.
Proof. Let B 1 = {Int(A) : A ∈ A n } and let B 2 = {Int(Cl(A j ∪ A k )) : A j , A k ∈ A n }. Then B 1 ∪ B 2 is a cover of K everywhere but at the locations where more than 2 elements of A n meet. Since A n is a finite set, then these locations are isolated and so can be covered by disjoint open sets. By Lemma 7.6 each of these open set intersects at most 4n + 6 elements of A n which grows linearly.
Proof. Assume that K is a polygon. We define a partition of K by A = {P H + ∩ K, (P H − \ PFL) ∩ K}. Then A n f = A ∨ f −1 (A) ∨ ... ∨ f −(n−1) (A) is the nth-common refinement of A with respect to f . By Lemma 7.7 there exists an open cover B such that each element of B intersects at most k(n) elements of A n . Let C be a minimal subcover of B m . Each element of C intersects at most (k(n)) m elements of (A n f ) m f n . The total number of elements of (A n f ) m f n is less than or equal to (#C)(k(n)) m . We get: