Fractals And The Julia Set

Second Year Essay Fractals and the Julia Set 0407950 April 20, 2006

Contents 1 Fractals 1.1 Topological Dimension . . . 1.2 Hausdorff Dimension . . . . 1.3 Fractals . . . . . . . . . . . 1.4 Typical Features of a Fractal

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2 Julia Sets 2.1 Introduction . . . . . . . . . . . . . . . 2.2 Normal Families and Montel’s Theorem 2.3 Normality of Iterates of f . . . . . . . 2.4 The Filled Julia Set . . . . . . . . . . .

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1

Fractals

To be able to pinpoint our idea of what we define as a fractal, we must first formalise our idea of dimension, as applied to a topological space.

1.1

Topological Dimension

Definition 1.1. The Topological Dimension, dimT , of a topological space, X, is defined to be the minimum value of n such that for every open cover C of X there exists some refinement of C, say C 0 , for which no point of X is contained in any more than n + 1 sets from C 0 For example, consider some line, X, in the R2 plane (not necessarily straight), with the standard topology. Intuitively, we would say that this line has a dimension of 1, so does the Topological Dimension agree with this? Consider some arbitrary open cover of this line in R2 . This open cover will have a refinement over the line which consists of a collection of open line segments along the line. From this stage, we can further refine this (intuitively) so as that every point along X is contained in at most 2 sets giving a dimension of 1. As it happens, the idea of Topological Dimension is in fact analogous to our intuitive idea of dimension on most normal sets. However the topological definition of dimension is not the only definition of dimension there is another important definition is that of the Hausdorff Dimension.

1.2

Hausdorff Dimension

To be able to explain the idea of the Hausdorff Dimension, we need to recap a couple of definitions of properties of sets: Definition 1.2. The Diameter of a set U is defined as |U | = sup{|x − y| : x, y ∈ U } Definition 1.3. A δ-cover of a set F is a collection of sets {Ui } such that S∞ F ⊂ i=1 Ui with 0 < |Ui | ≤ δ ∀i Now, if we suppose that our set F (for which we want to find the Hausdorff Dimension for) is a subset of Rn , and that s ≥ 0. We can then define: Definition 1.4. The s-dimensional Hausdorff Measure of a set F is defined to be Hs (F ) = lim Hδs (F ) δ→0

2

where Hδs

= inf{

∞ X

|Ui |s : Ui is a δ-cover of F }

i=1

For simple integer-dimensional subsets of Rn , we find that if F is a smooth (1-dimensional) curve, then H1 (F ) is the length of the curve, if F is a smooth (2-dimensional) surface, then H2 (F ) = ( π4 ) × area(F ), and if F is a smooth classical m-dimensional surface, then Hm (F ) = cm × volm (F ), where cm = m 2m ( m2 )!/π 2 . This of course all implies that, for example, that a smooth curve has a 2-dimensional Hausdorff Measure of 0, and that a smooth 2-dimensional surface has a 1-dimensional Hausdorff Measure which tends to infinity1 . This can be extended further, if we were to take a graph of Hs (F ) against s, as in Figure 1 (page 3), we would see that the Hausdorff Measure ‘jumps’ from ∞ to 0 at a particular value. At this value, the Hausdorff Measure can equal 0, ∞ or any other positive real. This value is defined to be the Hausdorff Dimension of F , dimH (F ). Definition 1.5. The Hausdorff Dimension of a set F is defined to be dimH (F ) = inf{s : Hs (F ) = 0} = sup{s : Hs (F ) = ∞} This, of course, appears to agree with the Topological Dimension of many simple sets, as outlined above. However, this definition also allows for noninteger dimensions, where the Topological definition does not. There is one futher very important property of the Hausdorff Dimension: 1

Since, for example, a line has area 0 and a unit disk has length → ∞

Figure 1: Graph of Hs (F ) against s ∞

Hs (F )

0

0

dimH (F ) s 3

Theorem 1.6. If F ⊂ Rn and λ > 0 then Hs (λF ) = λs Hs (F ) where λF = {λx : x ∈ F }.2 Proof. If {Ui } is a δ-cover of F , then {λUi } is a λδ-cover of λF . Hence X X s (λF ) ≤ |λUi |s = λs |Ui |s ≤ λs Hδs (F ) Hλδ since this holds for any δ-cover {Ui }. Letting δ → 0 gives that Hs (λF ) ≤ λs Hs (F ). Replacing λ by λ1 and F by λF gives the opposite inequality required.

1.3

Fractals

Definition 1.7. A set F is defined to be a Fractal if it has a Hausdorff Dimension greater than its Topological Dimension So let us consider an example of a set with this property Example 1.8. Let F0 be the unit interval [0, 1] as a subset of the reals. Let Fn+1 be equal to the set Fn with the middle third of each interval making it up removed. For example, F1 = [0, 31 ]∪[ 23 , 1] and F2 = [0, 91 ]∪[ 29 , 13 ]∪[ 32 , 79 ]∪[ 98 , 1]. This means that Fn consists of T 2n intervals, each of length 3−n . The middle third cantor set, F , is equal to ∞ n=0 Fn . The set F clearly has a topological dimension of 0, but what of its Hausdorff Dimension? First, it is prudent to notice that the set is self-similar. That is to say that if you were to take, for example, F ∩ [1, 13 ] you would find that it is the same as F . So, consider the two subsets of the cantor set Fl = F ∩ [1, 13 ] and Fr = F ∩ [ 23 , 1]. Clearly, F = Fl ∪ Fr and the two sets Fl and Fr are disjoint. Assume that the set F has Hausdorff Dimension d, then, since the sets Fl and Fr are dijoint and Borel, Hd (F ) = Hd (Fl ) + Hd (Fr ). Furthermore, by Theorem 1.6, 1 1 Hd (Fl ) + Hd (Fr ) = ( )s Hs (F ) + ( )s Hs (F ) 3 3 At this point, we make the assumption3 that at s = dimH F we have that log 2 0 < Hs (F ) < ∞, and so by dividing by Hs (F ), we get 1 = 2( 31 )s ⇒ s = log . 3 log 2 Note that dimH F = log 3 ≈ 0.63 > 0 = dimT F , and so the middle third cantor set is a fractal set. 2 3

From [1] This assumption is proven true in Example 2.7 on page 31 of [1]

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1.4

Typical Features of a Fractal Set

There are a number of characteristics which are typical to Fractal sets: 1. Self similarity is one of the main features that Fractal sets have4 , of which there are 3 different types: Exact self similarity is where the fractal appears identical at different scales - for example in the middle third cantor set, the set F is identical to F ∩ [0, 13 ] which is itself identical to F ∩ [0, 19 ] and so on. This tends to be typical of fractals formed by geometric replacement rules. Quasi-self similarity is where the fractal appears approximately identical at different scales - they contain small copies of the entire fractal in a distorted form. Statistical self similarity is where a fractal preserves a statistical measure at all scales, and is a feature of random fractals, but is an area I will not be going into in this essay. 2. Fractal sets also tend to have a fine structure - that is to say they have detail at arbitrary levels of viewing - for example, the middle third cantor set has as much detail in the section F ∩ 31n as it does as a whole. 3. Fractal sets tend to have straightforward definitions, even though they have a fine structure. 4. Fractal sets tend to5 have non-integer hausdorff dimension6 , meaning that if you were to measure their length or area (etc) in a classical sense, you would get an answer of either 0 or ∞. For example, the middle third cantor set is an uncountably infinite set (0-dimensional measure), with 0 length (1-dimensional measure). 5. Fractal sets tend to be defined by some sort of recursive procedure whether that be a geometrical rule, as in the middle third cantor set, or some form of iterated function system or probabalistic rule. The above features are typical of most Fractals, however to have these characteristics does not constitute a set as a fractal (although it is a good hint), and not all Fractal sets have all of the above characteristics. 4

Although note that being self similar does not constitute a set as a fractal - the straight line in euclidian space, for example, is exactly self similar, but is of course not a fractal 5 There are examples of fractals with integer dimensions, for example the path of brownian motion has topological dimension 1 and hausdorff dimension 2 6 The word Fractal suggests the idea of a fractional dimension, although many fractals have dimension in the set R \ Q

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2

Julia Sets

2.1

Introduction

First, we must define the polynomial f : C → C to be of degree n ≥ 2 and to have complex coefficients, i.e. f (z) = c0 + c1 z + . . . + cn z n . We must now remind ourselves of a few definitions related to interated function systems: Definitions 2.1. f k is the k-fold composition of the function f , i.e. fk = f ◦ . . . ◦ f | {z } k times

z is a fixed point of f if f (z) = z z is a periodic point of f if ∃p ≥ 1 st f p (z) = z The least p such that f p (z) = z is called the period of z Definition 2.2. A periodic point z, of period p, under a function f is called repelling if |(f p )0 (z)| > 1, and is called attractive if |(f p )0 (z)| < 1 We are now equipped to understand Gaston Julia’s own definition of his Julia Set: Definition 2.3. The Julia Set, J(f ), of a complex polynomial function f is defined to be the closure of the set of repelling periodic points of f . So, let us consider a simple example of a Julia set: Example 2.4. Let f (z) = z 2 . Clearly f k (z) = z 2k , so to find J(f ), we need to find the periodic points, and discover whether they are attractive or repelling. First, we shall find these periodic points. Clearly, we have to solve z 2k = z in the complex plane. The set of points satisfying each value of k are Jk = {exp(2πi 2kn−1 ) : n ∈ N, 0 ≤ n < 2k − 1}, with the extra special case of z = 0 which has period 1. z = 0 is of course attractive7 , since f (z) = z 2 ⇒ f 0 (z) = 2z ⇒ f 0 (0) = 0 and is therefore not in the Julia set. Note now that (f k )0 (z) = 2kz 2k−1 . Furthermore, the set Jk lies on the unit circle |z| = 1 for all k, and the infinite union of Jk for all k is equal to the unit circle. This means that |z m | = 1 for all z ∈ Jk , thus |(f k )0 (z)| = 2k, and is certainly greater than 1, thus making this points repelling. The Julia Set J(z 2 ) is therefore the unit circle |z| = 1. 7

In fact, this point is superattractive, since (f k )0 (z) = 0, but that is not relevant here

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This is, of course, one of the special cases where the Julia Set is not a fractal. In fact, there are only 2 such cases when looking only at Julia sets of functions of the form fc (z) = z 2 + c, which are the most interesting ones, and the ones I shall be using from here on8 . In this example however, the Julia set appears to be the boundary between 2 types of behaviour of the iterated function system. Inside the Julia set, upon repeated action of f0 , a point will tend toward 0. Outside the Julia set, however, a point will tend toward ∞ under repeated action of f0 .

2.2

Normal Families and Montel’s Theorem

To be able to analyse the basic properties of a Julia Set, we must introduce the ideas of normal families of analytic functions and Montel’s Theorem. Definition 2.5. Let U ⊂ C, and let U be open. Let gk : U → C be a family of complex analytic functions9 . The family {gk } is Normal on U if every sequence of functions from {gk } has a subsequence which converges uniformly on every compact subset of U , either to a bounded analytic function or to ∞. Remark 1. This is equivalent to saying that the subsequence from the family {gk } converges to a bounded analytic function or to ∞ on each connected component of U . Definition 2.6. The family of complex analytic functions {gk } are normal at the point w ∈ U if ∃ open V ⊂ U , such that w ∈ V and {gk } is normal on V. Remark 2. This is equivalent to saying that ∃ a neighbourhood V of w on which every subsequence from {gk } converges uniformly to a bounded analytic function or ∞. At this point we can now introduce Montel’s Theorem, which is the fundamental result on which the theory of Julia sets is based. Theorem 2.7 (Montel). Let {gk } be a family of complex analytic functions on an open domain U . If {gk } is not a normal family, then ∀w ∈ C, with at most one exception, we have gk (z) = w for some z ∈ U and some k. Proof. Too complicated for here, consult [6] for a proof. 8

The only other case where the Julia Set is not a fractal is J(z 2 − 2), which turns out to be a horizontal straight line 9 Functions which are complex differentiable on U

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2.3

Normality of Iterates of f

At this point I will form a new set which is made up of those points at which the function f is not normal. Definition 2.8. J0 (f ) = {z ∈ C : the family {f k }k≥0 is not normal at z} Definition 2.9. F0 (f ) = C \ J0 (f ) = {z ∈ C : ∃ open V with z ∈ V and {f k } normal on V } Remark 3. Note that F0 (f ) is trivially open, since it is the union of open sets. I am now going to show that this is an alternate definition of the Julia set for polynomials of the form fc . The following Lemma is needed by Lemma 2.11. Lemma 2.10. For any c there exists R > 0 such that |z| > R ⇒ |fc (z)| > 2|z| Proof. Let R = max(3, 2|c|), and let |z| > R. Then: |fc (z)| = |z 2 + c| ≥ ||z|2 − |c|| Now, since |z| > R, this implies |z| > max(3, 2|c|) ⇒ |z| > 2|c|, so: ||z|2 − |c|| ≥ ||z|2 −

|z| | 2

Similarly, |z| > 3, and so |z|2 > 3 × |z|, and so ||z|2 −

|z| 1 | > (3 − )|z| > 2|z| 2 2

This all gives that if you take R = max(3, 2|c|), then |z| > R ⇒ |fc (z)| > 2|z| as required. Lemma 2.11. J0 (fc ) is compact Proof. Since F0 (fc ) is open, its compliment, J0 (fc ) must be closed. By Lemma 2.10, fck (z) → ∞ uniformly on the open set V = {z : |z| > R}. This means that {f k } is normal on V , so V ⊂ F0 (fc ). Therefore, J0 (fc ) is bounded, and thus compact. 8

Lemma 2.12. J0 (fc ) is non-empty Proof. Assume that J0 (fc ) = ∅. Then, ∀r > 0 the family {fck } is normal on the open disc Br (0). Using R from Lemma 2.10, if we take r > R then ∃z ∈ Br (0) such that z → ∞. Furthermore, if we solve fc (w) = w and take R2 = |w|, then if r > max(R, R2 ), then ∃z ∈ Br (0) such that fck (w) = w∀k. Therefore, it is impossible for any subsequence of {fck } to converge uniformly to one of either a bounded analytic function or to infinity on any compact subset of Br (0) containing both z and w, contradicting the normality of {fck }. So J0 (fc ) 6= ∅. For the rest of this section, take the polynomial f to mean a polynomial of the form fc 10 . Lemma 2.13. J0 = f (J0 ) = f −1 (J0 ) Proof. Notice first that this lemma is equivalent to saying that F0 = f (F0 ) = f −1 (F0 ). Let V be an open set with {f k } normal on V . f continuous ⇒ f −1 (V ) is open. Now, take {f ki } as a subsequence of {f k }. Then {f ki +1 } has 0 a subsequence {f ki +1 } which is uniformly convergent on compact subsets of V 11 . Note that if D is a compact subset of f −1 (V ), then f (D) is a compact 0 subset of V . This means that {f ki +1 } is uniformly convergent on f (D), and 0 so {f ki } is uniformly convergent on D. Thus {f k } is normal on f −1 (V ), and so F0 ⊆ f −1 (F0 ). The other inclusions required can be obtained in a similar way, using the fact that f (V ) is open whenever V is open. Lemma 2.14. J0 (f p ) = J0 (f ) ∀p ∈ Z>0 Proof. The first thing to notice for this proof is that J0 (f p ) = J0 (f ) ⇔ F0 (f p ) = F0 (f ), and the right hand side is what we shall try to prove. Notice first that if every subsequence of {f k } has a subsequence which is uniformly convergent on a given set, then every subsequence of {f pk }k≥1 also has a subsequence which is uniformly continuous on that set, and so clearly F0 (f ) ⊂ F0 (f p ). Next, it is clear that if D is compact, and {gk } is a family of functions uniformly convergent on D either to a bounded function or to infinity, then the same must be true for {h ◦ gk } for a polynomial h. So, if {f pk }k≥1 is normal on V , then so is {f r f pk }k≥1 = {f pk+r }k≥1 for r ∈ {0, 1, . . . , p − 1}. Of course, any subsequence of {f k }k≥1 must contain an infinite subsequence of {f pk+r }k≥1 for some r ∈ {0, 1, . . . , p−1}, which, since {f pk+r }k≥1 is normal, 10 11

Although note that this can be extended to all polynomials f . By the definition of normal families of analytic functions

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contains a subsequence which is uniformly convergent to a bounded function or infinity on V . Thus, {f k } is normal, and so F0 (f ) ⊃ F0 (f p ), giving the result. Lemma 2.15. Let f be a polynomial. Let w ∈ J0 (f ). Let U be a neighS k bourhood of w. Let W := ∞ f (U ). Then W is the whole of C, with k=1 the possible exception of a single point. This point is not in J0 (f ) and is independent of w and U . Proof. By the definition of J0 , since w ∈ J0 (f ), it follows that {f k } is not normal at w. Montel’s Theorem (Theorem 2.7) then implies that W is the whole of C. Consider a point v ∈ / W . If moreover f (z) = w, then since f (W ) ⊂ W , it follows that z ∈ / W . As we showed above that C \ W consists of at most one point, then it follows that z = v. Thus f must be a polynomial of degree n, where the only solution of f (z) − v = 0. So, f (z) − v = c(z − v)n for some c ∈ R. Consider a z that is sufficiently close to v. It follows that f k (z) − v → 0 as k → ∞, this convergence being uniform on, say, {z : |z − v| < (2c)1/(1−n) }. This means that {f k } is normal at v, and so v ∈ / J0 (f ). It is clear that v depends only on f . Corollary 2.16. If U is an open set intersecting J0 (f ) then, for all z ∈ C with at most one exception, f −k (z) intersects U for infinitely many values of k. Proof. Taking z ∈ C not the exceptional point from Lemma 2.15, then z ∈ f k (U ), and so f −k (z) intersects U for some k. This can be repeated giving an infinite sequence of k for which f −k (z) intersects U . S −k (z)). Corollary 2.17. If z ∈ J0 (f ) then J0 (f ) = ( ∞ k=1 f Proof. For the second point, if you takeS z ∈ J0 (f ), then f −k (z) ⊂ J0 (f ) by −k Lemma 2.13. This of course means that ∞ (z), and thus its closure, is k=1 f contained in J0 (f ). If U is an open set containing z, then by Corollary 2.16, f −k (z) intersects U for some k. Therefore, by Lemma 2.15, z cannot be the exceptional point. Corollary 2.18. If f is a polynomial, then int(J0 (f )) = ∅. Proof. Suppose ∃U = int(J0 (f )) 6= U open. By Lemma 2.13, J0 (f ) ⊃ S∅, with k f (U ). By Lemma 2.15, J0 (f ) is the f k (U )∀k, and therefore J0 (f ) ⊃ ∞ k=1 whole of C with the exception of at most one point, and is so unbounded, which contradicts Lemma 2.11 that it is compact. Lemma 2.19. J0 (f ) is a perfect set12 and is therefore uncountable. 12

Meaning that it is closed and has no isolated points

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Proof. Let v ∈ J0 (f ) and let U be a neighbourhood of v. We must show ∃v 6= w ∈ J0 (f ), with w ∈ U . Consider 3 cases. 1. v is not a periodic point of f . Combining Corollary 2.17 and Lemma 2.13, it is clear that U contains a point from f −k (v) ⊂ J0 (f ) for some k ≥ 1, and since v is not a fixed point, this point must not equal v. 2. f (v) = v. If f (z) = v has no other solution that v, then, as in Lemma 2.15, v ∈ / J0 (f ). Therefore ∃w 6= v such that f (w) = v. By Corollary 2.17, U contains a point of f −k (w) for some k ≥ 1. This point, by Lemma 2.13, is in J0 (f ) and is distinct from v since f k (v) = v. 3. f p (v) = v, p > 1. By Lemma 2.14, J0 (f ) = J0 (f p ), so we can use the method for case 2 on f p instead of f , giving the result required. This case analysis shows that J0 (f ) has no isolated points, and it is closed since F0 (f ) is clearly open13 , and so is perfect. Theorem 2.20. If f is a polynomial, then J(f ) = J0 (f ) Proof. Let w be a repelling periodic point14 , period p, of f . It follows that w must therefore be a repelling fixed point of g := f p . Suppose that {g k } is normal at w. This implies that w has an open neighbourhood V on which a subsequence {g ki } of {g k } converges to a finite analytic function g0 , or infinity. Since g k (w) = w for all k, the analytic function that the subsequence converges to must map g0 (w) = w, and so cannot be infinite. By a result from complex analysis, the derivates must also converge15 , and so (g ki )0 (z) → g00 (z)∀z ∈ V . By the chain rule, |(g ki )0 (w)| = |(g prime (w))( ki )|. However, since w is a repelling fixed point, and the fact that follows that |g 0 (w)| > 1, |(g prime (w))( ki )| → ∞. This implies that g00 (w) must be infinite, which is a contradiction, and so {g k } is not normal at w. This means that w ∈ J0 (g) = J0 (f p ) = J0 (f ), by Lemma 2.14. Since J0 (f ) is closed, we can see that J(f ) ⊂ J0 (f ). Let us define a set L = {w ∈ J0 (f ) : ∃z 6= w, f (z) = w, f 0 (z) 6= 0}. Suppose w ∈ L. It follows that there is an open neighbourhood of w, V , on which we can find a local analytic inverse16 f −1 : V → C \ V such that f (f −1 (z)) = z∀z ∈ V . Define now a family of analytic functions {hk } by: hk (z) =

f k (z) − z f −1 (z) − z

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Discussed in Lemma 2.11 Note that this means that w ∈ J(f ), and in fact covers every w from J(f ). 15 To a non-infinite function. 16 By choosing values of f −1 (z) in a continuous manner.

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Let U be an open neighbourhood of w with U ⊂ V . Since w ∈ J0 (f ), the family {f k } and thus the family {hk } is not normal on U . By Montel’s Theorem (Theorem 2.7), hk (z) ∈ {0, 1} for some k and some z ∈ U 17 . If hk (z) = 0, then f k (z) = z for some z ∈ U . If hk (z) = 1, then f k (z) = f −1 (z), and so f k+1 (z) = z for some z ∈ U . Thus U contains a periodic point of f , so w ∈ J(f ). We have shown above that L ⊂ J(f ), and so L ⊂ J(f ) = J(f ). However, L contains all of J0 (f ) excepting a finite number of points. Since, by Lemma 2.19, J0 (f ) contains no isolated points, J0 (f ) = L ⊂ J(f ) as required.

2.4

The Filled Julia Set

We showed at the end of Section 2.1 how the Julia Set of f0 appears to be the boundary between two different types of behaviour of the system. We shall try to generalise this to all Julia Sets J(fc ). To do this, we need to formalise our ideas of what these behaviours are, and the sets produced by points exhibiting these behaviours. Definition 2.21. The Filled Julia Set, or Prisoner Set of a function fc : C → C is defined to be K(fc ) = {z : ∃N st ∀n, fcn (z) < N }. The Escape Set is the compliment of the Filled Julia Set, and is the set of points which tend to infinity under repeated action of fc . I shall now prove that the boundary of the Filled Julia Set is equal to the Julia Set18 , i.e. ∂K(fc ) = J(fc ). Lemma 2.22. If z ∈ / K(fc ) then there exists a neighbourhood, V , of z on k which fc tends uniformly to ∞ as k → ∞. In particular {fck } is normal at z and z ∈ / J(fc ). Proof. Using R from Lemma 2.10, take k such that |fcn (z)| > R. We can take a sufficiently small neighbourhood V of z so that fcn (V ) ⊂ {z : |z| > R}. Lemma 2.10 implies that fcm (V ) ⊂ {z : |z| > 2m−n R} for all m > n, and so fcm tends to ∞ uniformly on V , as required. Theorem 2.23. ∂K(fc ) = J(fc ) Proof. Note first, that if z ∈ int(K(fc )), then we can take an open neighbourhood U of z such that U ⊂ int(K(fc )). Furthermore, note that if we take R from Lemma 2.10, fcn (U ) ⊂ {z : |z| < R} for all n. 17

Note that we take 2 possible points here, in case either 0 or 1 is the one exception in Montel’s Theorem. 18 Thanks to Richard Oudkerk, Mathematics Institute, University of Warwick for the outline of this proof

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Now, consider the family of analytic functions {gk } = {fck } on the open domain U . Consider w1 , w2 ∈ {z : |z| > R}, with w1 6= w2 . Assume that {fck } is not a normal family. Then, by Montel’s Theorem (Theorem 2.7), for all w ∈ C, with at most one exception, fck (z) = w for some z ∈ U and some k. fck (z) < R as shown above, but w1 > R and so w1 must be the one exception. Similarly, w2 > R and so w2 must be the one exception, implying w1 = w2 , which is a contradiction, itself implying that {fck } is a normal family. So, z ∈ / J(fc ). This, with Lemma 2.22, implies that z ∈ / ∂K(fc ) ⇒ z ∈ / J(fc ), and so J(fc ) ⊂ ∂K(fc ). Assume that z ∈ ∂K(fc ) \ J(fc ). Thus, we can choose a neighbourhood U of z such that {fck } is normal. Since z ∈ K(fc ), the orbit of z is bounded, and so no subsequence will converge to infinity. Thus normality implies that there must exist a subsequence which converges to an analytic function F : U → C. However, U contains points in C \ K(fc ) which by Lemma 2.22 will tend to infinity under fck as K → ∞, so giving us a contradiction. This means our assumption was wrong, and so ∂K(fc ) \ J(fc ) = ∅ and thus ∂K(fc ) = J(fc )

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References [1] Falconer, K. (1999) Fractal Geometry: Mathematical Foundations and Applications, Chichester: John Wiley & Sons Ltd. [2] Peitgen, H-O., J¨ urgens, H. & Saupe, D. (1992) Fractals and Chaos: New Frontiers of Science, New York: Springer-Verlag [3] Bandt, C., Graf, S. & Z¨ahle, M. (eds.) (1995) Fractal Geometry and Stochastics, Basel: Birkh¨auser Verlag [4] Dekking, M., Vehel, J. L., Lutton, E. & Tricot, C. (1999) Fractals: Theory and Applications in Engineering, London: Springer-Verlag [5] Crownover, R. M. (199?) Introduction to Fractals and Chaos, ?: Jones and Bartlett Publishers [6] Krantz, S. G. (1999) Handbook of Complex Variables, Boston, MA: Birkhuser, Sections 8.4.3 and 8.4.4 [7] Wikipedia contributors (2006) Fractal, Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Fractal&oldid=48724032 [8] PlanetMath contributors (2006) Periodic Point, PlanetMath, http://planetmath.org/encyclopedia/AttractivePeriodicPoint.html

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