Ohrimenko+ / Barnsley. Superfractals

the notation sup S, where S R {−∞, +∞}, is defined to be the smallest number in R {−∞, +∞} that is greater than or equal to all the numbers in the set S.

P r o o f The proof of Theorem 2.3.15 is beautiful and subtle, and it may be found in most books on measure theory; see for example [31], Theorem 5, p. 180. The key steps are the following. (i) Define a function, called an outer measure, ν0 : S(X) → [0, ∞), by

(iii)Show that F (X) D and that ν0 agrees with ν on F (X).

(iv)Define ν to be ν0 restricted to F(X). Note that F(X) D.

wise continuous) density function, as discussed in Example 2.3.14. Then Theorem 2.3.15 tells us that there exists a unique measure vρ on the pixel σ -algebra that agrees with the measure v , defined in Example 2.3.14, on the pixel field F ( ). Since, as in Exercise 2.3.8, FGpixels ( ) = B( ), vρ is a Borel measure.

E x a m p l e 2.3.17 There exists a unique measure ν on the code space σ - algebra F( A) that agrees with the measure ν on the code space field F ( A), as described in Example 2.3.13. Since the cylinder sets generate the natural topology on A, the measure ν is actually a Borel measure.

E x a m p l e 2.3.18 Any digital picture defines a (vector of) Borel measure(s) in the following manner. Let α denote the area of the domain of the pixel Pw,h . The area of each pixel in the digital picture PW ×H is the same. Then we define a piecewise-constant density function by

We now use this measure to define a Borel measure on B( ) as in Examples 2.3.14 and 2.3.16. Then if we make a digital picture PW ×H of this measure, we will have PW ×H = PW ×H . The advantage of converting a digital picture into a Borel measure is that it can then be manipulated by continuous transformations and digitized, in a consistent manner.

A measurable set is one to which a measure may be assigned, that is, a member of a field or σ -algebra. Usually, for us, this will mean a Borel set. (Imagine that we could distribute one candlepower of luminous powder on a subset of that is not a Borel set. How would one make digital pictures of the resulting glowing thing?)

In Chapter 4 we will discover a multitude of interesting measures on . For now we need to know that there exist diverse measures on , that they define arrays of pictures, one for each W and H , namely digital pictures, and that they behave as nicely as picture functions under continuous transformations, as Theorem 2.3.19 below shows. Throughout this book we give many examples of digital pictures of measures.

Transformations of measures

Let us first describe intuitively what we would like to happen when a transformation is applied to a measure. Suppose that we are given a normalized Borel measure ν on R2, which we imagine to be a luminous picture in the euclidean plane. Perhaps it is embedded in infinitely thin flat material, like the skin of a vast balloon. Let f : R2 → R2 be a continuous transformation. Then we may think of f as deforming, stretching, shrinking and folding the luminous material. Regions that are stretched will tend to become less bright, regions that are compressed will become brighter and parts that are folded on top of one another will have a brightness that is the sum of the brightnesses of the parts. Figure 2.14 illustrates this idea. This is how we would like to think of the continuous transformation of a measure. We want the result to be a new luminous picture, that is, another Borel measure.

This inspires us to define below the action of a transformation on a measure in a certain obvious sort of way. But is the resulting transformed measure indeed always a Borel measure? Does the transformation process damage the underlying σ -algebra? No, wonderfully, it does not.

Theorem 2.3.19 defines the continuous transformation of a Borel measure and assures us that we obtain a new Borel measure. The key ideas are that the Borel sets are generated by the open sets and that the inverse images of open sets, under continuous transformation, are open sets.

T h e o r e m 2.3.19 Let v M(X) be a Borel measure and let f : X → X be continuous. Then there exists on X a unique Borel measure μ M(X) such that

We denote this measure μ by f (v) and also by f ◦ v.

D e f i n i t i o n 2.3.20 The measure f (v) is called the transformation of the measure v by the function f or the transformation f applied to the measure v.

Figure 2.14 Each image represents a coloured measure on a disk in R2. Each pair of measures is related by a Mobius¨ transformation that maps the disk to itself. So the total brightness of each image is in principle the same, as is the total brightness of each curvaceous square.

When v = (v1, v2, v3) M(X)3 we define

f (v) = ( f (v1), f (v2), f (v3)) M(X)3.

We may refer to a transformation of a measure where we mean a transformation of a vector of measures.

P r o o f o f t h e o r e m 2.3.19 We will show that (i) f ◦ v is defined on F , the field generated by the open subsets of X; (ii) f ◦ v is a measure on F , that is, it obeys Equation (2.3.1) in Definition 2.3.9; (iii) f ◦ v is a measure on FT(X), the smallest σ -algebra that contains F . In fact (iii) follows immediately from Theorem 2.3.15 once (i) and (ii) are established. So we need only to prove (i) and (ii).

Proof of (i): Suppose that S F . Then S can be written as a finite expression (that makes sense) involving unions and complements (with respect to X) of a

finite number of open sets, say S = E(O1, O2, . . . , ON ) where O1, O2, . . . , ON are open sets. Then, since f −1(V W ) = f −1(V ) f −1(W ) and f −1(X\V ) =

X\ f −1(V ) whenever V, W X, as you learnt in Exercise 1.3.2, it follows that

f −1(S) = E( f −1(O1), f −1(O2), . . . , f −1(ON )). Furthermore, because f is continuous it follows that each of the sets f −1(O1), f −1(O2), . . . , and f −1(ON ) is

open. Hence f −1(S) is a finite expression that makes sense, involving unions and

Figure 2.15 illustrates transformations of a Borel measure. The top left panel represents a vector (red, green and blue) v of Borel measures on R2. The other three panels represent three different projective transformations of the measure. Projective transformations map straight lines into straight lines and quadrilaterals into quadrilaterals. In particular, the total amount of light (component by component) given off by the points inside each quadrilateral should be the same. See also Figures 2.14 and 2.16.

E x e r c i s e 2.3.21 Let v M(X) be a Borel measure and let f : X → X be continuous. Show that

( f ◦ v)(X) = v(X)

and hence that f : P(X) → P(X).

E x e r c i s e 2.3.22 In our discussion of transformations of pictures we restricted our attention to one-to-one transformations but we do not do so in the case of transformations of measures. Why is this?

This measure is invariant under the transformation f : [0, 1] → [0, 1] defined by f (x) = 4x(1 − x).

In Figure 2.17 we show two pictures of (vectors of Borel) measures that are invariant under transformations that map R2 into itself. The left-hand panel is a picture of a measure that is invariant under any rotation Rθ : R2 → R2, where θ is a multiple of 36◦, about the origin, which corresponds to the centre of the picture. The right-hand panel is a picture of a measure that is invariant under any Mobius¨ rotation

and a = (−0.25, 0.15) C is the centre of the rotation. The transformation Ma maps the circle of radius 1 centred at the origin into itself, while mapping the point a to the origin. Such transformations are discussed in Section 2.6.

An example that illustrates a closely related picture and measure, both of which are invariant under the Mobius¨ rotation in Equation (2.3.4) , is illustrated in Figure 2.18. Clearly the same transformation may possess many different invariant pictures and invariant measures. Similarly Figures 2.19 and 2.20 contrast (parts of) pictures and measures that are invariant under the same transformation as that illustrated in Figure 2.6.

E x e r c i s e 2.3.26 Show that if f, g : X → X are both continuous and the mea-

sure μ M(X) is invariant under f then the measure g(μ) is invariant under g ◦ f ◦ g−1.

2.4Fixed points and fractals

D e f i n i t i o n 2.4.1 Let X be a space and let f : X → X be a transformation. Then a point a X such that

f (a) = a

is called a fixed point of the transformation f .

Let X be a metric space, or a topological space, so that H(X) is defined. Then an invariant set A H(X) of a transformation f : X → X is a fixed point of f : H(X) → H(X) because it obeys

f (A) = A.

An invariant measure μ P(X) for a transformation f : X → X is similarly a fixed point of f : P(X) → P(X) because

f (μ) = μ.

Also, an invariant picture P of a one-to-one transformation f : R2 → R2 is a fixed point of f : → . So in our search for an understanding of when sets, pictures and measures may be invariant under transformations it is natural to consider conditions relating to the existence of fixed points.

Contraction mapping theorem

D e f i n i t i o n 2.4.2 Let (X, d) be a metric space. A transformation f : X → X is said to be Lipschitz with Lipschitz constant l R iff

d( f (x), f (y)) ≤ l · d(x, y) for all x, y X.

A transformation f : X → X is called contractive iff it is Lipschitz with Lipschitz constant l [0, 1). A Lipschitz constant l [0, 1) is also called a contraction factor. A contractive transformation is also called a contraction mapping.

We may write Li pl (X) to denote the set of Lipschitz transformations F : X → X with Lipschitz constant l ≥ 0.

The following theorem, for all its formal elegance, is of great practical importance to us. We will use it over and over again to construct fractal sets, pictures, measures and superfractals.

T h e o r e m 2.4.3 (Contraction mapping theorem) Let X be a complete metric space. Let f : X → X be a contraction mapping with contraction factor l. Then f has a unique fixed point a X. Moreover, if x0 is any point in X and we have xn = f (xn−1) for n = 1, 2, 3, . . . then

and

lim xn = a.

n→∞

P r o o f The proof of this theorem is an enjoyable exercise. Start by showing that {xn }∞n=0 is a Cauchy sequence. Let a X be the limit of this sequence. Then use the continuity of f to yield a = f (a).

Figure 2.16 Projective transformations of a digital photograph, treated as a measure. Colour saturation effects can be seen here.

Figure 2.17 Two examples of pictures of measures that are invariant under transformations. The measure represented by the picture on the left is invariant by a rotation through 36◦ . The measure represented inside the disk, in the right-hand panel, is invariant under a Mobius¨ rotation, as in Equation (2.3.4). The picture fades where it expands and brightens where it contracts.

Equation (2.4.1) tells us an upper bound for the distance from x0 to the fixed point a that involves only d(x0, f (x0)) and l. We will use this bound in Chapters 4 and 5 to help construct fractal approximations to given sets, pictures and measures.

E x a m p l e 2.4.4 The transformation f : R → R defined by f (x) = 23 + 13 x is a contraction mapping in the euclidean metric with contractivity factor l = 13 . Let x0 = 0. Then

1

xn = 1 − 3n

and the fixed point is the limit of the sequence 0, 23 , 89 , 2627 , . . . , namely a = 1. In this case

d(x0, 1) = 1 ≤ d(x0, x1) = 1. 1 − l

E x a m p l e 2.4.5 Let f : {0,1} → {0,1} be defined by f (σ ) = 01σ . Then

1

d ( f (σ ), f (ω)) ≤ 23 d(σ, ω)

for all σ, ω {0,1}. Let us choose x0 = 0. Then xn = 010101 · · · 010 and it follows that a = 01 {0,1} is the unique fixed point.

There are many different examples and applications of the contraction mapping theorem, involving diverse transformations and spaces. But we are primarily interested in fixed points of transformations on spaces such as H(X), P(X) and .

Contractive transformations on (H(X), dH ) and the existence of fractal sets

In this subsection we show how contractive transformations on an underlying space X can be used as building blocks to construct contractive transformations on H(X). The fixed points of such contractive transformations on H(R2) are examples of the fractal sets that we shall explore in Chapter 4. For now it is important to understand the dependence on transformations on the underlying space, in order to help guide and motivate our investigation of Mobius¨ and projective transformations acting on R2 later in this chapter.

The next theorem tells us that the property of f : X → X of being contractive is inherited by f : H(X) → H(X).

T h e o r e m 2.4.6 Let f : X → X be a contractive transformation on the metric space (X, d) with contractivity factor l. Then f : H(X) → H(X) is a contractive transformation on the metric space (H(X), dH ) with contractivity factor l.

tractive transformation on (H(R), dH ) with contraction factor 3 . Its unique fixed

point is the nonempty compact set A = {1}. Also, if A0 H(R) and An = f (An−1) for n = 1, 2, 3, . . . then we must have limn→∞ An = A. For example, the sequence

of closed intervals 0, 12 , 23 , 56 , 89 , 1718 , . . . converges in the Hausdorff metric to {1}.

It is clear that if f : X → X is a contractive transformation on a complete metric space with unique fixed point a X then f : H(X) → H(X) is a contractive transformation on a complete metric space with unique fixed point A = {a}. It might appear that we have not gained much, with all our elaboration and inheritance. But actually we have achieved the start of a beautiful constructive theory for deterministic fractal sets, the first hint of which is provided by the following theorem. This theory, based on ideas in a visionary book, entitled Fractals: Form, Chance, and Dimension, by Benoit B. Mandelbrot, see [63], was first analyzed and presented in a general mathematical framework by John Hutchinson in [48]. See also [4] and [44].