Ohrimenko+ / Barnsley. Superfractals

invariant under homeomorphisms. We also continue to describe properties of code spaces.

D e f i n i t i o n 1.10.1 Let X be a topological space. Then X is said to be perfect iff it is equal to the set of its accumulation points.

For example, the space Rn is perfect in the natural topology, and so is [0, 1] R; but [0, 1] {2} R is not perfect because 2 is not a limit of any sequence of points in [0, 1].

T h e o r e m 1.10.2 When |A| > 1, the code space A is perfect.

P r o o f The natural topology is implied. Let σ A be given. Then we can choose ω A in such a way that ωn = σn for n = 1, 2, . . . Now define a sequence

{αn A}∞n=1 by (αn )m (i.e. the mth component of αn ) = σm for m = 1, 2, . . . , n and (αn )m = ωm for m = n + 1, n + 2, . . . Then it is easy to see that αp = αq

E x e r c i s e 1.10.3 Show that if f : X → Y is a homeomorphism then X is perfect iff Y is perfect.

D e f i n i t i o n 1.10.4 Let X be a topological space. Then X is said to be connected iff the only two subsets of X that are both open and closed are X and . A subset S X is said to be connected iff the space S with the relative topology is connected. S is said to be disconnected iff it is not connected. S is said to be totally disconnected iff the only nonempty connected subsets of S are those that contain single points.

E x e r c i s e 1.10.5 Let X be a space. Show that (X, Tdiscrete) is totally disconnected.

E x e r c i s e 1.10.6 Show that ( A A, d ) is totally disconnected in the natural topology.

D e f i n i t i o n 1.10.7 Let X be a topological space. Let S X. Then S is said to be pathwise connected iff whenever x, y S there is a continuous mapping f : [0, 1] R → S such that x, y f ([0, 1]).

Each property, of being connected, disconnected, totally disconnected or pathwise connected, is invariant under homeomorphism. They are topological properties. For example, if f : X → Y is a homeomorphism between topological spaces and S X then S is a connected subset of X iff f (S) is a connected subset of Y. See Figure 1.22.

If X is pathwise connected then it is connected. But the converse is not true. For example, let g(x) = sin(x/(10 − x)), let Gg denote the graph of g : [0, 10)

Figure 1.22 Homeomorphisms preserve topological properties such as being connected, being the boundary of a subset, being the interior of a subset, being an accumulation point of a subset and so on. Here the action of a certain homeomorphism acting on an elliptical subspace of R2 is illustrated by showing how it acts upon various coloured subsets. See Chapter 2 for more precision on what it means for a transformation to act upon a picture.

R → R2 and let L := {(10, y) R2 : −1 ≤ y ≤ 1} be a line segment. Then the subset S = L Gg R2, illustrated in Figure 1.23, is connected but not pathwise connected (see [70], p. 141): there does not exist any curve, homeomorphic to [0, 1] R, that passes through both the points (0, 0) and (10, 0) S. Clearly, by following Gg one can find a curve that connects (0, 0) to a point lying arbitrarily close to (10, 0). But one cannot find a curve that ‘crosses the divide’.

In Figure 1.24 we have illustrated variants of the previous example. Part (i) of the figure shows the graph Gl of a piecewise linear function l : (0, 1] → [0, 1]. The set S = Gl L1, where L1 := {(0, y) R2 : 0 ≤ y ≤ 1}, is a connected but not pathwise connected subset of R2.

We note that S has the following property:

S = w1(S) w2(S),

where the transformations w1, w2 : R2 → R2 are given by

w1(x, y) = (0.7x, −y + 1),

(1.10.1)

w2(x, y) = (0.3x + 0.7, x).

We have taken the origin of coordinates to be at the lower left-hand corner of Figure 1.24(i) and the width of S to be one unit. The transformation w1 shrinks the

x-coordinates of S by a factor 0.7 and reflects the result in the line y = 0.5, so that w1(S) is all S minus the line segment L2 := {(x, 13 (10x − 7)) : 0.7 ≤ x ≤ 1.0}.

−0.8

−1

Figure 1.23 The graph of sin(x /(10 − x )) for 0 ≤ x < 10 and for 0 ≤ 9.9 < 10. This graph, completed with a line segment parallel to the y -axis at x = 10, makes a connected set that is not pathwise connected.

The transformation w2 maps S onto L2. (In IFS theory it is known that the attractor of an IFS of two strictly contractive maps on R2 is either connected or totally disconnected; see Chapter 4). In the present case one map is not strictly contractive, and the attractor is neither connected nor totally disconnected.

In Figure 1.24(ii) a further variant of the connected-but-not-pathwise-connected type is illustrated. This time the figure is made of four transformations of itself: can you spot the transformations? Now the set is quite a bit more broken up; it is not pathwise connected at a countable infinity of places.

In Figure 1.24(iii) we zoom to a comb-shaped part of the curve. See also Figure 1.25.

D e f i n i t i o n 1.10.8 Let S be a subset of a topological space X. Then the boundary of S is the set of points in X such that every neighbourhood of x contains a point in S and one in X\S.

The boundary of the open disk {(x, y) R2 : x2 + y2 < 1} as a subset of R2 is the circle of radius 1, centred at the origin. The boundary of the set R\{x = 0} as a subset of R is the point x = 0, and as a subset of R2 it is R. The boundary of a closed set is always contained in the set.

The boundary of an open set is the empty set. If f : X → Y is a homeomorphism then ∂ S is the boundary of a set S X iff f (∂ S) is the boundary of f (S) Y. That is, the concept of a boundary is a topological one. Note that a topological space has an empty boundary.

E x e r c i s e 1.10.9 Show that the boundary of A in ( A A, d ) is A.

D e f i n i t i o n 1.10.10 Let S be a subset of a topological space X. Then the interior of S is the set of points each of which belongs to an open set contained in S.

The interior of the closed interval [0, 1] R is the open interval (0, 1). The interior of the open ball

B(x0, ) := {d(x0, x) < : x X}

in the metric space (X, d) is the open ball itself, where > 0. The interior of B(x0, ) is also B(x0, ). The interior of an open set is the set itself. If f : X → Y is a homeomorphism then S◦, say, is the interior of a set S X iff f (S◦) is the interior of f (S). That is, the concept of the interior of a set is a topological one.

E x e r c i s e 1.10.11 Show that in ( A A, d ) the interior of A is A and the interior of A is empty.

1.11Compact sets and spaces

Over time, some mathematical concepts become clearly established as being of key importance. They are concepts that can be expressed concisely, occur often and are powerful ingredients of theorems. Continuity is such a concept, and so is the topological property of compactness, which we shall introduce in the present section.

Many fractal objects with which we will deal are compact, and indeed owe their very existence to the compactness of the spaces in which we seek them. So here we are anxious not only to define compactness but also to provide ways of knowing when a set is compact. Therefore we need to mention sequential compactness, closedness and boundedness in Rn , total boundedness in metric spaces and the compactness of code spaces.

Let X be a space. A sequence {yn }∞n=1 X is called a subsequence of the sequence {xn }∞n=1 X iff there is an increasing sequence of positive integers

{nk }∞k=1 R such that xnk = yk for all k = 1, 2, . . . We may write {xnk }∞k=1 to denote this subsequence.

Let X be a topological space. A collection of sets {Oi X : i I} is called a cover for or covering of S X iff every point in S lies in at least one of the Oi .

That is, S {Oi X : i I}. The collection of sets {Oi X : i I} is called an open covering of S iff it is a cover for S and each of the sets Oi is open. A cover is called a finite covering iff it consists of finitely many sets.

D e f i n i t i o n 1.11.1 A topological space X is compact iff every open cover of X contains a finite cover of X. X is said to be sequentially compact iff every infinite sequence {xn }∞n=1 X contains a subsequence {xnm }∞m=1 that converges to a point x X. A subset S X is said to be (sequentially) compact iff it is (sequentially) compact in the relative topology.

The property of being (sequentially) compact is invariant under homeomorphism and so is indeed a topological property. A simple source of compact sets is provided by the closed subsets of compact spaces.

T h e o r e m 1.11.2 Let X be a compact space, and let S X be closed. Then S is compact.

Sequential compactness and compactness are equivalent in the case of metric spaces.

T h e o r e m 1.11.3 When X is a metric space, a subset S X is compact iff

A rich source of compact sets is the set of closed bounded subsets of Rn , as the following theorem attests.

T h e o r e m 1.11.4 Let X be a subspace of Rn with the natural topology. Then the following three properties are equivalent.

(i)X is compact.

(ii)X is closed and bounded.

(iii)Each infinite subset of X has at least one accumulation point in X.

One of the main ways of establishing that a metric space is compact involves the following concept.

D e f i n i t i o n 1.11.5 A metric space (X, d) is said to be totally bounded iff, for every given > 0, there is a finite set of points {x1, x2, . . . , xL } such that

%

X = {B(xl , ) : l = 1, 2, . . . , L}.

We have given no proofs of compactness results so far. But the proof of the following key theorem gives a good idea of how such proofs are constructed.

T h e o r e m 1.11.6 Let (X, d) be a complete metric space. Then X is compact iff it is totally bounded.

P r o o f Suppose that X is totally bounded. Then, for some finite integer L,

X = {B(yl , 1) : l = 1, 2, . . . , L}

for some points yl X, for l = 1, 2, . . . , L. Let {xn }∞n=1 X be an infinite sequence of points. Since there are infinitely many points in {xn }∞n=1, one of the

B(yl , 1) must contain an infinite subsequence, which we denote by {xn1,k }∞k=1. Let us call this ball X1. Then X1 is totally bounded because X is. So we can re-

n2,k with no loss of generality. We continue in this manner to obtain a decreasing sequence,

X1 X2 X3 · · ·

where Xn is a ball of radius 1/2n−1. We also obtain the sequence of points {xnm,1 Xm : m = 1, 2, 3, . . . }, where n1,k < n2,k < n3,k < · · · , which is a subsequence of {xn }∞n=1. Since the diameter of the Xm tends to zero as m tends to infinity it follows that {xnm,1 }∞m=1 is a Cauchy sequence and, since X is complete, converges to a point x X. So X is compact.

Conversely, suppose that X is compact but not totally bounded. Then for some> 0 we can find an infinite sequence of points {yl }l∞=1 such that d(yl , ym ) > whenever l =m. But since X is assumed to be compact, it must possess a

One way in which fractals are constructed is by means of decreasing sequences of subsets. In Section 1.4 we claimed that the decreasing sequence of real closed intervals in Equation (1.4.3) converges to a point x R. Here is the justification of that claim.

T h e o r e m 1.11.7 Let (X, d) be a complete metric space and let {Cn X}∞n=1 be a decreasing sequence of nonempty compact sets, that is

P r o o f C is compact, because if we have any open cover of C we can extend it to an open cover of C1 by adding to it the open set X\C. Then this open cover contains a finite subcover of C1. This subcover also covers C. If this subcover

contains X\C then we remove X\C from the subcover. The resulting set of open sets continues to cover C and is now a finite subset of the original open covering of C. So we have found a finite subcover of the original open cover of C.

To show that C is nonempty, make an infinite sequence {xn Cn }∞n=1 by choosing one point from each of the Cn . This sequence must contain a convergent subsequence, and it is easy to show that the limit must belong to each of the Cn and hence must belong to C.

Of great importance to us is the fact that code space is compact.

T h e o r e m 1.11.8 The code space = A A is compact.

P r o o f The natural topology is implied, and it suffices to work with the metric d . We already know from Theorem 1.7.5 that is complete, so we merely need to prove that is totally bounded. It suffices to prove that A is totally

m}, where we note that each cylinder has diameter less than /2. Thus, A can be covered by 2m balls each of radius , each centred on a point in a different cylinder set.

1.12The Hausdorff metric

In this section we develop and explore a wonderful metric, the Hausdorff metric. It measures the distances between nonempty compact subsets of a metric space. Later we will use the Hausdorff metric to describe the convergence of sequences of approximations to fractals.

In order to help form our intuition about how the Hausdorff metric works, we will explain it in several stages and explore some examples in detail. We also mention connections between optical processes and the Hausdorff metric. These connections lead us to speculate that in the future the metric may be computed by optical means.

This section also illustrates how we can build a new space of mathematical objects out of an underlying space. In the present case the underlying space is a complete metric space. The mathematical objects are the compact nonempty subsets of this space. A metric on the new space is derived from that on the underlying space. What properties of the new metric space are inherited from the original metric space?

The distance from a point to a set

To define the Hausdorff metric, first we need the concept of the distance from a point to a set.

T h e o r e m 1.12.1 Let (X, d) be a complete metric space and let H(X) denote the space of nonempty compact subsets of X. Let x X and let B H(X). Then there exists at least one point &b = &b(x) B such that

is continuous and B is compact. Hence there exists at least one point in B where

Theorem 1.12.1 enables us to make the following definition.

D e f i n i t i o n 1.12.2 Let (X, d) be a complete metric space. Let H(X) denote the space of nonempty compact subsets of X. Then the distance from a point x X to B H(X) is defined by

DB (x) := min{d(x, b) : b B}.

We refer to DB (x) as the shortest-distance function of the set B.

E x e r c i s e 1.12.3 Let X = = {(x, y) R2 : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}. Let dmax ((x1, y1), (x2, y2)) = max{|x1 − x2| , |y1 − y2|}. Let B = {(x, y) : x2 + y2 = 0.25}. Calculate DB ((0.6, 0.8)).

E x e r c i s e 1.12.4 Show that

DB (x) ≤ d(x, y) + DB (y) for all x, y X. Use this to show that DB (x) is a continuous function of x X.

E x e r c i s e 1.12.5 Prove that if C, D H(X) with C D then DC (x) ≥ DD (x) for all x H(X).

For given d ≥ 0 we call the set of points

Ld := {x X : DB (x) = d}

a level set of DB (x). All points on Ld are at the same distance d from B. In R2 these level sets {Ld : d ≥ 0} may form a graceful family of curves, like patterns of ripples, shaped like B, produced by simultaneous disturbances on a water surface or like the wavefronts of light at successive equally spaced time intervals after tiny coherent light pulses are emitted by the points of B at an initial time.

We can imagine optical devices, based on the latter idea, that generate approximate level sets of DB (x) when B R2 . For example, schematically, we can imagine a collection of light-emitting diodes organized in two dimensions to form a discrete model for B. We suppose that these diodes are turned on and off rapidly,

Figure 1.26 The space around a fern image is painted using different colours for different level curves of the shortest-distance function DF (x , y ). These level curves do not possess well-defined tangents at all their points. Also, the perturbation P0, a small purple disk, makes no difference to the shortest-distance function close to the fern but modifies it further away.

while an array of ultra-fast and sensitive charge-coupled devices, something like the CCD chip in a digital camera, in the same plane as the diodes is used to ‘photograph’ the wavefront at different times.

In Figure 1.26 we show for comparison DF (x) and DF P0 (x), where F is a fernlike subset of R2 and P0 R2 is a small disk. From left to right: the subset F R2 and a small disk P0; some level curves of DF (x); some level curves of DF P0 (x) (the outermost contour, red, contains points equidistant from F and P0); the same as the preceding image but more contours are shown. We see that DF (x) = DF P0 (x) whenever DF (x) is sufficiently small but that P0 provides a serious perturbation to the shortest-distance function at points sufficiently far away from F, in some directions. In Figure 1.27 we show a close-up of the level sets of DF (x) in the vicinity of the subset F R2. It is fascinating to imagine these lovely patterns at higher resolutions. In Figure 1.28 we show an artificial artistic work. It was made using the shortest-distance function associated with the euclidean metric. Four objects were drawn and coloured, then level sets of the shortest-distance function for the coloured points were computed and rendered.

Paths of steepest descent

In this subsection we continue to discuss shortest-distance functions.

Let B H(R2). At those points (x, y) R2 where the shortest-distance function DB (x, y) is differentiable,

is a vector pointing along the path of steepest descent from x to the nearest point on B. When the underlying metric is the euclidean metric, and the level sets of DB (x, y) are differentiable curves, this vector is oriented perpendicular to the level set through the point (x, y). In this case, paths of steepest descent for