Ohrimenko+ / Barnsley. Superfractals

completeness, defined in Section 1.7, and compactness, defined in Section 1.11, are needed to establish the existence of fractal objects. We describe the conditions under which these properties occur.

Over and above the themes of code spaces, properties of spaces that are preserved under transformations and the nature of euclidean space, a central focus of this chapter, which will carry on throughout the book, is the idea that the points in a space may themselves be mathematical objects. For example, they may be mathematical pictures, or measures, defined in Chapter 2. Or they may simply be the nonempty compact subsets of another underlying space.

Thus, the points of a space HX may be constructed using sets of the points of an underlying space X . Organizational principles such as addresses, metrics and topologies may be inherited from X and provide structure to HX . Properties of the underlying space X such as compactness and completeness may also be inherited by the space HX . Moreover, transformations acting on X may be used to define transformations on HX . These inheritances are important because they enable us to establish the existence of diverse types of fractal in later chapters.

For example, in Section 1.13 we show that the property of being a compact metric space may be inherited from X by a certain space H(X ). The inherited metric, the Hausdorff metric, is discussed earlier, in Section 1.12, with a view to developing our intuition about how it works. This remarkable inheritance continues from generation to generation, from X to H(X ) to H(H(X )) and so on. It enables us to establish the existence of superfractal sets in Chapter 5.

1.2Points and spaces

In this section we introduce the notation and nomenclature for points, sets and spaces that we shall use throughout the book.

A space is a set. The elements of the set are called the points of the space. We use the notation X to denote a space. The expression x X means that x belongs to the set X or equivalently that x is a point of the space X. Similarly the expression x, y X means that both x and y are points of X. We say that two points x, y X are distinct if x =y, that is, x is not equal to y. When we consider several spaces at once, we may denote them by X, Y, . . . A space may be empty, that is, it may contain no points.

For illustration, some spaces are shown in Figure 1.2. An important example of a space is R, the set of all finite real numbers. A point x R is simply any number, positive or negative, that can be expressed by a decimal expansion, either finite as in x = 1.5 or unending as in x = −7.93121059912791101 · · · . We can write

R = {x : −∞ < x < +∞}.

Figure 1.2 Shown here are illustrations of spaces: (i) a cube in R3; (ii) a fractal subset of R2; (iii) a line segment; (iv) a subset of R2 that looks like a leaf; (v) the space of subsets of a set; (vi) a code space.

We use the notation {elements : conditions} to mean a set of elements, or objects, on the left of the colon, that obey the conditions on the right of the colon. We may think of the points of R as being organized to lie on a straight line, the x-axis in coordinate geometry; see Section 1.4.

We denote the four intervals defined by a, b R, with a < b, by [a, b] = {x

R : a ≤ x ≤ b}, [a, b) = {x R : a ≤ x < b}, (a, b] = {x R : a < x ≤ b} and (a, b) = {x R : a < x < b}. Each interval is an example of a space.

An important space is the euclidean plane, which we denote by R2. It should be familiar to you from calculus and geometry. It is sometimes called the xy-plane. It is the place where straight lines and circles exist and where one imagines graphs of functions like y = x2 + 1. Each point in the euclidean plane can be represented by a pair of coordinates (x, y), x and y being finite real numbers. We can write

R2 = {(x, y) : −∞ < x < +∞, −∞ < y < +∞}.

If X and Y are spaces then X × Y denotes the space of ordered pairs of points,

which are denoted by either x × y or (x, y), where x X and y Y. We write X1 = X, X2 = X × X and Xn+1 = Xn ×X, for n {1, 2, . . . }. So for example we

have the space R2 = R × R. Note that we can write

We use any of the notations x, (x1, x2), (x, y) to denote a point in R2 and the obvious extension of this notation in R3, R4, . . .

E x e r c i s e 1.2.1 Express R3 in a similar way to R2 in Equation (1.2.1).

The spaces R, R2, R3, . . . occur throughout the mathematical sciences and serve numerous purposes, many related ultimately to models of reality. With R we model distance, time, mass, temperature and other scalar physical quantities. Using R2 we model observations of flat things, patterns for making clothes, pictures, maps, photographs and so on. And R3 is the oldest model for the physical space about us, in which we live, design buildings and fly space missions. Also, the spaces R, R2, R3, . . . are the underlying mathematical fabric from which are constructed prime examples in topology, geometry, measure theory and many other areas; in them we formulate the basic equations of physics. They are incredibly rich in structure and properties.

Most fractals that we study in this book are either subsets of R, R2, R3, . . .

or else built upon them, and many properties of fractals are inherited from these spaces. We learn something new about these spaces, the fabric of which they are made, by studying fractals.

The spaces that interest us most are those that are in some way self-similar. In this book we describe the euclidean plane as R2. But we may consider this space unadorned by coordinates, so that we have a blank space, like an endless, perfectly flat, homogeneous sheet of paper. Then one part of the space is like any other and we have no way of knowing whether, for example, a circle inscribed on this plane is big or small, or even where it is. The space is just like itself everywhere and at all scales of observation.

An example of a space with a finite boundary is the unit square

:= {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}.

The symbol ‘:=’ means ‘is defined to be’. Imagine an empty picture that represents. Its homogeneous quality represents the uniformity of the euclidean plane before it is invaded by theories and marks, like a new beach after the tide has gone down on which no one has yet walked. One mathematician looking at it might imagine open sets, topology and connected paths; another, lines, triangles and intersections; and yet another, myriads of points of some algebraic variety. But let us, just for a moment, imagine nothing.

Let S(X) denote the set of all subsets of the space X; then S(X) is also a space. In S(X) both the empty set , the subset of X that contains no points of X, and X itself are single points!

Some spaces that we shall consider, such as sets of points or sets of circles in the euclidean plane, have an explicit geometrical character while others, such as S(X), are more abstract. But we will try to think geometrically about spaces, for

Figure 1.3 A green image in a white surround may be thought of as an approximate description of a set of points in the space . All points that are not white belong to the set. A set of points in represents a single point in S( ), the space whose points are the subsets of .

example by assigning distance functions or imagining pictorial representations; we could think of the space S( ) as the set of all green drawings on . In this way of thinking, green of varying strength (the strongest green is the lightest in appearance) replaces white and each green dot in such a drawing represents a point in , that is, an element of a set in S( ). A blank white image, where no drawing has occurred, represents the empty set, and an entirely green image represents the point S( ). A green line from the lower left corner to the upper right corner ofrepresents the point {(x, y) : x = y} S( ); and an image such as Figure 1.3 serves as an approximate description of a single point in S( ).

1.3 Functions, mappings and transformations

In this section we introduce notation and definitions related to functions. We use the notation

f : X → Y

to denote a function f that acts on the space X to produce values in the space Y;

fassigns to each point x X a unique point f (x) in Y. The graph of f is defined by

G f := {(x, f (x)) X × Y : x X}.

Figure 1.4 The space X × Y, where both X R and Y R are unions of intervals. Also shown is the graph Gf of a function f : X → Y. This picture is a reminder that X × Y may be much ‘bigger’ than X or Y and that complicated domains and ranges may occur and yield fragmented graphs.

To know f is equivalent to knowing G f . That is, to specify a function f : X → Y is equivalent to specifying a certain type of subset of X × Y, one whose ‘shadow’ or ‘projection’ on X is all X and such that for each x X there is a unique ‘height’ value in Y. In Figure 1.4 we illustrate the graph of such a function in the case where both the domain and the range of the function are disconnected subsets of R.

We also call f : X → Y a transformation from the space X to the space Y or a mapping from the space X to the space Y. We define the domain D f of the function f to be the set of points upon which it acts. If f : X → Y then D f = X. The range of f is defined by

R f := {y Y : f (x) = y for some x X} =: f (X).

Let S X denote a subset of X. Note that S might be the empty set or it might be X itself. Let f : S → Y. Then D f = S. In such cases we might use the notation f : S X → Y and when we also wish to refer explicitly to the range of f we will sometimes write f : S X → R f Y.

Generally, we will extend the definition of a function f : X → Y to encompass a function f : S(X) → S(Y) defined by

f (S) = { f (x) : x S},

for S S(X), where S(X) is the space of all subsets of X. We intend that it should be clear from the context whether we mean a point-valued or set-valued function.

We say that a function f is one-to-one if and only if (iff) for each y R f there

is a unique point x D f such that f (x) = y. In this case the inverse function f −1 : R f Y → D f X is defined by f −1(y) = x.

When f : X → R f Y is one-to-one we will sometimes call f an embedding function. Then we may use the points of f (X) = R f to represent the points of X. We think of X as being embedded in the space Y, where it is represented by the set f (X).

We say that f : D f X → Y is onto when R f = Y. Even when f : X → Y is neither one-to-one nor onto, we define the set-valued inverse function

f −1 : S(Y) → S(X)

by

f −1(S) = {x X : f (x) S},

for all S S(X). We will sometimes write f −1(x) in place of f −1({x}) when {x} is a singleton set, that is, the set consisting of the single point x X. For us, such an inverse function always exists but its values may consist of a set of more than one point or the empty set.

this exercise? Define ξ : R → R by ξ (x) = (x, 0). Show that ξ is an embedding function.

Now we introduce the union symbol and the intersection symbol ∩. The expression X Y means the set that consists of all the points in X and all the points in Y:

X Y = {x : x X or x Y}.

Note that a point that belongs to both X and Y also belongs to X Y. Let I denote an index set, that is, a set of objects that we call indices. Let Si be a set for each

16 Codes, metrics and topologies

i I. Then we will use the notation

Si := {x : x Si for at least one i I}.

i I

Similarly we define the intersection of two sets by

X ∩ Y = {x : x X and x Y},

and we write

Si := {x : x Si for all i I}.

i I

When A, B X, the notation A\B means the set of points of X that are in A and not in B.

E x e r c i s e 1.3.2 Let f : X → Y, let S, T X and let V, W Y. Prove, and learn forever, that:

(i)f (S T ) = f (S) f (T );

(ii)f (S ∩ T ) f (S) ∩ f (T );

(iii)f −1(V W ) = f −1(V ) f −1(W );

(iv)f −1(V ∩ W ) = f −1(V ) ∩ f −1(W );

(v)f −1(Y\V ) = X\ f −1(V ).

Let f : X → Y and let S X. Then we can define a function f |S : S → Y by f |S (x) = f (x) for all x S. f |S is called the restriction of f to S. We will often denote f |S simply by f .

1.4Addresses and code spaces

In this section we describe how the points of a space may be organized by means of addresses. Addresses are themselves members of certain types of spaces that we call code spaces.

When a space consists of many points, as in the cases of R and R2, it is often convenient to have addresses for the points in the space. An address of a point is a means to identify the point, just as a postal address identifies a mailbox. It is in effect an algorithm or formula for locating the point precisely. It may be a string of numbers or symbols, either finite or infinite, that uniquely specifies the point, via some procedure that is implicitly understood and unstated. For example, the address of a point x R may be its decimal or binary expansion. Points in R2 may be addressed by ordered pairs of decimal expansions.

A single point may have more than one address; for example the same point in R has the two binary addresses 1.0 = 1.0000 · · · and 0.1 = 0.1111 · · · . Here an overbar means that the symbol or finite string of symbols is repeated endlessly,

We shall introduce some useful spaces of addresses, namely code spaces. These spaces will be needed later to represent sets of points on fractals. An address is made from an alphabet of symbols. An alphabet A consists of a nonempty finite set of symbols such as {1, 2, . . . , N }, {0, 1, . . . , N } or { A, B, . . . , Z } where each symbol is distinct. The number of symbols in the alphabet is |A|. For example,

|{0, 1, 2, . . . , N }| = N + 1.

Let A denote the set of all finite strings made of symbols from the alphabet A. The set A includes the empty string . That is, A consists of all expressions of the form

σ = σ1σ2 · · · σK ,

where σn A for all n {1, 2, . . . , K } with K a positive integer, as well as . We will write |σ | to denote the length of the string σ A.

Examples of points in { A,B,...,Z } are A, D O O R, A A A A A A, and Y OU . Examples of points in {1,2,3} are 1111111, 123, 1231111 and 3. A convenient

address for a point σ A is σ itself. An example of a point in {0,1}is σ = 1011010111 , which we refer to as a finite binary string. Notice that {0,1} is a convenient space for addressing the space of all computer files.

Throughout the book we will often refer to the spaces {1,2,...,N } and {1,2,...,N }. Make sure now that you really do know what these symbols signify.

Given two strings σ, ω A we will write σ ω to denote the concatenated string

σ ω := σ1σ2 · · · σ|σ |ω1ω2 · · · ω|ω|.

So for example if σ, ω {0,1} with σ = 000 and ω = 11 then σ ω = 00011 and ωσ = 11000. And if σ = 000 and ω = then ωσ = σ ω = 000.

An important space, which we denote by A, consists of all infinite strings of symbols from the alphabet A. That is, σ A if and only if it can be written

σ = σ1σ2 · · · σn · · ·

where σn A for all n {1, 2, . . . }. An example of a point in {0,1} is σ = 1011010111 · · · . A point in { A,B,C} is A.

is countable but, when |A| > 1, A is uncountable.

D e f i n i t i o n 1.4.1 Let ϕ : → X be a mapping from A A onto a space X. Then ϕ is called an address function for X, and points in are called

addresses. is called a code space. Any point σ such that ϕ(σ ) = x is called an address of x X. The set of all addresses of x X is ϕ−1({x}).

Figure 1.5 Points in the space X may be assigned addresses. The function f : → X that maps from code space , the space of addresses, to points in X is called an addressing function. Each point in X has at least one address.

Figure 1.5 illustrates the concepts in this definition.

E x e r c i s e 1.4.2 Define a code space and address function for each of the following spaces:

(i)X = [0, 1] = {x : 0 ≤ x ≤ 1};

(ii)X = {(x, y) R2 : x2 + y2 = 1} ∩ {(x, y) R2 : y > 0};

(iii)X = {(x, y) R2 : x2 + y2 ≤ 1};

(iv)X = Z+ = {1, 2, 3, . . . }, the set of positive integers;

(v)the set of real numbers that can be written in the form x = m/2n for some m {0, 1, . . . , 2n − 1} and n {0, 1, 2, . . . }. How many addresses does the point x = 0.25 have, according to your addressing scheme?

Addresses of points on a line

In this subsection we illustrate an addressing scheme for the points on a line segment in the euclidean plane. One goal is to demonstrate how coordinates may depend on geometrical properties of the space. But also we illustrate how real space may be broken up into smaller and smaller similar parts.

Let A and B denote a pair of distinct points in the euclidean plane. Let L[A, B] denote the set of points in the line segment that joins A and B. Then L[A, B] is a space.

Figure 1.6 A binary ruler makes finding an address of a point in the line segment L [ A , B ] easy! Or does it?

Each point x L[A, B] can be represented by an address in {0,1,...,9}, and each address in {0,1,...,9} defines a unique point in L[A, B]. A simple way to see this is to identify L[A, B] with the unit interval [0, 1] := {x R : 0 ≤ x ≤ 1}. Simply take A to be the origin of coordinates, let the x-axis pass through B and define B to be the point x = 1. Then an address of x for 0 ≤ x < 1 is the sequence of digits after the decimal point in a decimal expansion of x, and the point x = 1 is assigned the address 9 {0,1,...,9}. Alternatively, we may use an address function ϕ : {0,1} → L[A, B] defined by using the binary expansion of x.

These addressing schemes and others like them will be used often later on. So here we describe a bit more deeply the construction of the address function ϕ : {0,1} → L[A, B]. The description in the previous paragraph assumed that we already have a ruler or measuring stick, namely the unit interval addressed by real numbers; see Figure 1.6. But this ruler can be constructed using a straight-edge and compass, which reveals the geometrical origin of such addresses.

L := L[A, B] may be bisected, as illustrated in Figure 1.7, by constructing two circles, one centred at A and passing through B and the other centred at B and passing through A. Denote the two points of intersection of these circles by C and D. Then construct the line segment L[C, D], and let this meet L[A, B] at the point E. Then E is the bisection point of L[A, B]. The result is two intervals, which we denote by L0 and L1, with, say, L0 to the left of L1.

This latter assertion is of a geometrical kind – it derives from axiomatic properties of line segments. See for example [26], p. 22, the end of the first paragraph. We have

L = L0 L1 and E = L0 ∩ L1;

see Figure 1.8. Both L0 and L1 contain the midpoint of L . We next similarly bisect L0 to produce two intervals L00 and L01, where L00 lies to the left of