Ohrimenko+ / Barnsley. Superfractals

Figure 5.5 This illustrates successive pairs of images on the two output screens after a certain number L > 20 of iterations. The red image is a close-up of another image in the stationary state distribution. Such pictures are typical of the ‘stationary state’ at the printed resolution.

(iv)Repeat step (iii) many times, to allow the system to settle into its ‘stationary state’.

In Figure 5.4 we observe the start of the sequence of pairs of images obtained in a particular trial, for the first seven iterations. Notice that some pairs are the same! Then in Figure 5.5 we show three successive pairs of computed screens, obtained after more than twenty iterations. These latter images are typical of those obtained after twenty or more iterations. Notice how the two images in the second pair of panels in Figure 5.5 consist of the union of smaller affine copies of the images in the top pair of panels. We observe that these images are very diverse but that they always appear to represent continuous ‘random’ paths in R2 tethered at A and C; they correspond to the stationary state, at the resolution of the images. More precisely, with probability 1, the empirically obtained distribution of such images over a long experimental run corresponds to the stationary state distribution.

In order to illustrate the intricate structure of the observed curves, which in general possess both disordered and ordered aspects, Figure 5.5 includes a closeup in red of one such curve. We also observe that the images produced in the stationary state are independent of the starting images. For example, if the initial images in the example had been dots or lines instead of fish, and the same sequence of random choices had been made, then the images corresponding to those in Figure 5.5 would have been the same at the printed resolution.

Figure 5.6 illustrates the attractors of the IFSs F1 and F2.

Figure 5.6 The attractors of the two IFSs used in the computational experiment are illustrated in red and green. The black curve is a 1-variable fractal set while the colours lavender and yellow indicate two 2-variable fractal sets associated with the same superIFS.

5.3 SuperIFSs and superfractals

Our computational experiment suggests that we begin our mathematical treatment of superfractals by defining a compact metric space X together with a collection of hyperbolic IFSs {Fm : m = 1, 2, . . . , M} with probabilities, where

and M ≥ 1 is an integer, to be a superIFS. We denote it by

{X; F1, F2, . . . , FM } or {X; F1, F2, . . . , FM ; P1, P2, . . . , PM }, (5.3.1)

where the Pm are probabilities, with

M

Pm = 1, Pm ≥ 0 for all m {1, 2, . . . , M}.

m=1

Notice that a superIFS is not an IFS: each IFS Fm defines a transformation on H(X), C(X) and P(X), see Equations (4.3.1), (4.8.2) and (4.3.2), but we do not specify how it might map X into itself.

We will use a given superIFS to define IFSs acting on higher-order spaces such as H(X), P(X) and

H(X)V = H(X) × H(X) × · · · H(X), where V {1, 2, . . . }.

The metric space (H(X), dH(X)) consists of the nonempty compact subsets of X, with the Hausdorff metric. The metric space (P(X), dP(X)) consists of the normalized Borel measures on X, with the Monge–Kantorovitch metric. The metric of each higher-order space is deduced from the metric of the space from which it is built, in an obvious way, as discussed in Chapter 1.

The IFSs on these new higher-order spaces arise in a very natural manner. The attractors of these IFSs provide the sets that we call superfractals. A superfractal is thus a collection of fractal objects, such as fractal sets, generalized orbital pictures, relative tops or measures. The structure of the higher-order space provides the variability V of the superfractal, ties its elements together and constrains the underlying code space, as we will explain. Superfractals have useful properties, not the least of which is that they may be sampled by means of the chaos game in various different settings. We describe the objects by adjectives such as V - variable, 2-variable and 1-variable. A certain superfractal may consist of 2-variable projective fractal sets while another might consist of 1-variable orbital pictures.

5.41-variable IFSs

The superIFS in Equation (5.3.1) may be used to define the hyperbolic IFS

How do we know that F(1) is indeed a hyperbolic IFS? Firstly, we know that (H(X), dH ) is a compact metric space, by Exercise 1.13.3. Secondly, here each of the IFSs Fm acts as a transformation

Fm : H(X) → H(X)

defined by

Lm

Fm (B) = flm (B) for m = 1, 2, . . . , M.

l=1

Each transformations is strictly contractive on H(X) with respect to the Hausdorff metric, by Theorem 2.4.8. That is,

dH (Fm (X ), Fm (Y )) ≤ λm dH (X, Y ) for all X, Y H(X),

for some λm [0, 1) and all m = 1, 2, . . . , M. For reasons that will become increasingly clear, we call F(1) a 1-variable IFS.

5.5The set attractor A(1) of the 1-variable IFS F(1)

The set attractor A(1) of the IFS F(1) must be an element of H(H(X)). That is, it must be a nonempty set, compact with respect to the metric dH(X), whose elements are themselves compact nonempty subsets of X. Furthermore, this set attractor must be given by

A(1) = φF(1) {1,2,...,M} ,

where

φF(1) : {1,2,...,M} → H(X)

is the code space mapping associated with F(1). Following the definition of φ in Theorem 3.3.12, we see that the elements of A(1) must be precisely the points Aσ H(X) that can be written in the form

Aσ = φF(1) (σ ) = lim Fσ1 ◦ Fσ2 ◦ · · · ◦ Fσk (X)

k→∞

for σ {1,2,...,M}. We call Aσ a 1-variable fractal set. The set

A(1) = Aσ : σ {1,2,...,M}

is an example of a 1-variable superfractal.

Next we look at two examples of 1-variable fractal sets. Setting σ = 1 we find

the attractor of the IFS F1. Supposing that N ≥ 3 and setting σ = 1312 we find that

By considering such examples you will see that A(1) contains the attractors of all the IFSs that can be constructed by composing finite sequences of the Fm . It also contains the images of these attractors under such finite compositions of the Fm . To be precise, let Fσ denote the hyperbolic IFS defined, in an obvious

Here F : H(X) → H(X) is taken to be the identity transformation. It is straightforward to prove that A(1) is the closure of Ar(1)ational in the metric dH(X). You should not find the latter too hard to envisage, after your experiences with H(X) at the

end of Chapter 1.

In order to see what 1-variable fractal sets look like, and to obtain a better understanding of the corresponding superfractal, we can use the chaos game.

5.6 Chaos game reveals 1-variable fractal sets

The 1-variable IFS F(1) is a hyperbolic IFS with probabilities. Hence it possesses a unique invariant probability measure μ(1) P(H(X)). Here P(H(X)) is the space of normalized Borel measures defined on H(X). The chaos game, adapted to the present setting, yields sequences of ‘points’ {Xk }∞k=1, Xk H(X), that almost always converge to the measure attractor μ(1) of F(1). The manner in which this convergence occurs is governed by Theorem 4.5.1. Here it is, transcribed to the present setting.

T h e o r e m 5.6.1 Let F(1) = {H(X); F1, F2, . . . , FM ; P1, P2, . . . , PM } be a 1-variable IFS and let μ(1) P(H(X)) denote its measure attractor. Specify a

starting set X1 H(X). Define a random orbit of the IFS to be {Xk }∞k=1, where Xk+1 = Fm (Xk ) with probability Pm independently of all other choices. Then for almost all random orbits {Xk }∞k=1 we have

for all B B(H(X)) such that μ(∂ B) = 0, where ∂ B denotes the boundary of B.

This tells us that, almost always, the random orbit {Xk }∞k=1 is asymptotically distributed according to μ(1). In practice, working to some level of approximation, say to within the accuracy specified by a parameter > 0, the sequence of sets {Xk }∞k=1 will be such that, after a readily estimated number of steps K , each of the sets X K +1, X K +2, . . . will lie to within the Hausdorff distance of an element of A(1).

Let us think more precisely how this occurs. Since A(1) is a point in the metric

space (H(H(X)), dH(H(X))) we can find a finite set of points A1, A2, . . . , AN( ) in H(X) such that every point of A(1) is contained in one of the balls B(An , ) of

radius , each of which is centred on one of the points An . What does such a ball B(An , ) look like? An is a fractal set belonging to the set attractor A(1). The ball B(An , ) is the set of all nonempty compact subsets of X that, when dilated by , contain An and moreover are such that each is contained in An dilated by .

Now suppose that the probabilities { Pm : m = 1, 2, . . . , M} are strictly positive. Then the support of μ(1) is A(1). Theorem 5.6.1 then, generally speaking, tells us that our random orbit {Xk }∞k=1 will visit B(An , ) at a proportion of its random steps that is equal asymptotically to μ(1)(B(An , )) > 0. Exactly how this occurs, of course, is stated exactly in Theorem 5.6.1.

So, in practical two-dimensional pictorial examples, we may choose > 0 to be sufficiently small that compact sets, when represented as images on the screen or on paper in our chosen experimental set-up, are visually indistinguishable when they are at a distance less than apart in the metric dH(R2). Then we may estimate a number K > 0 such that the elements of {Xk }∞k=K +1 are all indistinguishable, at viewing resolution, from elements of A(1).

If we discard sufficiently many initial iterates of the random orbit {Xk }∞k=1 then we should find that our chaos-game orbit produces, one after another, fractal sets that, to within viewing resolution, are all indistinguishable from elements of A(1), and our orbit will dance wildly about yielding at each step one of a finite but probably huge number of representatives of the actual elements of A(1) over and over again, with relative frequencies controlled by μ(1). And this is exactly what does happen! It has the same flavour as our computational experiment.

In Figure 5.7 we show some pictures of approximations to some of the 1- variable fractal sets produced by following just such a chaos-game orbit in a case for which M = 2. The attractor A1 of the IFS F1 is illustrated at the top right in Figure 5.23, from which you can deduce the form of the three affine transformations that comprise F1. The IFS F2 consists of four affine transformations on R2 and its attractor A2 is illustrated at the top left of Figure 5.23. You can see that the individual set attractors in Figure 5.7 look like semi-random mixtures of A1 and A2. The images in this figure were rendered by colour-stealing, with relative tops functions produced, quite remarkably, using the chaos game, as we will explain in Section 5.9.

In Figure 5.8 we show examples of 1-variable fractal interpolation functions, again produced using the chaos game. Here M = 2 and we have used two IFSs, F1 and F2. The attractor of each is the graph of a standard fractal interpolation function; see Section 5.14. Both graphs have the same endpoints. It is easy to show that the elements of the corresponding 1-variable superfractal are the graphs of continuous functions that connect the same pair of endpoints.

Another example is illustrated in Figure 5.9. Here M = 2 and Fm = αm ◦ F ◦ αm−1, where αm , for m = 1 and 2, is an affine rescaling in the x-direction and F is the IFS given in Table 0.1 in the Introduction. Here too the successive 2-variable fractal sets have been rendered by colour-stealing, to demonstrate how successive random iterates of approximate tops functions converge towards accurate tops functions; see Section 5.9. What you should notice, at this juncture, is how the silhouettes of the pictures, which represent successive 1-variable fractal

Figure 5.7 Some 1-variable fractals belonging to the same superfractal, computed by the chaos game. The sets are rendered by a variant of IFS colouring. Look closely, and do not hurry by. Note the differences.

sets produced by the chaos game, change from one step to the next. In this manner, over time we can obtain a visual sample of 1-variable fractal sets belonging to the corresponding superfractal, distributed according to the probability measure μ(1).

5.7 Hausdorff dimension of some 1-variable fractal sets

D e f i n i t i o n 5.7.1 The superIFS {X; F1, F2, . . . , FM ; P1, P2, . . . , PM } is said to obey the uniform open set condition if there exists a nonempty open set O X such that

Fm (O) O

Figure 5.8 A few of the uncountably many 1-variable fractal interpolation functions belonging to a single superfractal. The graphs at the top and bottom are the attractors of the two IFSs that comprise the superIFS. Such graphs are produced in rapid succession by the chaos game. Can V -variable fractal interpolants be used to improve simulations of stockmarket portfolio performance, as proposed by Mandelbrot [65]?

5.8 The underlying IFS of a superIFS

In order to discuss sets of 1-variable tops and other objects associated with F(1), as well as to prepare the way for V -variable fractals, we introduce some notation related to the IFS

which we call the underlying IFS. We will let Aunderlying denote its attractor. We also write

Funderlying = X; f1, f2, . . . , fN1 , fN1+1, fN1+2, . . . ,

fN2 , . . . , fNM−1+1, fNM−1+2, . . . , fN ,

where N = L1 + L2 + · · · + L M ,

N0 = 0, N1 = L1, N2 = L1 + L2, . . . , NM = L1 + L2 + · · · + L M

Figure 5.9 A few elements of a random sequence of 1-variable tops, {τk }∞k=1, rendered by colour-stealing. Observe the differences in the silhouettes.

and

fNm−1+l = flm for l = 1, 2, . . . , Lm and m = 1, 2, . . . , M. We also write

Im = {Nm−1 + 1, Nm−1 + 2, . . . , Nm } for m = 1, 2, . . . , M, (5.8.1)

to denote the members of the obvious partition of the set {1, 2, . . . , N }.

All the V -variable fractal objects discussed in this chapter are built on or in some way connected to subsets of the attractor Aunderlying of the underlying IFS. You will know that you are dealing with the underlying IFS by the occurrence of the symbol N , which we reserve in this chapter for the number of functions fn in the underlying IFS, whereas the symbol M is used to define the number of IFSs Fm in the superIFS. In particular, we will define relative tops functions of 1-variable fractal sets and establish homeomorphisms between them with the aid of the underlying IFS. We do this next.

5.9Tops of 1-variable fractal sets

It follows from the discussion in Section 5.5 that the set attractor A(1) of the 1-variable IFS F(1) can be written as

A(1) = Aσ : σ {1,2,...,M} ,