Ohrimenko+ / Barnsley. Superfractals

Figure 5.31 Two sets of superfractal interpolation functions, each with equally spaced interpolation points. The vertical scaling factor in the top set is 0.45 while in the lower set it is 0.25. The colours red and green indicate the attractors of the two IFSs; black indicates a 1-variable IFS and blue indicates a 2-variable IFS.

functions corresponding to two different values of the Hausdorff, or fractal, dimension. Each set includes the attractors (red and green) of the two IFSs used to create the corresponding superfractal, together with a 1-variable graph (black) and also 2- variable graph (blue). In each set the members have the same vertical scaling factor dlm . For some recent developments in V -variable fractal interpolation, see [85].

5.15V -variable space-filling curves

Space-filling curves may be constructed with the aid of IFS theory; see for example [84], Chapter 9. These curves have many applications, including adaptive multigrid methods for the numerical computation of solutions of PDEs and the hierarchical watermarking of digital images. Here we note that interesting V -variable space-filling curves, and finite-resolution approximants to them, can be produced.

and

f12(x, y) = 23 y, 12 x , f22(x, y) = − 23 y + 23 , − 12 x + 1 , f32(x, y) = 13 x + 23 , −y + 1 .

See Figure 5.32. Neither F1 nor F2 is strictly contractive but each is contractive on

average, for any assignment of positive probabilities to the constituent functions.

−→

An initial image consisting of the line segment OC is chosen on both screens, and the random iteration algorithm is applied; typical images produced after five iterations are illustrated in Figure 5.33; an image produced after seven iterations is shown in Figure 5.34.

5.16 Fractal transformations between the elements of V -variable superfractals of ‘maybe-not-tops’

It is not true, in general, that the chaos game preserves local tops in the sense of Theorem 5.13.1. To convince yourself of this, randomly iterate a few steps of Equation (5.13.1) in a decently overlapping case with say M = 2, L1 = L2 = 2 and V = 2. You can readily construct examples in which ‘higher’ code values become buried under ‘lower’ ones.

This has the consequence that the analogues of the sequences of 1-variable orbital pictures described in Section 5.11 do not have a tidy structure, as illustrated

Figure 5.33 Low-order approximants to two 2-variable space-filling curves generated by the IFS illustrated in Figure 5.32. Both images represent elements of the same superfractal.

Figure 5.34 Finite-resolution approximation to a 2-variable space-filling curve generated by the superIFS of Figure 5.32.

Figure 5.35 Three successive fractal measures belonging to a 2-variable superfractal. The pixels in the support of each 2-variable measure are coloured either black or a shade of green. The intensity of the green of a pixel is a monotonic increasing function of the measure of the pixel.

P r o o f See [16].

Everything works analogously to the case of F(V ) but now the underlying space consists of measures instead of sets. The set attractor AP(V ) of F is a set of measures in P(X)V. The set of components of the elements of AP(V ) is a superfractal, which we may denote by AP(V ). It consists of V -variable fractal measures. The elements of AP(V ) are distributed on P(X)V according to the probability measure μP(V ) P(P(X )V ), which is the measure attractor of FP(V ). The measure μP(V ) defines a marginal probability distribution μP(V ), obtained by projecting it onto a single component, and this measure describes the asymptotic distribution of measures obtained, almost always, by following chaos-game orbits for FP(V ) and keeping only the first components.

In Figure 5.35 we show some examples of 2-variable fractal measures, rendered in shades of green according to pixel mass. This example corresponds to the same superIFS as that used in Figure 5.24. The probabilities of the functions in the IFSs are p11 = p12 = 0.74 and p21 = p22 = 0.26. The IFSs are assigned probabilities P1 = P2 = 0.5.

I think that by now you will have got the idea. There are many fascinating kinds of V -variable objects that may be defined and explored both mathematically and experimentally. If you do this in the context of either scientific or engineering applications, rich rewards may be obtained. This is new territory!

5.18Code trees and (general) V -variability

Let H( {1,2,...,N }). That is, is a nonempty compact subset of {1,2,...,N }. Then we define the code tree of to be the set {1,2,...,N } given by

= σ1σ2 · · · σk {1,2,...,N } : σ1σ2 · · · and k {0, 1, 2, . . . } .

We use k = 0 here to say that the empty string is an element of . If we know then we know because the latter is the set of accumulation points of , that is,

= ∩ {1,2,...,N }.

Thus, each element of H( {1,2,...,N }) can be represented by its code tree.

A code tree can be described precisely in botanical terms. To do this we treat the tree as being embedded in R2, much as we did in Section 2.8. See Figure 5.36. The bottom of the tree consists of a single point, which we call the level-1 node. This node is connected, by a finite set of upward reaching straight-line segments, to a set of level-2 nodes, each of which in turn is similarly connected to a finite set of level-3 nodes, and so on without end, as illustrated. We call the straightline segments limbs. Optionally we include an additional limb, which we call the trunk, below the level-1 node. The figure makes clear what we mean when we refer to ‘level-1 limbs’, or to ‘the set of level-k limbs attached to a particular level-k node’. The set corresponding to the code tree seen in Figure 5.36 contains only strings that commence with the numbers 1, 2 or 5. At the next level of precision, it contains only strings that commence with 12, 14, 24, 25, 51, 52 or 53.

Each limb except the trunk is labelled by one of the indices {1, 2, . . . , N } . The trunk is labelled by the symbol for the empty set. The labels of the level-k limbs attached to a level-k node are all distinct. To provide a unique representation we write the labels of the level-k limbs from a particular level-k node in increasing order, from left to right. The tree spreads upwards without limit. We call the part of the tree which is connected to a node and which lies above the node a level-k branch of the tree. Notice that each branch defines a code tree. The elements of the code space {1,2,...,N } corresponding to the tree are provided by all sequences of labels that may be obtained by starting at the base of and steadily climbing up the tree from one level to the next, from one node to another that is connected to it by a limb, and reading off the string of codes that is encountered.

Now we note the following. Let S : {1,2,...,N } → {1,2,...,N } be the shift trans-

The shift transformation maps any code tree into a finite union of distinct code trees.

Level-1 limbs

Level-1 node

Ø Level-0 limb = trunk

Figure 5.36 Any element of H( {1,2,...,N}) may be represented by a code tree such as this. The strings constituting points of may be discovered by climbing up the tree, along all possible connected paths. The sets of dotted lines on the left and on the right each enclose a branch of the code tree. It should be noted that any branch of a code tree is itself a code tree. The dark blue vertical object at top right represents the code tree of the point σ {1,2,,...,N} .

positive integer V such that Kk ≤ V for all k then is called a general V -variable subset of {1,2,...,N } and is called a general V -variable code tree.

Figure 5.37 illustrates a general 1-variable code tree.

For V = 1, 2, . . . let H(V )( {1,2,...,N }) denote the set of all general V -variable subsets of {1,2,...,N }. Then H(V )( {1,2,...,N }) H(H( {1,2,...,N })) and

H(1) {1,2,...,N } H(2) {1,2,...,N } · · · H {1,2,...,N } .

Is1

Ø

Figure 5.38 Illustration of a code tree corresponding to an element σ of Hsuper( {1,2,...,N} ). The purple arrow shows the ordering of the nodes of the tree. The example at top right corresponds to the case N = 5 and M = 2, so that M = σk = 1 or 2 and therefore there are only two possible labels, I1 and I2; here we have chosen I1 = 3 and I2 = 2.

hold this superIFS fixed for the rest of this section. We continue to use the notation introduced in Section 5.8.

We define the set Hsuper( {1,2,...,N }) to be the set of H( {1,2,...,N }) such that, given any level-k node of the code tree , there is an m {1, 2, . . . , M} such that the level-k limbs, which are connected to the node, are labelled from left to right Nm−1 + 1, Nm−1 + 2, . . . , Nm . Figure 5.38 illustrates the code tree of an element of Hsuper( {1,2,...,N }). In this figure each σk belongs to {1, 2, . . . , M} and thus plays the role of m above.

Notice that if Hsuper( {1,2,...,N }) then S( ) is a finite union of elements of

Hsuper( {1,2,...,N }).

D e f i n i t i o n 5.18.3 Let {X; F1, F2, . . . , FM } be a superIFS. Then, for V = 1, 2, . . . , the space

(V ) = Hsuper {1,2,...,N } ∩ H(V ) {1,2,...,N }

is called the set of V -variable subsets of {1,2,...,N }. A point A H(X) is said to be V -variable iff it can be written in the form

A = φFunderlying ( )

for some (V ). We denote the set of all V-variable points in H(X) by A(V ).

What do V -variable points in H(X) look like? How can we compute them? Actually you already know the answer. A(V ) is precisely A(V ). But by now we have some understanding of the nature of the associated code space. The transformation T illustrated in Figures 5.2 and 5.22 is just the manifestation of the shift transformation in Equation (5.18.1), acting on the underlying code trees.

T h e o r e m 5.18.4 For all V {1, 2, . . . } we have A(V ) = A(V ). That is, the set of all V -variable points in H(X) is the same as the set of all first components of the points belonging to the attractor of the hyperbolic IFS F(V ).

and slm : {1,2,...,N } → {1,2,...,N } is the contraction mapping slm (σ ) = (Nm−1 + l)σ

for all l = 1, 2, . . . , Lm and all m = 1, 2, . . . , M. Let the associated V -variable IFS be denoted by

S(V ) = H {1,2,...,N } V ; sa , Pa , a A ,

where A is the index set defined in Equation (5.12.1). Let (V ) denote the corresponding superfractal of V -variable fractal sets and ρ(V ) denote the corresponding invariant measure, namely the projection of the measure attractor ρ(V ) P(H( {1,2,...,N })) onto one component.