Ohrimenko+ / Barnsley. Superfractals

is a nested sequence of segments of P0 and so converges to a unique element of

is called the space of limiting pictures associated with the orbital picture of P0 under the IFS semigroup S{ f1, f2,..., fN }.

Let us define the closure Qσ of a panel Qσ Ppanels by

Qσ = fσ fσ−1(Qσ ) .

Then each point in ( P0 ) corresponds to a set of panels in Ppanels whose closures are homeomorphic. Indeed if σ , ω P0 with (σ ) = (ω) then

Qω = fω fσ−1(Qσ ) .

The following theorem tells us that corresponding to each periodic orbit of the

←− shift transformation acting on the space P0 there is at least one point in P0 .

T h e o r e m 3.5.25 Let (X, d) be a compact metric space. Let the domain of P0 = C(X) be compact. Let {Qσ : σ P0 } denote the set of panels of the orbital picture of P0 under the IFS semigroup S{X; f1, f2,..., fN }. Let ρ P0 be a periodic point for the shift transformation S : P0 → P0 of period k {1, 2, . . . }. That is,

Figure 3.46 The panels of an orbital picture, illustrated using various colours to distinguish them, together

address 2?

E x e r c i s e 3.5.26 Find the IFS used to generate the orbital picture whose panels are illustrated in Figure 3.46.

at the end of the proof of Theorem 3.5.25 can be obtained from the code space{1,X } {1,X } by replacing the symbol X , wherever it occurs, by the string 12. The symbol X has been used here, rather than the symbol 2, to help you to avoid confusions when making the replacements.

In Figure 3.47 we have illustrated some parts of some boundaries of segments belonging to the space of limiting pictures in the case of the orbital picture

Figure 3.47 The internal boundaries within this picture demarcate parts of boundaries of segments in the space of limiting pictures, in relation to the orbital picture in Figure 3.42.

illustrated in Figure 3.42. In Figure 3.48 we have illustrated some of the segments belonging to the space of limiting pictures corresponding to buttercup-field orbital pictures like those illustrated in Figure 3.32. Let us denote these limiting pictures by Qω (λ), where ω P0 is the address and λ {0.7, 0.8, 0.9} is a parameter that specifies the IFS,

{R2; f1(x, y) = (λx, λy + 1 − λ), f2(x, y) = (λx + 1 − λ, λy + 1 − λ)}.

Look at the top left and bottom right pictures in Figure 3.32. You will notice that the panels on the left-hand side and right-hand side of each picture, which look something like half buttercup-plants, seem to have converged after few iterations, so that

Q1111(λ) = Q111···1(λ) and Q2222(λ) = Q222···2(λ).

In Figure 3.48 the limiting pictures Q1212(λ) and Q2121(λ) become more fragmented, into torn-up fragments of yellow petals, as λ increases. We note from Figure 3.48 that the domain of a panel of an orbital picture may be disconnected even though the domain of P0 is connected.

Figures 3.43 and 3.44 provide further illustrations of the wide variety of pictures that we can expect to find in the space of limiting pictures. In this case, is the space of limiting pictures finite or infinite?

Figure 3.48 Elements of the space of limiting pictures associated with some buttercup fields. The parameter values are 0.7, 0.8 and 0.9 and the corresponding addresses are 0000, 0101, 1010 and 1111. The domain of which of these segments possesses the greatest number of connected components?

It seems clear that the size of the space of limiting pictures, |LP0 |, is an interesting parameter both mathematically and descriptively, as a means to capture the visual complexity of some orbital pictures. But when |LP0 | = ∞ we need a finer parameter, so we make the following definition.

D e f i n i t i o n 3.5.29 Let (X, d) be a compact metric space, let P(P0) be the orbital picture of P0 C(X) and let LP0 denote the associated space of limiting pictures. The diversity of the orbital picture is |LP0 | {1, 2, . . . } {∞}. When

((

(LP0 ( = ∞ the (exponential) rate of growth of diversity (in the orbital picture) is defined to be

have |LP0 | = 1 while the orbital pictures with entropies 0.15 and 0.32 clearly have |LP0 | > 1. Indeed, for the family of IFSs considered in connection with Figure 3.48 it appears that for some values of the parameter λ (0.5, 1) the value of |LP0 | is finite while for others, related to ‘β-numbers’, which have certain number-theoretic properties, |LP0 | is infinite; the growth rate of diversity may be the same as the growth rate of periodic cycles, namely the symbolic entropy, in these cases. See for example [17]. The growth rate of diversity seems to provide an independent measure of the visual complexity of some orbital pictures.

Code spaces of orbital pictures, tree-like or not tree-like

We digress briefly here to illustrate how the code space of an orbital picture may have the structure of a ‘pruned tree’ and how in other cases it may not be tree-like. This digression serves to increase our familiarity with orbital pictures.

In some cases the structure of P0 P0 is tree-like, in the sense that A A is tree-like, as seen in Figure 1.15. Consider the following examples, associated with the family of IFSs

where 0 < λ < 1. The fractal set, the attractor of this IFS, is the line segment A that connects the pair of points (0, 1) and (1, 1) in R2. We choose P0 to be a picture of a block, with domain

DP0 = (x, y) R2 : 0 ≤ x ≤ 0.97, 0 ≤ y ≤ 13 .

The resulting patterns of blocks, the orbital pictures, for λ = 0.6, 0.66, 0.7 and 0.8, are illustrated in Figure 3.49.

In Figure 3.50 we have labelled the visible blocks, the panels, by their addresses. Each ‘tree of gaps between blocks’ converges to the line segment A and provides a different coding or addressing system for the unit interval. These codings all have the following property: if σ P0 P0 then 1σ P0 P0 and, if also σ1 = 2, then 2σ P0 P0 ; it follows that in these cases the code spaces of the orbital pictures are ‘pruned trees’, the trees of gaps between the blocks.

In Figures 3.51 and 3.52 we show examples which are not tree-like. The IFS used in these figures belongs to the family of examples

f1(x, y) = (−λy, −λx + 1), f2(x, y) = (λx + 1, λy + 1 − λ) (3.5.18)

with λ = 0.6. It is quite easy to see that the code space includes the codes { , 1, 2, 11, 12, 22, 111, 112, 122, 211, 221, 222} but not the code 21 because

Figure 3.49 Examples of different sets of panels of orbital pictures, using the family of IFSs in Equation (3.5.17), for (i) λ = 0.6, (ii) λ = 0.66, (iii) λ = 0.7 and (iv) λ = 0.8. The domain of P0 is the rectangle at the bottom of each picture. In the limit, each different orbital picture is associated with a different addressing scheme for the points in the interval [0, 1]. See also Figure 3.50.

the corresponding picture in the orbit of P0 is hidden underneath P0. Hence the code space P0 is not tree-like in this case; see Figure 3.53.

Figure 3.52 illustrates the relationship between the orbital picture, the underneath picture and the attractor of the associated IFS.

E x e r c i s e 3.5.30 Write down the addresses of the larger panels in Figure 3.51. Identify some addresses in {1,2} that do not correspond to panels in this orbital picture. Show that the set of panel addresses P0 in this case is not tree-like.

Picture tilings and panellings

We now distinguish between picture tilings and panellings. The idea of a IFS semigroup picture tiling is the same as that of an IFS semigroup tiling: non-overlapping picture tiles are obtained by applying all the elements of the semigroup to the condensation picture. Illustrations of IFS semigroup picture tilings are provided by Figures 3.3, 3.25, 3.54 and 3.55. In Figure 3.54 the picture tiles are leafy annuli.

Figure 3.52 The code space structure is not tree-like for this example or the example illustrated in Figure 3.46. Here the IFS is that given in Equation (3.5.18). (i) The orbital picture for the condensation picture P0, which looks like a square tile with a leaf on it; (ii) the orbital picture and the set A P0 ‘peeking out from underneath’; (iii) the underneath picture; (iv) the underneath picture plus the attractor A of the IFS.

Figure 3.53 Points σ P0 with |σ | ≤ 3 associated with Figure 3.52 are here represented as some of the nodes on a tree-like structure, as defined in graph theory. The presence of the nodes with addresses 2 and 211 and the absence of the nodes corresponding to the address 21 means that P0 is not tree-like.

D e f i n i t i o n 3.5.31 Let S{ f1, f2,..., fN }(R2) be an IFS semigroup and let P0 be a picture with domain DP0 R2. Let the orbit O(DP0 ) be a semigroup tiling

of the set O(DP0 ). Then the orbit O(P0) of P0 is called a semigroup tiling of

the picture O(P0) or a picture tiling. Each picture fσ (P0), for σ {1,2,...N }, is called a semigroup picture tile and σ is called the address of the picture tile

fσ (P0). We say that the semigroup, acting on the picture P0, generates the picture tiling O(P0).

Figure 3.54 Here an orbital picture is tiled by leafy annuli. Notice how this picture also looks like an underneath picture. Underneath pictures can be used to help find tilings!

In Figure 3.56 we show three examples of IFS semigroup picture tilings. These are especially interesting. In each case, let the IFS that generates the semigroup be called IFS#1. Then, in each case, the domain of the orbital picture is the attractor of a just-touching IFS (see Chapter 4) IFS#2, such that IFS#1 IFS#2. Let IFS#3 = IFS#2\IFS#1, meaning the IFS whose transformations consist of those in IFS#2 that are not in IFS#1. Then the domain of the condensation picture consists of the union of the sets obtained by applying the transformations in IFS#3 to the attractor of IFS#2. For example, in the case of the fern picture in Figure 3.56, IFS#2 is given by the four projective transformations represented, as in Equation (3.5.9), by the data in the following table: