Ohrimenko+ / Barnsley. Superfractals

Figure 3.84 This floral pattern is an orbital picture in which the panels have different colour tones. It was generated by the Mobius¨ IFS in Equation (3.7.3).

shows the orbital picture generated by an IFS group Ghyperbolic(C), using the same condensation picture as in the right-hand image. Ghyperbolic(C) corresponds to the IFS

where Ma,θ (z) denotes a member of the family of transformations defined in Equation (2.6.10). The transformations in Ghyperbolic(C) are hyperbolic and map the unit disk onto itself; each has two fixed points, one repulsive and one attractive, located on the boundary of the disk.

Another more artistic picture generated using the Mobius¨ IFS in Equation (3.7.3) is shown in Figure 3.84. We have illustrated only a very few orbital pictures associated with Mobius¨ IFS semigroups and groups, however. A wealth of others can be imagined. To obtain families of IFS objects associated with Mobius¨

geometry, consider IFS groups and semigroups of transformations that share fixed points, or map from a fixed point of one to a fixed point of another, or share an invariant circle, or have invariant circles that are tangent to one another. See [73] for inspiration.

Code space geometries

Klein certainly had in mind that the underlying space for a geometry should be something like a surface, say of a sphere, or R3, and that the transformations should be quite ‘geometrical’ too. We can invent many other geometries, however; they may not really be quite so geometrical as the ones we have described and that were in Klein’s mind. For example, we might work on R2 but take the group of transformations to be the set of homeomorphisms of R2 into itself. This geometry is relevant to fractal geometry, as we will see in Section 4.14.

It is useful to think about code space in geometrical terms. We introduced various families of transformations on code spaces in Chapter 2. Most of these, such as the shift transformation, are not invertible and do not give rise to geometries. But any homeomorphism f : A → A generates a group of transformations that conserve topological properties such as compactness, connectivity, boundaries, and so on. One example of a group of homeomorphisms is the group of permutations. This group is relevant to orbital pictures and fractal tops, both of which depend on the ordering of the functions in the IFS that produces them.

Let GA denote the permutation group for the alphabet A. For each p GA define f p : A A → A A by

f p(σ ) = p(σ1) p(σ2) p(σ3) · · ·

= { f p : A A → A A : p GA} is called the permutation group on code space. It is easy to see that each permutation f p is a homeomorphism, that the topological entropy of a point in code space is invariant under each permutation and that shift-invariant subspaces are mapped into shift-invariant subspaces by each permutation.

E x e r c i s e 3.7.18 Let p : {1, 2} → {1, 2} obey p(1) = 2 and p(2) = 1. Let and denote the code spaces for orbital pictures generated by the IFS semigroups S{ f1, f2}(R2) and S{ f2, f1}(R2) respectively, acting on the same condensation picture. Suppose that f2(x, y) := − f1(−x, −y) and that the condensation picture is invariant under the transformation (x, y) → (−x, −y). Show that f p( ) = .

Figure 4.1 The two mathematical ferns represented here are not topologically conjugate because their branching structures are different. Hence by the fractal homeomorphism theorem (see Section 4.14) their code space structures are different. But there exist transformations from one to the other that are ‘nearly continuous’. See also Figure 4.2.

serious analysis. We define the tops dynamical system and an associated symbolic dynamical system; we show how pictures produced by tops plus colour-stealing are analogous to set attractors and measure attractors because they are fixed points of a contractive transformation defined using the IFS; and we establish a relationship with orbital pictures and other material in this chapter. Finally, inspired by what we have learnt, we introduce directed IFSs, which generalize IFSs in a very natural way.

In the back of your mind, as you read or scan this chapter, keep alive the theme of bioinformatics. What does this new material suggest in the way of new models in the biological science? Does it just look biological but really is not? Or is there something very deep here in the idea of treating protoplasmic things in the language of topology and sets of sets in code space?

Read on, and enjoy.

4.2 Hyperbolic IFSs

An iterated function system or IFS, as explained earlier, consists of a finite sequence of transformations fi : X → X for i = 1, 2, . . . , N where N ≥ 1 is an

Figure 4.2 There exist transformations that map from one mathematical fern to the other which are ‘nearly’ homeomorphisms. These transformations are easy to implement and have diverse applications. Carefully study these images to see how the form and colour are shifted.

integer and X is a space. It may be denoted by

{X; f1, f2, . . . , fN } or {X; fn , n = 1, 2, . . . , N }.

We use such terminology as ‘the IFS {X; f1, f2, . . . , fN }’ and ‘Let F denote an IFS’. We first introduced IFSs in the Introduction and Chapter 2. Typically, the space X is a metric space, the transformations are Lipschitz and there is more than one transformation.

An IFS with probabilities consists of an IFS together with a sequence of probabilities p1, p2, . . . , pN , positive real numbers such that p1 + p2 + · · · + pN = 1. An IFS with probabilities may be denoted

{X; f1, f2, . . . , fN ; p1, p2, . . . , pN }.

The probability pn is associated with the transformation function fn for each n {1, 2, . . . , N }.

fn : X → X, for n = 1, 2, . . . , N . Then {X; f1, f2, . . . , fN } is called a strictly contractive IFS or a hyperbolic IFS.

Recall that a transformation fn : X → X is strictly contractive iff there exists a number ln [0, 1) such that d( fn (x), fn (y)) ≤ ln d(x, y) for all x, y X. The number ln is called a contractivity factor for fn and the number

l = max{l1, l2, . . . , lN }

is called a contractivity factor for the IFS.

We use such terminology as ‘Let F denote a hyperbolic IFS with probabilities’. Although we often deal with hyperbolic IFSs, we tend to drop the adjective ‘hyperbolic’. We may use an adjective such as affine, projective or Mobius¨ when we want to describe the geometry to which the transformations of the IFS belong.

Rθ denotes an anticlockwise rotation through angle θ about the origin.

4.3 The set attractor and the measure attractor

Recall, from Theorems 2.4.6 and 2.4.8, that a hyperbolic IFS F possesses a unique set attractor, A H(X). The space H(X) is the set of nonempty compact subsets of X.

The set attractor A is the unique fixed point of the strictly contractive transformation F : H(X) → H(X) defined by

The transformation F : H(X) → H(X) is strictly contractive with respect to the Hausdorff metric, with contractivity factor l. Note that we use the same symbol F for the IFS and for the transformation F : H(X) → H(X).

The set attractor A obeys the self-referential equation

A = f1( A) f2( A) · · · fN ( A).

An example of a set attractor, when the transformations are two similitudes on R2, is illustrated in Figure 4.3. The transformations are given in Table 4.1. This

Table 4.1 Mobius¨ IFS code for Figure 4.3. The attractor of this IFS is pictured below, as in the tables that follow. These transformations are actually similitudes, in contrast with those in Table 4.6

Figure 4.3 The top two images are pictures of the set attractor of an IFS and a measure attractor of the same IFS, given in Table 4.1. You can see on the left how the set attractor can be regarded as the union of two scaled copies of itself. The measure illustrated on the right is a superposition of two measures, each rescaled. Zooms are shown at the bottom of the figure.

attractor is known as the Heighway dragon. You can see quite clearly how it is the union of two scaled copies of itself. As described in the Introduction and justified in Section 4.5, we can use algorithms based on the chaos game to compute such pictures.

Closely related to the set attractor is the measure attractor. In dealing with measure attractors we restrict our attention to the case where (X, d) is a compact

metric space, because this implies that (P(X), dP ) is also a compact metric space. We do this purely for simplicity. There are many cases where this restriction is not needed. For example, given a hyperbolic IFS for which the metric space X is locally compact, that is, closed balls of finite radius are compact, we can redefine the IFS to act on a new space X X that is compact; see Exercise 4.4.1. The space R2 is locally compact. So we will sometimes treat a hyperbolic IFS as though the underlying space were compact although in fact the specified underlying space X is not compact.

From Theorems 2.4.19 and 2.4.21, there exists a unique normalized measure μ P(X), which is the fixed point of the transformation F : P(X) → P(X) defined by

for all ξ P(X). Notice that we use the same symbol F for the IFS, for the transformation F : H(X) → H(X) and for the transformation F : P(X) → P(X). The interpretation of F should to be clear from the context.

The transformation F : P(X) → P(X) is a strict contraction, with contractivity factor

l = p1l1 + p2l2 + · · · + pN lN

with respect to the metric dP on P(X). It is also strictly contractive with contractivity factor l with respect to the metric dP .

D e f i n i t i o n 4.3.1 Let X be a compact metric space and let

F = {X; f1, f2, . . . , fN ; p1, p2, . . . , pN }

be a hyperbolic IFS with probabilities. Then the unique fixed point μ P(X) of F : P(X) → P(X) is called the measure attractor of the IFS.

The measure attractor μ of a hyperbolic IFS with probabilities obeys the selfreferential equation

N

This says that the measure is a weighted sum of the transformations of the IFS applied to it.

An example of a measure attractor, represented as a picture, is given on the right in Figure 4.3. It can be seen that this picture is a superposition of two transformed copies of itself, weighted by the probabilities in Table 4.1. In Section 4.5 we explain how, with the aid of the chaos game, this picture was computed.