Ohrimenko+ / Barnsley. Superfractals

The following theorem provides a convenient necessary and sufficient condition for a layered orbit O(C) to yield a semigroup tiling of P = O(C). It is relevant to image compression. Given a set of the form P, we would like to represent it as efficiently as possible; therefore we may wish to choose the condensation set C to be as small as possible, without changing the orbital set P. This theorem fills me with wonder because it seems almost magical that all the tiles fit together neatly once the initial ones do.

T h e o r e m 3.4.6 Let S{ f1, f2,..., fN }(X) be an IFS semigroup of one-to-one transformations. Let O(C) be a layered orbit of a nonempty set C X. Let

because fn (P) contains fn (C) for all n {1, 2, . . . , N }. We substitute from Equation (3.4.3) into itself to obtain

sion within braces for some k, which in turn implies that x fσ (C0) for some σ

We proceed by induction. Let us assume that, for some integer K ≥ 1, we have

Figure 3.17 Two examples of IFS semigroup tilings. The triangle on the left is tiled with the orbit of a six-sided figure, under an IFS of two affine transformations. The limit set of the set of triangular tiles, on the right, is the attractor of an IFS of three transformations.

Figure 3.18 The left-hand picture illustrates the points in the orbit of a set, the flower picture at centre left, under a semigroup generated by a single Mobius¨ transformation. This orbital set is in fact a semigroup tiling, as illustrated by the image at centre right. The image on the far right is the initial tile.

Using the assumed one-to-oneness of the transformations and the fact that fn (P) ∩

tion (3.4.4) holds with K replaced by K + 1. This completes the inductive step and

forms a semigroup tiling of P. To prove the converse, we simply note that if any statement in Equation (3.4.2) is false then the corresponding pair of putative tiles ‘overlap’ one another, which implies that O(C) is not a semigroup tiling of P.

If the ranges of the transformations fn (X) are disjoint then the second set of conditions in Equation (3.4.2) is satisfied. Such IFSs, whose transformations have disjoint ranges, occur in connection with set attractors that are ‘just touching’ or totally disconnected, as will be discussed in Chapter 4. Also, semigroup tilings associated with a dynamical system f : X → X may occur when there is an IFS associated with the inverse of f . For example, in Section 3.5 we show that the Henon transformation generates fascinating tilings because it is one-to-one and onto.

Also, in complex analytic dynamics, when f is a rational function on the Riemann sphere the ranges of the branches of f −1 intersect only at certain isolated points. These inverse branch transformations generate astonishing tilings. For example, let J be the Julia set of the dynamical system {C; f (z) = (z − λ)2}, where λ C is a parameter. Then J is essentially the fractal set, the attractor, of the IFS

{C; f1(z) = λ + √z, f2(z) = λ − √z};

see for example [9]. In this case f1(P) ∩ f1(P) = whenever P C and 0 / P. It follows that if one chooses C C\{0} such that C ∩ J = , which ensures that the orbit of C is layered, then it follows from Theorem 3.4.6 that the set C0 generates a semigroup tiling under the IFS semigroup. In fact, pictures of J are often produced by means of the ‘escape time’ algorithm; see for example [78] or [9]. Points z C are assigned colours according to the least integer n such that f ◦n (z) C0, where C0 C has been chosen so that C0 ∩ J = . The resulting beautiful pictures, artistic and harmonious, illustrate the Julia set by colouring points that do not belong to the Julia set; see Figure 3.19, for example. For us, however, such pictures comprise two different, more substantial, mathematical entities: semigroup tilings, and pictures that are invariant, or mapped into themselves, under certain transformations.

E x e r c i s e 3.4.7 Identify the addresses of the tiles in the IFS semigroup tilings illustrated in Figure 3.15. State your assumptions.

E x e r c i s e 3.4.8 Let O(C) denote the orbit of a set C X under the semigroup S{ f1, f2,..., fN }(X). Assume that the orbit is layered and that the transformations are one-to-one. Define

Figure 3.19 Semigroup tiling associated with a Julia set. The condensation set, the first tile, is the outer necklace. Each of the two ‘second-generation’ tiles, whose addresses are 0 and 1, is formed from a copy of the original necklace that has been cut so that it forms a strand instead of a loop. These two cut necklaces meet each other to make the second necklace, which has twice as many beads as the first one. The different colours indicate different tiles, for the first four generations.

Show that

O C C = O(C)

and illustrate this result using an elegant overlapping orbit of a set C under a nontrivial semigroup of transformations on R2.

3.5 Orbits of pictures under IFS semigroups

The underlying ideas in this section are not the same as those in Section 3.4 for this reason: whereas the union of two sets is a new set, the union of two pictures is not defined. This leads us to use the tops union to define orbital pictures. An orbital picture is a picture that is specified, uniquely and naturally, in terms of an orbit of pictures under an IFS semigroup, using the tops union. But the tops union is not commutative. This has the consequence that an orbital picture may be more intricate mathematically than a corresponding orbital set.

In this section we define, establish properties of and illustrate orbital pictures; we often find illustrations of them to be visually exciting and beautiful. Orbital pictures have applications to fractal image compression and computer graphics. We believe that they also have applications in image recognition, cryptography, number theory and bioinformatics. We will mention these applications in later sections.

Although the theory of orbital pictures has been inspired by the idea of IFSs acting on pictures, it is important to remember, as you read the mathematics, that this theory is simply mathematics and may be read as such, without regard for physical pictures.

After defining orbital pictures we establish the theoretical basis for the deterministic computation of them. We show how orbital pictures obey their own special type of self-referential equation and define the panels and the code space of an orbital picture. We prove that the shift transformation maps this space into itself, and this provides us with a symbolic dynamical system and various topological invariants of orbital pictures, including the orbital growth rate and a certain symbolic entropy. We illustrate the corresponding dynamics on the panels of an orbital picture and mention the critical relationship between an orbital picture and the attractor of an IFS, when both are defined. We also illustrate and discuss a family of examples, related to pictures of flowers in a field that stretches to the horizon, for which we can say something about the symbolic entropy and the diversity. The diversity of an orbital picture is another invariant and, together with the orbital growth rate, provides quantitative information, which is invariant under homeomorphism, about the way that orbital pictures look. When the diversity equals infinity we define another quantity, the growth rate of diversity, which is bounded above by the growth rate of periodic cycles. We introduce the space of limiting pictures associated with an orbital picture, which is used to define the diversity and the growth rate of diversity. Finally we introduce the concept of orbital tiling (in contrast with semigroup tiling by pictures, which is essentially the same as tiling a set by images of the set under a semigroup of transformations) and underneath pictures. Applications of orbital pictures are mentioned. A transformation called the Henon mapping is used to illustrate that the orbital picture generated by a single transformation can be quite complicated.

We restrict our attention to semigroups of one-to-one transformations, because only invertible transformations can be applied to pictures. We will tend to think of X as being a ‘flat’ space such as R2, so that pictures whose domains lie in X may be illustrated. The symbol = C(X) denotes the space of pictures whose domains lie in X and whose ranges lie in a fixed colour space C.

D e f i n i t i o n 3.5.1 Let P0 denote a picture with its domain in X. Then the orbit of the picture P0 under the semigroup S(X) is defined to be the set of pictures

O(P0) = { f (P0) : f S(X)}.

Notice that the set of domains of the pictures in an orbit of a picture P0 is the orbit of a set, the domain of P0. In general these domains will overlap, so we are led to use the tops union to define a picture of the orbit of a picture. In turn this

means that we need the set of pictures in the orbit to be countable, so we restrict our attention to orbits of pictures generated by IFS semigroups.

The orbital picture P = O(P0)

A picture of the orbit of a picture P0 under an IFS semigroup S{ f1, f2,..., fN }(R2) is obtained as follows. (i) Arrange the pictures in the orbit as a sequence, {Pn }∞n=0, starting with P0. (ii) Let P1 be the picture whose domain is the union of the domains of P0 and P1 and whose colour values agree with P0 on DP0 , the domain of P0, and agree with P1 on the rest of its domain. (When the domains of P0 and P1 overlap, the resulting picture P1 looks like P0 with the picture P1 sticking out from underneath it.) (iii) Next, similarly combine P1 with P2 to produce P2, which may look like P1 with P2 sticking out from underneath it. (iv) Continue in this manner to make a sequence of pictures {Pn }∞n=1, which, in turn, defines a limiting picture that we denote by P = O(P0).

Now we will describe this construction more specifically.

D e f i n i t i o n 3.5.2 Let S{ f1, f2,..., fN }(X) be an IFS semigroup and let P0 be the space of pictures with domains in X. The canonical sequence of pictures

{Pn }∞n=0 in the orbit of P0 is defined by

{0, 1, 2, . . . } to each element of the code space, and it is invertible. So, for example, when N = 2 we have

T h e o r e m 3.5.3 Let {Pn }∞n=0 denote the canonical sequence of pictures in the orbit of P0 under the IFS semigroup S{ f1, f2,..., fN }(X). Let

for n = 0, 1, 2, . . . Then there exists a unique picture P = P(P0) such that DP =

O(DP0 ) and such that, given x DP, P(x) = Pn (x) for some n {0, 1, 2, . . . }.

P r o o f Notice that, by construction,

∞∞

picture of the orbit of the picture P0 under the IFS semigroup S{ f1, f2,..., fN }(X). It is denoted by

P := O(P0)

and we can write

P = P0 P1 P2 · · ·

We may refer to P = O(P0) as the orbital picture of P0 (under the IFS semi- group S{ f1, f2,..., fN }(X).)

An example of a picture of an orbit of a picture P0 under an IFS semigroup is shown in Figure 3.20. The IFS consists of three transformations and its associated fractal set, the set attractor of the IFS, is a Sierpinski triangle. Notice how the domain of P0, the part corresponding to the red flower with stamens, overlaps the attractor of the IFS. As a result, all the pictures in the orbit of P0 overlap the Sierpinski triangle.

Other simple examples of pictures of orbits of pictures under IFS semigroups are shown in Figures 3.21–3.25 and 3.55. The manner in which these images were computed is explained below. The two affine transformations for the orbital picture of buttercups in Figures 3.21, and the close-up in Figure 3.22, are illustrated in Figure 3.36; they are given by

together with their inverses, and the viewing window is specified by −1 ≤ x ≤ 1 and −1 ≤ y ≤ 1. An invariant set for the IFS in this case is a circle; N = 4 and

are used in Figure 3.25.

Notice that there was potential arbitrariness in the choice of the integers that we assigned to the elements of the IFS semigroup. We made a convenient choice, once

Figure 3.20 Picture of an orbit of a condensation picture, of flowers and a ribbon, under an IFS semigroup with N = 3. This is not a picture tiling, according to our definition, because bits of the condensation picture are missing from the ‘tiles’. Can you see how the original picture, lower left, overlaps the attractor of the IFS?

and for all, and we will keep to it because it has some wonderful consequences. One consequence is that P obeys a self-referential equation; see Theorem 3.5.8. Another is that we can construct a code space and a symbolic dynamical system for the orbital picture, as described in a later subsection. Yet another is that we can compute, efficiently, approximations to P = O(P0), as we describe next.

Computation of orbital pictures

Notice that Pn is a function of P0. Specifically Pn : → is given by

Pn (P0) = fσ (0)(P0) fσ (1)(P0) fσ (2)(P0) · · · fσ (n)(P0)

Figure 3.21 Picture P of the orbit of a picture of a buttercup P0 under an IFS semigroup S{ f1, f2 }(R2). See also Figures 3.22 and 3.36. Can you find the two transformations for which P = P0 f1(P) f2(P)? How would the picture of the orbit of P0 under the IFS semigroup S{ f2 , f1}(R2) differ from P?

for all P0 and for all n {0, 1, 2, . . . }. Recall that f denotes the identity transformation.

T h e o r e m 3.5.5 Let P = O(P0) denote the picture of the orbit of P0under the IFS semigroup S{ f1, f2,..., fN }(X). Let {Pn }∞n=0 denote the canonical

P r o o f We prove this result for the case N = 2. We will use induction on k in Equation (3.5.7). To keep the notation clean let us write F = PN . Then we are trying to show that, for all k = 0, 1, 2, . . . ,

P N (N k −1) (P0) = F◦k (P0),

N −1