Ohrimenko+ / Barnsley. Superfractals

Figure 3.62 The orbit of the measure represented by the flower at the bottom right is represented by the sequence of successively brighter flowers going up on the right. The successive flowers cease to become brighter after approximately six iterations because of saturation effects, which also cause changes in colour. Does the corresponding sequence of measures converge to a limiting measure?

the measure υ under the semigroup S(X) is the set of Borel measures

O(υ) = { f (υ) : f S(X)}.

Some pictures of measures belonging to orbits of measures under a semigroup generated by an affine transformation in R2 are illustrated in Figure 3.62. Notice that this image contains pictures of the measures in the orbit of the measure represented by the flowers in the bottom row and the left-hand column.

How can we make a single measure out of an orbit of measures? The natural and simple thing to do is to ‘add them all up’ with appropriate weights. To be able to do this easily we restrict our attention to orbits of measures generated by IFS semigroups.

D e f i n i t i o n 3.6.2 An IFS with probabilities is an IFS {X; f1, f2, . . . , fN } together with a set of probabilities, non-negative real numbers p1, p2, . . . , pN such

that p1 + p2 + · · · + pN = 1. The probability pn is associated with the function fn for n = 1, 2, . . . , N . An IFS with probabilities may be denoted

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

The following theorem is notable because it applies in very general circumstances. It is not required that the space X is compact or even complete; nor is it required that the transformations in the IFS be contractive, or even contractive on average.

T h e o r e m 3.6.3 Let X be a topological space and let {X; f1, f2, . . . , fN ; p1, p2, . . . , pN } be an IFS with probabilities, where fn : X → X is continuous for each n {1, 2, . . . , N }. Let 0 < p0 ≤ 1 and let υ0 P(X), the space of normali-

P r o o f Let B B(X) be a Borel subset of X. Then the value υ(B) is well

consists of non-negative terms and is bounded above, term by term, by the absolutely convergent series

Hence υ : B(X) → [0, 1]. Notice that υ(X) = 1. Let us define

ρ0 = υ0 and ρn = pσ1 pσ2 . . . pσ|σ | fσ (υ0) for n = 1, 2, . . .

σ {1,2,··· ,N } ,|σ |=n

Then it is readily verified that ρn P(X), and we can rewrite Equation (3.6.1) as

∞

υ = p0(1 − p0)n ρn .

n=0

Now, referring back to Definition 2.3.9, let {Om B(X) : m = 1, 2, . . . } be a sequence such that ∞m=1 Om B(X) and

υ(Om ) = p0(1 − p0)n ρn (Om ).

Notice that the expressions above could have been written down and handled more succinctly by introducing the linear operator L : P(X) → P(X) defined by

L acts linearly on the space of all possible linear combinations of Borel measures on X. We call L the Markov operator associated with the IFS. Using this notation, the self-referential equation (3.6.2) reads

μ = p0υ0 + (1 − p0)Lμ,

and the series expansion in Equation (3.6.1) can be written as

μ = p0(1 − (1 − p0)L)−1υ0

∞

= p0 (1 − p0)m Lm υ0.

m=0

We did not introduce L earlier because we wanted to display and manipulate the full series expansions, show the parallels and distinctions between orbital pictures and orbital measures and specifically illustrate how the probability p0(1 − p0)|σ | is associated with the measure fσ (υ0). When represented as a picture, each term in the series corresponds to a contribution or component of the picture; for example, each term in the series may correspond to a distinct ‘semigroup measure tile’, as in Figure 3.63. This suggests how one might define an IFS semigroup measure tiling.

Pictures of orbital measures corresponding to various simple IFS semigroups acting on R2 are illustrated in Figures 3.63–3.67. The manner in which these pictures were computed is described below.

Figures 3.64 and 3.65 relate to condensation measures that are drawn by the IFS towards the ‘horizon’, namely a line segment in R2, the set attractor of the IFS. Figure 3.65 is particularly interesting because it illustrates not only how elementary orbital measures can be used to produce synthetic, real-looking, pictures but also how subtle changes in these pictures can be produced by making small changes in the probabilities. On the right p1 = p2 = 0.5, on the left p1 is approximately 0.4 and p2 is approximately 0.6, and in both cases p0 is very close to zero, see below. The horizon on the left in Figure 3.65 looks threatening in contrast with the bright distant sky on the right.

It is worth comparing Figure 3.66 with Figure 2.23. The latter illustrates the convergence of the sequence of measures {Ln μ0}∞n=1 to the measure attractor of the same IFS with slightly different probabilities, where μ0 P(R2) is similar to the condensation measure used in Figure 3.66.

Figure 3.63 Picture of an orbital measure of an IFS semigroup generated by two contractive similitudes. The condensation measure is represented by the bottom shield-shaped tile. The probabilities on the maps are such that successive shields on the left are darker and darker, while those on the right are successively lighter.

The set attractor is the filled unit square with lower left corner at the origin. The support of the orbital measure represented in Figure 3.67 is contained in . A comparison of Figure 3.67 and Figure 3.42 provides a striking contrast between an orbital measure and a closely related orbital picture.

Conclude that we do not obtain, in the limit, a solution to υ = f (υ) with υ P(X). What happens if the interval [0, 1) is replaced by [0, 1]?

Next we describe the type of method that we used to compute the approximate

pictures of orbital measures shown in Figures 3.63–3.67. Let υ0 P(X), 0 < p0 < 1, and an IFS {X; f1, f2, . . . , fN ; p1, p2, . . . , pN } be given, where X = R2. Let F : P( ) → P( ) be defined by

F(υ) = p0υ0 + (1 − p0)Lυ for all υ P( ).

Then, by what we have been saying above, the sequence {F◦k (υ0) P( )}∞k=1 converges to the orbital measure υ; namely, given any > 0 there exists an integer l such that |F◦k (υ0)(B) − υ(B)| < for all k > l, uniformly for all Borel subsets B B( ).

It follows that we can compute a sequence of approximations to the value of υ for any array of pixels, successively, one step at a time. Specifically, let a resolution W × H be selected and construct the discretization { w,h : w = 1, 2, . . . , W , h =

1, 2, . . . , H } of , as discussed in Section 2.2. Then observe that the sequence of digital pictures {P(k) : → [0, ∞)}∞k=0, whose pixels are P(w,k)h = F◦k (υ0)( w,h ) for k = 0, 1, 2, . . . , satisfies

P(k+1) = P(k+1)( w,h ) = F(F◦k (υ0))( w,h )

w,h

N

= p0υ0( w,h ) + (1 − p0) pn fn (F◦k (υ0))( w,h ).

n=1

Notice that P(0)w,h = υ0( w,h ). Given P(k), we can form approximations to each term inside the last summation and thus produce an approximation to P(k+1).

where Q(n, w, h) is the set of indices (w , h ) corresponding to pixel domainsw ,h whose centre points, say, are mapped into w,h , that is,

Q(n, w, h) = {(w , h ) {1, 2, . . . , W } × {1, 2, . . . , H } : fn (cw ,h ) w,h },

where cw,h denotes a selected representative point in w,h . This type of approximation produces pictures that are accurate to viewing resolution when the transformations are sufficiently contractive. In other cases we use the inverse of the maps fn to provide approximations for the contribution fn (F◦k (υ0))( w,h ) in terms of P(k); for example, in some cases we use the approximation

288 Semigroups on sets, measures and pictures

where w (n, w, h), h (n, w, h) is the index of the pixel domain in which lies the point fn−1(cw,h ). Here we may approximate the ratio of areas using the Wronskian of the transformation fn , as described in Section 2.7 for the case of projective transformations. In general, a good understanding of the specific way in which the transformations of the IFS deform the space, as described in Chapter 2, is very helpful in the construction of good approximations to pictures of orbital measures. Some problems in the discretization of IFSs have been analyzed in [79].

In working with sequences of approximate digital pictures of orbital measures, we also run into effects caused by the finite range of values in the colour space C. The expressions above assume that the colour space is of the form [0, ∞) R. In practice C may be {0, 1, 2, . . . , 255}. To deal with this, we not only discretize the values of P(k) but also replace those that exceed 255 by 255, which leads to colour

This expression represents an ‘unbounded measure’ because

)

υ0(X) + pσ fσ (υ0)(X) = ∞.

|σ |≥1

Nonetheless, it is straightforward to make approximate pictures of this ‘unbounded measure’ using the same techniques as above, because saturation effects stop the divergence. This allows us to make approximate pictures of orbital measures when p0 is very small. The two pictures in Figure 3.65 are of this kind; the bright horizon on the right would be utterly dazzling if not for saturation. Imagine it.

3.7Groups of transformations

A group of transformations is a special type of semigroup – every transformation possesses an inverse that is also in the group. A group of transformations acting upon a picture of a seahorse is illustrated in Figure 3.68. An important difference between Figure 3.68 and Figure 3.3 is that each seahorse is the image of another seahorse under some transformation in the group. In Figure 3.3, however, one flower has no pre-image. Another example of a group of transformations, this time acting on subsets of R2, is illustrated in Figure 3.69.

We have chosen to introduce groups of transformations with the complicated and initially slightly confusing image in Figure 3.68 in order to emphasize the

Figure 3.68 A group of Mobius¨ transformations acts on a leafy seahorse on the Riemann sphere C. Think of the picture as a map of most of the surface of the sphere. Then you may imagine that the source of the seahorses is the centre of a two-dimensional reverse whirlpool. Seemingly, they grow as they swirl outwards from the source, and some are hidden from view, on the other side of the sphere. Eventually they appear to be caught by a second whirlpool. But which is the source and which is the sink?

richness and visual complexity that may be associated with the underlying simple idea of a group – a parade of identical horses prancing round a carousel, say, hardly has the same intricacy. In our example, not only is each seahorse a different size, it is also a different shape.

D e f i n i t i o n 3.7.1 A group (G, ) is a semigroup with the following properties:

(i) there is a unit element I G with the property

I g = g I = g for all g G;

(ii)given any g G there is an element g−1 G, called the inverse of g, with the property

g−1 g = g g−1 = I .

A subgroup of (G, ) is a group of the form (G, ), where G G.