Ohrimenko+ / Barnsley. Superfractals

80 Codes, metrics and topologies

Proof of (v): From (iv) we have that A H(X). Let > 0 and choose N so large

that n, m ≥ N implies dH (An , Am ) ≤ /2 and Am BAn ( /2). Let n ≥ N and y An . There exists an increasing sequence of integers greater than n, {Ni }∞1=1, such that for m, k ≥ N j , Am BAk ( /2 j+1). Note that An BAN1 ( /2). Since y An

dX (xN j , xN j+1 ) ≤ /2 j+1. It follows that {xN j AN j }∞j=1 is a Cauchy sequence that converges to a point x A and that d(y, xN j ) ≤ for all j. The latter implies

that d(y, x) ≤ . Hence An BA( ) for all n ≥ N . But, by (iii), A BAn ( ) for all sufficiently large n. It follows that dH (An , A) ≤ for all sufficiently large n.

Hence A = limn→∞ An .

A simple example of a Cauchy sequence of points in H(X) is {BA(1/n)}∞n=1 for A H(X). Clearly {BA(1/n)}∞n=1 converges to A, whether or not X is complete.

Figure 1.39 shows a sequence of images that represents a Cauchy sequence of compact subsets of R3. Read the images from left to right and from top to bottom. The intensity of green represents the z-component of the set. The base of each image is taken to lie on the x-axis. In such cases we can infer the existence of the limiting fractal fern from the existence of the Cauchy sequence and the completeness of R3.

E x e r c i s e 1.13.3 Show that if (X, dX ) is a compact metric space then (H(X), dH ) is a compact metric space. Hint: Assume that X is nonempty. Define An = X for all n = 1, 2, . . . Then { An H(X)}∞n=1 is a Cauchy sequence that converges to X. Now look back at the proof of Theorem 1.13.2.

E x e r c i s e 1.13.4 Show that H(R) is pathwise connected.

E x e r c i s e 1.13.5 In Figure 1.40 we show the first four generations of shield subsets of R2. Let An denote the union of the boundaries of the 2n−1 shields belonging to the nth generation. Show that { An }∞n=1 converges in the Hausdorff metric to a line segment.

The space (H(H(X)), dH(H))

It is at first sight amazing. But it is true. The space H(H(X)) is highly nontrivial: it is fascinating, rich, at least as interesting as is H(X) relative to X and it has significant applications to superfractal sets.

As we showed in Theorem 1.12.13, the condition that (X, d) is a metric space implies that (H(X), dH ) is a metric space. It follows that (H(H(X)), dH(H)) is also a metric space, where H(H(X)) is the space of compact subsets of the set of compact subsets of the metric space (X, d) and dH(H) is the Hausdorff metric on H(H(X))

Figure 1.39 These images represent a sequence of compact subsets of R3 that converges in the Hausdorff metric. The intensity of green represents the z-component of the set, which in each case lies in a plane parallel to z = 0. The base of each image lies on the x -axis.

implied by the Hausdorff metric dH on H(X). That is, for all α, β H(H(X),

dH(H)(α, β) = max DαH (β), DβH (α)

where

DαH (β) = max min dH (A, B).

B β A α

We summarise the basic inheritance properties of (H(H(X)), dH(H)) in the following theorem.

T h e o r e m 1.13.6 Let (X, d) be a metric space. Then (H(H(X)), dH(H)) is a metric space. If (X, d) is complete then (H(H(X)), dH(H)) is complete. If (X, d) is compact then (H(H(X)), dH(H)) is compact.

metric on R . If αn denotes the set of boundaries of the nth generation shields, to which set of sets does the sequence {αn }∞n=1 converge, in the metric dH(H)? See also Figure 1.9.

Figure 1.41 illustrates, in its four panels, four different related points in H(H(R2)). These points may be imagined to belong to a sequence of similarly constructed points and to converge to a set of subsets of R2, which, taken together, constitute a single point in H(H(R2)). The first point, represented in the top left panel of Figure 1.41, contains four sets that look like leaves (green). We will refer to these sets as leaf sets. In the same way, let us refer to calyx sets (mauve) and flower sets (yellows, dark purple and pale mauve). Furthermore, from time to time elsewhere in this book we will use a similar abbreviated nomenclature to describe sets represented by parts of images. Then we can say that the points represented successively in the other panels of Figure 1.41 contain more and more, smaller and smaller, copies of leaf sets, calyx sets and flower sets. We may suppose that the point in H(H(R2)) to which the sequence converges is {{x} R2 : x } wheredenotes a certain filled triangle. Then any neighbourhood, however small, of any such x would contain a set of sets that contains at least one minute leaf set, at least one minute calyx set and at least one minute flower set, all belonging to a point in the sequence.

E x e r c i s e 1.13.7 Calculate dH(H)(α, β) for the case when the underlying

space is (R2, deuclidean), α = { A, B} and β = {C, D}, where A = {(x, y) R2 : x2 + y2 = 1, x ≥ 0}, B = {(x, y) R2 : x2 + y2 = 12 }, C = {(0, y) R2 :

−2 ≤ y ≤ −1} and D = . Compare this distance with dH (A B, C D).

Figure 1.41 Four points in H(H(R2)) are represented here. The first point consists of four leaf sets (green), a calyx set (mauve), and a flower set (yellows, dark purple and pale mauve). The points represented in the other panels contain more and more, smaller and smaller, copies of leaf sets, calyx sets and flower sets. Assume that these points belong to a sequence of points in H(H(R2)) in the implied progression, which converges to the set of all singleton sets {x }, where x belongs a filled triangle. Then any neighbourhood, however small, of any such {x } would contain a set of sets that contains at least one minute leaf set, at least one minute calyx set and at least one minute flower set, all belonging to a single point in the sequence.

A significant difference between the relationship of H(H(X)) to H(X) and the relationship of H(X) to X is that if α H(H(X)) is a finite set of sets thenA α A H(X); it is not true in general that if A H(X) is a finite set of points then a A a X. That is, we can often ‘project’ from H(H(X)) to H(X) in a way that cannot analogously be used to link H(X) to X. This leads us to the comparisons in Theorems 1.13.8 and 1.13.9 below. Theorem 1.13.8 asserts that the metric dH(H) is a ‘stronger’ metric than dH .

Figure 1.42 Illustration of the falling leaves theorem. For an explanation of the symbols, see the main text immediately before Theorem 1.12.20.

Theorem 1.13.8 enables us, in some cases, to think about approximation in H(X) in terms of approximation in H(H(X)). According to Theorem 1.13.6, a sequence {αn H(H(X))}∞n=1 converges to a point α H(H(X)) iff for each A α there is a sequence of sets An αn such that { An H(X)}∞n=1 converges to A. So in the case of R2 we may for example discuss the possible convergence of a sequence of approximations to a tree set in terms of sequences of sets that contain sequences of leaf sets that converge to leaf sets, sequences of foliage sets that converge to foliage sets and a sequence of trunk sets that converges to a trunk set. We obtain a richer view of convergence in the Hausdorff metric.

Falling leaves theorem

Leaves fall from the sky, the sun is setting, and the shadows of three leaves float down a white wall. At one instant t the set of leaf shadows is represented by α = { A, B, C} H(H(R2)) while at a later instant t it is represented by α = { A , B , C } H(H(R2)). Here A and A represent the shadows of a given leaf, B and B represent the shadows of the second leaf and C and C represent the shadows of a third leaf; see Figure 1.42.

We will suppose that the leaf shadows A, B, C are disjoint. The following theorem tells us that when t and t are sufficiently close the Hausdoff distance between the union of the shadows at time t and the union of the shadows at time t is the same as the distance dH(H)(α, α ). We have framed this result for three leaves, but you will easily see how it is true for any finite set of leaves.

T h e o r e m 1.13.9 (Falling leaves theorem) If A, B, C H(X) are disjoint and A , B , C H(X) are such that dH (A, A ), dH (B, B ) and dH (C, C ) are all sufficiently small then

dH(H)({ A, B, C}, { A , B , C }) = dH (A B C, A B C ).

P r o o f We can suppose that

dH (A, A ) < min 12 dH (A, B), 12 dH (A, C) , dH (B, B ) < min 12 dH (B, A), 12 dH (B, C)

and

dH (C, C ) < min 12 dH (C, A), 12 dH (C, B) .

Now we start from the triangle inequality dH (A, B ) ≥ dH (A, B) − dH (B, B )

It now follows that D{HA ,B ,C }({ A, B, C}) = D{HA,B,C}({ A , B , C }) and hence that

dH(H)({ A, B, C}, { A , B , C }) = max{dH (A, A ), dH (B, B ), dH (C, C )}. (1.13.3)

often but not always coincides with the values provided by other definitions, when they apply.

The following two definitions are discussed in [34], pp. 25 et seq.

D e f i n i t i o n 1.14.1 Let S X, δ > 0 and 0 ≤ s < ∞. Let

where |Ui |s denotes the sth power of the diameter of the set Ui , and where a δ- cover of S is a covering of S by subsets of X of diameter less than δ. Then the s-dimensional Hausdorff measure of the set S is defined to be

H s (S) = lim Hδs (S).

δ→0

The limit exists but may be infinite, since Hδs (S) increases as δ decreases. Moreover H s (S) is non-increasing as s increases from zero to infinity. For any s < t we have Hδs (S) ≥ Hδt (S), which implies that if Hδt (S) is positive then H s (S) is infinite. Thus there is a unique value, given by the following definition.

D e f i n i t i o n 1.14.2 The Hausdorff dimension or fractal dimension of the set S X is defined to be

dimH S = inf{s|H s (S) = 0}.

There is much written about fractal dimensions in many sources, including Fractals Everywhere [9]. It is important to read Mandelbrot’s book [64] to understand his vision of why fractal dimension is important. Other useful references are [34] and [96].