Ohrimenko+ / Barnsley. Superfractals

T h e o r e m 5.18.5 For all V {1, 2, . . . } we have (V ) = (V ). That is, the set of all V -variable points in H( {1,2,...,N }) is the same as the set of all first components of the points belonging to the attractor of the hyperbolic IFS S(V ).

P r o o f is easy. The other direction is more subtle but should not cause you much difficulty. Otherwise consult [16].

5.19V -variability and what happens as V → ∞

Here we complete the explanation of how V -variable fractals provide a bridge between deterministic fractal sets and fully random fractal sets.

We can address the elements of Hsuper( {1,2,...,N }) by means of the continuous onto-mapping

ξsuper : {1,2,...,M} → Hsuper {1,2,...,N }

defined by

ξsuper(σ1σ2 · · · ) = Iσ1 Iσ2 ··· for all σ {1,2,...,M},

where the code tree of Iσ1 Iσ2 ... Hsuper( {1,2,...,N }) is the unique one whose kth node, reading up from the bottom of the tree in the ordering shown in Figure 5.38, is associated with the set of integers Iσk for k = 1, 2, . . .

We use ξsuper to define a probability measure ρsuper on Hsuper( {1,2,...,N }), according to

ρsuper = ξsuper(ρ).

Here ρ P( {1,2,...,M}) is uniquely defined by its values on the cylinder sets Cω

{1,2,...,M}:

ρ(Cω ) = Pω1 Pω2 · · · Pω|ω|

for all ω {1,2,...,M}.

T h e o r e m 5.19.1 Let (V ) denote the set of V -variable subsets of {1,2,...,N }, and let ρ(V ) denote the associated probability distribution for the IFS S(V ) defined at the end of Section 5.18. Then

lim (V ) = Hsuper {1,2,...,N }

V →∞

with respect to the metric dH(H( {1,2,...,N } )) and

lim ρ(V ) = ρsuper V →∞

with respect to the metric dP(H( {1,2,...,N } )).

P r o o f See Theorem 12 of [16], which provides the full proof in the case

Then in the spirit of Falconer [34], [35], Graf [41] and Mauldin and Williams [67] we make the following definition.

D e f i n i t i o n 5.19.2 We refer to the set of fractal sets A(∞) distributed according to the probability distribution μ(∞) as the random fractals associated with the superIFS {X; F1, F2, . . . , FM ; P1, P2, . . . , PM }.

Finally, we state the main result.

T h e o r e m 5.19.3 Let the superIFS {X; F1, F2, . . . , FM ; P1, P2, . . . , PM } be given. Let A(V ) denote the corresponding superfractal of V -variable sets and

denote the corresponding probability distribution. Then

lim A(V ) = A(∞)

V →∞

with respect to the metric dH(H(X)), and

lim ρ(V ) = ρ(∞)

V →∞

with respect to the metric dP(H(X)).

What’s the point? Simply this. We can compute approximations to, and study, random fractals by working with V -variable fractals. The latter can be explored by means of the chaos game on superfractals and lead to a wealth of insights into random fractals. In particular, we see how random fractals may be thought of as V -variable fractals, but of infinite variability.

Similar results also relate V -variable fractal measures to the random fractal measures introduced by Arbeiter [1]; see [16]. In [19] the theory of V -variable fractal sets and measures, as presented in this chapter, is strengthened to admit the uniform Prokhorov metric in place of the Monge–Kantorovitch metric, in order to allow a separable complete metric space X to replace the compact metric space used here, and to admit IFSs that are ‘on average’ contractive. The Hausdorff dimensions of some V -variable fractals and other recent developments are discussed in [20] and [21].

5.20Final section

So, dear Diana and Rose and gentle reader, there you have it! When I started, three years ago, I hoped to weave more closely a relationship between art, biology and mathematics, to exhibit a new geometry of colour and space and to make the vision so compelling that it would almost leave the abstract world where it lives and become instead part of yours. You must be the judge of how far this book fulfils my aim; and I will keep trying to develop these ideas further at www.superfractals.com.

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