Ohrimenko+ / Barnsley. Superfractals

Figure 3.35 Illustration of the dynamical system T : P → P , showing with red arrows the action on some of the panels. The panels are parts of flowers with a small part missing; each flower is mapped to the next flower out around the spiral, while the last flower is mapped to itself. The blue arrow represents a related transformation on the attractor of the IFS.

where h(P) is defined in Equation (3.5.14). We expect that the same relationship will hold when T : DP → DP is not continuous and the set of points where T is discontinuous makes no contribution to htop.)

Two examples of the dynamics of T : Ppanels → Ppanels are given in Figures 3.35 and 3.37; see Figure 3.36 for an illustration of the mappings used in connection with Figure 3.37.

In Figure 3.35 the panels are flowers; each flower is mapped to the next flower out along the spiral, while the last flower is mapped to itself. So T : DP → DP maps each point in a flower to the corresponding point in the next flower. The blue arrow represents an extension of the dynamics of T : DP → DP to the attractor of the IFS. Similarly, in Figure 3.37, the action of T on the domain of the visible parts of buttercups has been extended to define an action, again indicated by blue arrows, on a limit set, the horizon.

You can get an intuitive feel for this extension of T : DP → DP to the limit set by looking at Figure 3.37 and analysing how the dynamics of T : Ppanels → Ppanels acts on panels close to the horizon. In Chapter 4 we will show that this intuitively

Figure 3.36 Illustration of the action of the transformations, in Equation (3.5.4), used in Figures 3.21, 3.22 and 3.37. We have f1(A B C ) = A F D and f2(A B C ) = G B E . It is helpful to think of the triangle A F D as lying on top of triangle G B E .

glimpsed dynamical system is actually the ‘tops’ dynamical system restricted to the subset AP0 of the attractor A of the IFS, which ‘peeks out from underneath the orbital picture’, as illustrated in Figure 3.38. AP0 is defined with the continuous addressing function φ : → A from Theorem 3.3.12 by

AP0 := φ P0 .

E x e r c i s e 3.5.19 Figure 3.38 illustrates orbital pictures for the IFS semigroup generated by the three transformations

Mark some arrows between panels in the orbital picture at the top right of Figure 3.38 to illustrate the action of the dynamical system { Ppanels , T }. Deduce a

Figure 3.37 Illustration of the dynamical system T : P → P ; see also Figure 3.36. The action of T on visible parts of the buttercups may be extended to define an action, represented by the blue arrows, on the limit set, the horizon. The topological entropy of this limiting system, − log2 0.7, is a measure of the complexity of the orbital picture. See the main text.

consistent action for T on the ‘limit set’ of the orbital picture, illustrated in various colours in the top left panel of Figure 3.38.

E x e r c i s e 3.5.20 Prove that AP0 , as defined above, is a closed set.

E x e r c i s e 3.5.21 Consider the set-up in Exercise 3.5.12. Show that A P(A C) and AA C = A. So in this case we have AA C P(A C), and none of AA C would be seen ‘peeking out from underneath the orbital picture’. Show also that P(A) = A and AA = .

Further examples of panels and the associated dynamical systems are illustrated in Figures 3.39–3.45.

Figures 3.39–3.41 illustrate the panels {Qσ : σ P0 } in Figures 3.21 and 3.22. The colours of the panels have been modified to produce a new set of panels

Figure 3.38 (Top right) The orbital picture of the condensation picture P0, as in Figure 3.1; (bottom left) the underneath picture and, in colours different from green, the attractor of the IFS; (top left) the first few generations of the orbital picture, with the attractor ‘peeking out from underneath’; (bottom right) the orbital picture when a smaller condensation set P0 is used. Do the visible parts of the leaves in the bottom left image represent a picture tiling?

{Qσ : σ P0 } with the aid of a semigroup of homeomorphisms {Cσ : C → C : σ P0 }, according to

We should notice the diversity of the shapes and forms of the panels, and the emergence of new patterns, as we zoom in deeper and deeper towards the distant horizon. We will formalize this intuition in the next part of the subsection. This sequence of figures illustrates how orbital pictures may be used in graphics for video games to produce, in a simple way, scenery which possesses rich patterns that change as the user ‘travels towards the horizon’.

In Figure 3.42 we have illustrated the panels of the orbital picture of a brightgreen leaf silhouette, P0, situated inside the attractor set , a filled square, under an IFS of four similitudes, each of which maps onto one of its four quarters. Different colours are used to illustrate the panels (otherwise the orbital picture would look like a green .) Let us say that a panel is larger or smaller than a second panel if it is a segment of a leaf that is respectively larger or smaller than the leaf of which the second panel is a segment. Then the transformation T : Ppanels → Ppanels maps the largest segment, P0, to itself and every other panel to one of the next larger panels. Notice that there are various different-shaped panels of the same size. In this case the limit set AP0 includes the boundary of together with various fractal crosses that project into the interior of . Clearly there is a great diversity of panels in any neighborhood of AP0 .

It is interesting to compare Figure 3.42 with Figure 3.43. In the latter the attractor is again but this time the four maps in the IFS are the similitudes

These similitudes are such that fi ( ) ∩ f j ( ) has a nonempty interior for each i, j {1, 2, 3, 4}. A close-up of Figure 3.43 is shown in Figure 3.44. In this case the limit set AP0 is simply the boundary of and the growth rate of periodic cycles is lower than for the situation in Figure 3.42. But Figure 3.43 seems more complicated than Figure 3.42. Is it? In the next part of the subsection, which now follows, we will show a way in which such pictures may be compared.

The space of limiting pictures and the diversity of segments in the orbital picture

The code space P0 provides an addressing scheme for the panels of the orbital picture. But what is the significance of P0? Can we find pictures, some sort of

magnified limiting panels, that correspond to sequences of points in ? Can

P0

we find such pictures that also correspond to periodic cycles of the dynamical system {S, P0 }? And can we find a way to discuss the number of fundamentally ‘different’ panels that occur in an orbital picture?

To answer these questions we construct a wonderful new metric space whose elements are, essentially, segments of P0 that are homeomorphic either to panels of the orbital picture or to certain limiting pictures. We will restrict our attention to the case where (X, d) is a compact metric space. But the main ideas are much more generally applicable.

We need a few definitions and concepts first. Let P0 = C(X) have compact domain DP0 X. Then we define Ssegments (P0) to be the space of segments of P0 whose domains are compact and nonempty. Given any segment R of P0 we can form a corresponding segment R Ssegments (P0), which we will call the

Figure 3.42 Orbital picture of a leaf silhouette P0, taken from a photo, with the individual panels shown in different colours. Notice the diversity of visible coloured shapes. In this case the attractor of the IFS is ‘just touching’, in contrast with that used in Figure 3.43, which is ‘overlapping’.

closure of the segment R, by taking the domain of R, D R, to be the closure of the domain of R. We define

R(x) = P0(x) for all x D R = DR.

Then it is easy to see that (Ssegments (P0), d) is a compact metric space, where

d(R1, R2) = dH(X)(DR1 , DR2 ) for all R1, R2 Ssegments (P0)

and where dH(X) denotes the Hausdorff distance function defined in Chapter 1. Let us say that {Rn Ssegments is a nested sequence of segments iff

R1 R2 R3 · · ·

Figure 3.43 The panels have been assigned various colours. The IFS is given by Equation (3.5.16) and is ‘overlapping’ in contrast to that used in Figure 3.42. A close-up of this picture is shown in Figure 3.44. See also Figure 3.45. In the limit of infinite magnification, what shapes might you see?

Then any nested sequence of segments of P0 converges to a unique element of Ssegments (P0), because the corresponding sequence of domains forms a decreasing (nested) sequence of compact sets.

Now let P denote the orbital picture of P0 under the IFS semigroup S{ f1, f2,..., fN }. Then we define a mapping

: Ppanels → Ssegments (P0)

by

In other words, (Qσ ) is the closure of the unique segment of P0 which is transformed to the panel Qσ under a transformation that belongs to the set of transformations { fσ : σ P0 }.

Figure 3.45 Some of the panels in Figure 3.43. Can you identify a panel here whose domain is disconnected? Roughly, how many distinct shapes are shown here?

Furthermore, we have

where S : P0 → P0 denotes the shift transformation.

P r o o f We notice that when σ P0 the definition of is straightforward. It follows at once that

for each given σ P0 . Hence, on applying the transformation