THEORY AND APPLICATIONS OF FRACTAL TOPS MICHAEL BARNSLEY Abstract. We consider an iterated function system (IFS) of one-to-one con- tractive maps on a compact metric space. We define the top of an IFS; define an associated symbolic dynamical system; present and explain a fast algorithm for computing the top; describe an example in one dimension with a rich his- torygoingbacktoworkofA.Rényi[Representations for Real Numbers and Their Ergodic Properties, Acta Math. Acad. Sci. Hung.,8 (1957), pp. 477- 493]; and we show how tops may be used to help to model and render synthetic pictures in applications in computer graphics. 1. Introduction It is well-known that an iterated function system (IFS) of 1-1 contractive maps, mapping a compact metric space into itself, possesses a set attractor and various invariant measures. But it also possesses another type of invariant object which we call a top. One application of tops is to modelling and rendering new families of synthetic pictures in computer graphics. Another application is to information theory and data compression. Tops are mathematically fascinating because they have a rich symbolic dynam- ics structure, they support intricate Markov chains, and they provide examples of IFS with place-dependent probabilities in a regime where not much research has takenplace.Inthesenoteswe(i)define the top of an IFS; (ii) define an associated symbolic dynamical system; (iii) present and explain a new fast algorithm for com- puting the top, based on the dynamical structure; (iv) describe an example in one dimension with a rich history; (v) explain and demonstrate that tops can be used to define and render beautiful pictures. This work was supported by the Australian Research Council. 2. The Top of an IFS Let an iterated function system (IFS) be denoted (2.1) := X; w0, ..., wN 1 . W { − } This consists of a finite of sequence of one-to-one contraction mappings (2.2) wn : X X,n=0, 2,...,N 1 → − acting on the compact metric space (2.3) (X,d) with metric d so that for some (2.4) 0 l<1 ≤ Date: March 9, 2005. 1 2 MICHAEL BARNSLEY we have (2.5) d(wn(x),wn(y)) l d(x, y) ≤ · for all x, y X. ∈ Let A denote the attractor of the IFS, that is A X istheuniquenon-empty compact set such that ⊂ A = wn(A). n [ Let the associated code space be denoted by Ω = Ω 0,1,...,N 1 . This is the space { − } of infinite sequences of symbols σi ∞ belongingtothealphabet 0, 1,...,N 1 { }i=1 { − } with the discrete product topology. We will also write σ = σ1σ2σ3... Ω to denote th ∈ a typical element of Ω, and we will write ωk to denote the k element of the sequence ω Ω.WeordertheelementsofΩ according to ∈ σ<ωiff σk <ωk where k is the least index for which σk = ωk. Let 6 φ : Ω A → denote the associated continuous addressing map from code space onto the attrac- tor of the IFS. We note that the set of addresses of a point x A,defined to be 1 ∈ φ− (x), is compact and so possesses a unique largest element. We denote this value by τ(x).Thatis,τ : A Ω is defined by → τ(x)=max σ Ω : φ(σ)=x . { ∈ } We call τ the tops function oftheIFS.WealsocallGτ := (x, τ(x)) : x A the graph of the top of the IFS or simply the top of the IFS. { ∈ } The top of an IFS may be described as follows: consider the lifted IFS X Ω : W0,W1,...,WN 1 { × − } where Wn(x, σ)=(wn(x),nσ) where, for the avoidance of any doubt, nσ := ω where ω1 = n and ωn+1 = σn for n =0, 1, ..., N 1. Let the unique attractor of this IFS be denoted by A. Then the − projection of A on the X-direction is X,andintheΩ-direction it is Ω. The top of the original IFS is related to A according to: b b Gτ = (x, σ) A :(x, ω) A = ω σ . { b ∈ ∈ ⇒ ≤ } This latter formulation is useful because we can use the chaos game algorithm (also called a Markov Chain Monte Carlob (MCMC)b algorithm or a random iteration algorithm) to compute approximations to A and hence to Gτ . Accordingtothis method we select a sequence of symbols b σ1σ2σ3... 1, 2,...,N ∞ ∈ { } with probability pn > 0 for the choice σk = n, independent of all of the other choices. We also select X0 X and let ∈ Xn+1 = Wσn+1 (Xn) for n =0, 1, 2, ... Then, almost always, lim Xn : n = k,k +1, ... = A k →∞ { } b FRACTAL TOPS 3 where S denotes the closure of the set S. This algorithm provides in many cases asimpleefficient fast method to compute approximations to the attractor of an 2 IFS, for example when X = ¤,acompactsubsetofR . By keeping track of points which, for each approximate value of x X, have the greatest code space value, we ∈ can compute approximations to Gτ . We illustrate this approach in the following example which we continue in Section 6. Example 1. Consider the IFS (2.6) [0, 1] R; w0(x)=αx, w2(x)=αx +(1 α) { ⊂ − } We are interested in the case where 1 <α<1, 2 which we refer to as "overlapping" because w0([0, 1]) w1([0, 1]) contains a non- empty open set. The two maps of the IFS are illustrated∩ in Figure 1. In Figure 2 Figure 1. Graphs of the two transformations of the overlapping IFS in Example 1. See also Figure 2. **tifs1.gif we show the attractor A of the associated lifted IFS, and upon this attractor we have indicated the top with some red squiggles. Figure 2 was computed using random iteration: we have representedb points in code space by their binary expansions which are interpreted as points in [0, 1]. Since the invariant measure of both the IFS 4 MICHAEL BARNSLEY Figure 2. The attractor of the IFS in Equation 2.7. This repre- sents the attractor A of the lifted IFS corresponding to Equation 2.6. The top of the IFS is indicated in red. The visible part of the "x-axis" representsb the real interval [0, 1] and the visible part of the "y-axis" represents the code space Ω between the points 000000000.... and 111111111..... **graph2.gif and the lifted IFS contain no atoms, the information lost by this representation is irrelevant to pictures. Accordingly, the actual IFS used to compute Figure 2 is 1 1 1 (2.7) [0, 1] [0, 1] R; W0(x, y)=(αx, y),W2(x, y)=(αx +(1 α), y + ) { × ⊂ 2 − 2 2 } 2 with α = 3 . 3. Application of Tops to Computer Graphics Here we introduce the application of tops to computer graphics. There is a great deal more to say about this, but to serve as motivation as well as to provide an excellent method for graphing fractal tops, we explain the basic idea here. 2 A picture function is a mapping P : DP R C where C is a colour space, 3 3 ⊂ → for example C =[0, 255] R .ThedomainDP is typically a rectangular subset of ⊂ R2:weoftentake 2 DP = := (x, y) R :0 x, y 1 . ¤ { ∈ ≤ ≤ } The domain of a picture function is an important part of its definition; for example asegmentofapicturemaybeusedtodefine a picture function. A picture in the usual sense may then be thought of as the graph of a picture function. But we will use the concepts of picture, picture function, and graph of a picture function FRACTAL TOPS 5 interchangeably. We do not discuss here the important questions of how such functions arise in image science, for example, nor about the relationship between such abstract objects and real world pictures. Here we assume that given a picture function,wehavesomeprocessbywhichwecanrenderittomakepictureswhich may be printed, viewed on computer screens, etc. This is far from a simple matter in general. Let two IFS’s := ¤; w0, ..., wN 1 and := ¤; w0,...,wN 1 W { − } W { − } and a picture function f e e P : C ¤ → be given. Let A denote the attractor of the IFS and let A denote the attractor W of the IFS .Let e W τ : A Ω e → denote thef tops function for .Let W φ : Ω A → ⊂ ¤ denote the addressing function for the IFS .Thenwedefine a new picture e e function W P :A Cf → by P =P φ τ. ◦ ◦ This is the unique picture function defined by the IFS’s , ,andthepicture W W P. We say that it has been producede bye tops + colour stealing. We think in this way: colours are "stolen" from the picture P to "paint" codef space; that is, wee make a code space picture, that is the function P φ : Ω C,whichwethen use together with top of to paint the attractoreA. ◦ → W Notice the following points. (i) Picture functions havee e properties that are deter- mined by their source; digital pictures of natural scenes such as clouds and sky, fields of flowers and grasses, seascapes, thick foliage, etc. all have their own distinctive palettes, relationships between colour and position, "continuity" and "discontinu- ity" properties, and so on. (ii) Addressing functions are continuous. (iii) Tops functions have their own special properties; for example they are continuous when the associated IFS is totally disconnected, and they contain the geometry of the un- derlying IFS attractor A plus much more, and so may have certain self-similarities and, assuming the IFS are built from low information content transformations such as similitudes, possess their own harmonies. Thus, the picture functions produced by tops plus colour stealing may define pictures which are interesting to look at, carrying a natural palette, possessing certain continuities and discontinuities, and also certain self-similarities.