VSFCF - CVM 1.1

# Mathematical Background

We'll consider finite collections {f0,f1,...,fn-1} of similarities of Rm (usually with m=2) that are contractions - for each fk there is a shrinking factor sk between 0 and 1 such that for all x and y we have |fk(x)-fk(y)|=sk|x-y|. In fact, all our functions will have the same contraction, s=sk. This is an example of an iterated function system (IFS) and must have a unique compact attractor: a set A with the property that it equals the union of all the fk(A). (For details, see [Ba].) In addition, our IFS's will always satisfy the open set condition (also called Moran's open set condition: see [Sag, p. 158], [Ed, p. 160], [Hu, p. 735], or [Ba, p. 129]); there will be an open set U such that:
1. For k different from j, fk(U) and fj(U) are disjoint, and
2. For all k, fk(U) is a subset of U.
(The open set condition prevents "overlap" among the functions of the IFS.) In this situation one defines the similarity dimension of the attractor A of our system of n functions with contraction constant s to be the unique number D such that
n s D=1
So D = -log(n)/log(s). To see that this definition of dimension makes sense, note these examples:
1. In R1, A=[0,1], f0(x)=x/2, f1(x)=1-x/2, U=(0,1), s=1/2, 2·s1=1, so D=1 and a segment has similarity dimension 1.
2. In R2, A=[0,1]×[0,1], and using complex variable notation, f0(z)=z/2, f1(z)=(z+1)/2, f2(z)=(z+i)/2, f3(z)=(z+1+i)/2, U=(0,1)×(0,1), s=1/2, 4·s2=1, so D=2 and a square region has similarity dimension 2.
3. In R3, we cut a cube (A) into 8 congruent cubelets with half the edge length, so n=8, s=1/2, 8·s3=1, so D=3 and a cube has similarity dimension 3.
This is perhaps the easiest definition of dimension. Its drawback is that it only applies to sets defined as attractors in this way. However, for such sets, it gives the same value as the very general concept of Hausdorff dimension ([Hu]).

In all our examples, U can be taken to be the interior of the convex hull of A. For example, for the snowflake curve, U will be the inside of a triangle, while for the Hilbert curve U will be the inside of a square.

The Cantor set will play a role in many of our constructions. The standard Cantor set, C, can be constructed by starting with the unit interval, C0=[0,1], deleting the open middle third to get C1 the union of [0,1/3] and [2/3,1], deleting the open middle thirds of these two intervals to get C2, etc. Then C is the intersection of all these Ck. All homeomorphs of C are called Cantor sets and are characterized as nonempty metric, compact, totally disconnected, perfect (no isolated points) sets ([HY, p. 99]). In particular, as we will use here, instead of deleting thirds we can delete any part between 0 and 1 and still get a Cantor set. And the Cartesian product of two of these Cantor sets will be a Cantor set. Figure 5 in Section 5 shows such Cantor sets.

Communications in Visual Mathematics, vol 1, no 1, August 1998.