Iterated Function Systems

Following M. Barnsley [2], under fractals we mean compact subsets of a complete metric space.

A deterministic Iterated Function System (IFS) [2] is a structure

F = ((X,d), f₁, f₂, ..., f_N), (1)

where (X,d) is a complete metric space, with metric d, $f_i:X \rightarrow X$ - a contiguous function $\forall i$. An IFS F induces an operator $F:{\cal H}(X) \rightarrow {\cal H}(X)$, where ${\cal H}(X)$ - the space of all compact subsets of X:

$$F = \bigcup_{i=1}^q f_i.$$ (2)

The metric space $({\cal H}(X),h)$, with Hausdorff metric h, is also complete. When functions f_i are contractions, the IFS is hyperbolic. In this case, there is a compact subset A_F of X which is the fixed point of the operator F:

$$\exists A_F \in {\cal H}(X): \; A_F = F(A_F).$$ (3)

Such a set is called the attractor of the IFS F. The pair $({\cal H}(X), F)$ is a set dynamical system whose attractive point is a set (rather than a point) - the attractor A_F. Realizing the dynamics of such a dynamical system, we can build a fractal set.

The Hausdorff metric is defined as follows:

$$h(A,B)=\max(d_{\rm s}(A,B),d_s(B,A)), \qquad A,B \in {\cal H}(X),$$ (4)

where

$$d_{\rm s}(A,B)=\max_{x \in A} \min_{y \in B} (d(x,y)).$$

We are dealing here with hyperbolic IFSs with all f_i affine transformations.