| The general setting for deterministic IFS is
contraction maps |
| Ti:R2 → R2 |
| for i = 1, ..., N. |
| The collage map T is defined on the set
K(R2) of compact subsets of R2 by |
| T (C) = T1(C) ∪ ... ∪ TN(C), |
| where Ti(C) = {Ti(x, y): (x, y) in C}. |
| We show there is a unique compact set A satisfying |
| T (A) = A |
| Moreover, for any compact set B, |
| limk → ∞T k(B) = A |
| where we must describe how this limit makes sense. |
| This whole process has several steps. |
| First, we need a way to measure distances between compact subsets of
the plane. As a preliminary step, we introduce ε-thickenings
of compact subsets. |
| Using ε-thickenings, we define the Hausdorff metric
h on compact subsets of the plane. |
| Here is an example |
| Now we relate the Euclidean contraction factor of Ti to the
Hausdorff contraction factor of Ti. |
| Next we show if each Ti is a Eucidean contraction, then
T is a Hausdorff contraction. |
| From this we show T has a unique fixed point A, and
for any compact set B, limk → ∞h(T k(B), A) = 0. |
| Here is an exercise. |
|