Convergence of Deterministic IFS

Now we are ready to prove the fundamental convergence theorem for deterministic IFS.

Theorem For any collection {T1, ..., TN} of Euclidean contraction maps,

(1) there is a unique compact set A satisfying

A = T (A).

(2) For any compact set B in the plane,

limk→∞h(T k(B),A) = 0.

Proof The proof splits into four pieces.

First, we construct the set A. Next, we show T (A) = A. That is, A is a fixed point of T. Then we show A is unique. Finally, we show all other compact sets iterate to A under T.

Return to Convergence of determinisitc IFS.