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. | ||||
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Return to Convergence of determinisitc IFS.