FractionalGeometry-Chap2

FractionalGeometry-Chap2

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Unformatted text preview: 1 , T2 , and T3 are r1 = 1/4, and r2 = r3 = 1/8. Consequently, h(T (A), T (B )) = 1/4 can be read as h(T (A), T (B )) = (1/4) · h(A, B ) = max{r1 , r2 , r3 } · h(A, B ). Is this true in all cases? What do you think? Proposition 2.3.2 For all A, B ∈ K(R2 ), h(T (A), T (B )) ≤ r · h(A, B ) where r = max{r1 , . . . , rn }, and ri is the d-contraction factor of Ti . Now we are ready for the fundamental theorem of IFS. Theorem 2.3.1 (a) For any collection {T1 , ..., Tn } of contraction maps, Ti : R2 → R2 , there is a unique, nonempty compact set A ⊂ R2 satisfying A = T1 (A) ∪ ... ∪ Tn (A) = T (A) (b) Moreover, for any compact set D ⊂ R2 , lim h(T k (D), A) = 0. k→∞ This last result, showing that limk→∞ h(T k (D), A) = 0, is the reason for calling A the attractor of the IFS {T1 , ..., Tn }. When the attractor A is a fractal, the sets T (D) are called prefractals. Generating the sequence D, T (D), T 2 (D), ... is called producing the limiting fractal A by the deterministic IFS algorithm....
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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