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Unformatted text preview: The translations of the IFS guarantee that
(B3 )1/4 ∩ A1 = ∅, for example. In Fig. 2.22 in light grey we see the ǫ = 1/4nbhds of A1 , A2 , and A3 . Note that B ⊆ (Ai )1/4 for i = 1, 2, 3. In darker grey
we see the ǫ = 1/8nbhds of A1 , A2 , and A3 . Note that B ⊆ (Ai )1/8 for i = 2, 3,
but not for i = 1. Similar observations hold for A and the Bi . Consequently,
the larger Euclidean contraction factor, 1/4, of the Ti determines the Hausdorﬀ
contraction factor of the collage map. That is, h(T (A), T (B )) = 1/4.
Before beginning our proof of convergence of IFS, ﬁrst we recall some facts
from analysis. A sequence {A1 , A2 , . . . } of sets is nested if A1 ⊇ A2 ⊇ A3 ⊇ · · · .
Standard results about continuous functions and compact sets include 2.3. CONVERGENCE OF DETERMINISTIC IFS 45 A B Figure 2.22: Relating Euclidean and Hausdorﬀ contraction factors.
Proposition 2.3.1 (1) A continuous function takes compact sets to compact
sets.
(2) Contractions are continuous.
(3) The union of a ﬁnite collection of compact sets is compact.
(4) The intersection of a nested sequen...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.
 Fall '14
 AmandaFolsom
 Geometry, Fractal Geometry, Limits, The Land

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