FractionalGeometry-Chap2

# Why do we exclude only the interior of s4 because

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Unformatted text preview: 3/5. Note the eﬀect on the apparent density of the various parts of the gasket. Figure 2.25: Three renderings of the gasket by the random algorithm. Left: Σ contains 500 points. Middle: 5000. Right: the random algorithm with diﬀerent probabilities A natural question is what happens if the initial point (x0 , y0 ) is not a point of the attractor A? To answer this, we must introduce another topological construction, the limit set Λ(X ) of a set X . A point y belongs to Λ(X ) if for every ǫ &gt; 0, there is x ∈ X , satisfying 0 &lt; d(x, y ) &lt; ǫ. These points y are the limit points of the set X . For instance, it is easy to see the limit set of the interval (0, 1) ⊂ R is the interval [0, 1], and the limit set of {1/n : n = 1, 2, 3, . . . } ⊂ R is 0. The ﬁrst instance points out similarities between the limit set and the closure; the second instance points out the main diﬀerence: isolated points of X belong to the closure, but not to the limit set, as guaranteed by the condition x = y in the deﬁniti...
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## This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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