FractionalGeometry-Chap2

So we shall keep track of the locations of the points

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Unformatted text preview: e of Fig. 2.27. But there are more consequences of never applying T4 . For example, no point of Σ lies in int(S34 ), because S34 = T3 (S4 ). Similarly, no point of Σ lies in int(S24 ) and int(S14 ), so we see Σ lies in the shaded region in the middle of Fig. 2.27. Continuing in this way, we see Σ is disjoint from every region with address containing 4. In fact, it is not difficult to see the limit set of Σ is a right isosceles gasket. After all, T1 , T2 , and T3 are the IFS rules for that gasket. In general, when Tj can be applied immediately after Ti we shall say the transition i → j is allowed; otherwise, it is forbidden. These notions extend to longer compositions, and for later applications we emphasize that the composition i → j → k can be forbidden even when both i → j and j → k are allowed. To avoid a thicket of special cases, we make the simplifying assumption that regions of the same address length have disjoint (nonempty) interiors; that is, they intersect at most at boundary points. In m...
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