Brownian Motion on Nested Fractals by Tom Lindstrom

By Tom Lindstrom

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11 Proposition. ,s 1 p such that . Then there is a strict 1-walk x=s , y=s 1 p and s $F for all i, l

A) n be the result of applying this times. 1 Theorem. set Assume that k EC R such that set, the iterates v>1. Then there is a unique compact E=¥(E) . Moreover, if n ¥ (A) converge to E k AC R is any compact in the Hausdorff metric. To understand the relevance of this result, let us go back to the snowflake fractal in the previous section. If we let 4>, , . , c|>7 be affine maps sending the hexagon in Figure 5a) to each 25 T O M LINDSTR0M 26 one of the smaller hexagons in Figure 5 b ) , it's easy to check that the only fixed point of the system fractal S.

L l+l l l+l To prove that p is continuous, we shall need the following simple lemma. 3 Lemma. ,p )6j) of basic transition probabilities is used in the construction of Bi. TOM LINDSTR0M -1 P r o o f : Let k = ( J - , m. ) . ,p is d e c r e a s i n g , r 1=. p. < ( . Z 1=1 and hence p, >k 1 1 1=1 whenever us that if we just choose F is connected to than N, and thus if l m. ,p ) € y. Proposition IV. 11 N large enough, any element u tells in F- {x } by a strict 1-walk of length less p is the multiplicity of E, we get P^{w: Tu(u))>N}0-(k/p)N By induction P"{GJ: for all Tu(a))>mN}<(l-(k/p)N)m m£ IN and from this the lemma follows easily.

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