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Page 204
boxes containing 2/27 percentage of point population is 3. Since n and k tend to infinity, it is more convenient to define a value ω, where ω=k/n or, alternatively, k=nω. Thus Equations (5.17) and (5.18) each yield:
(5.19) |
and
(5.20) |
Using Equation (5.5), the fractal dimension of a specific sub-box containing ρω percentage points can be calculated in terms of:
(5.21) |
where the sub-box length l is defined as 2−n, and N(ω) is derived from:
(5.22) |
Details of the derivation of N(ω) are given by Tso and Mather (1999). The fractal dimension D(ω) is derived from:
(5.23) |
Since
(5.24) |
one finally obtains
(5.25) |
The Lipschitz-Holder exponent a is more generally employed rather than ω. The value of α a is given by:
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