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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 ρ_ω 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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