Note that the expression for β(q), |
p1qr1β(q) + p2qr2β(q) + ... + pNqrNβ(q) = 1 |
gives this when q = 0 |
p10r1β(0) + p20r2β(0) + ... + pN0rNβ(0) = 1 |
That is, |
r1β(0) + r2β(0) + ... + rNβ(0) = 1 |
This is just the Moran equation with β(0) = d, the dimension of the attractor of the IFS. |
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Next, note for q = 0 |
f(α(q)) = q⋅α(q) + β(q) |
becomes |
f(α(0)) = 0⋅α(0) + β(0) = β(0) |
That is, f(α(0)) is the dimension of the attractor of the IFS. |
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Finally, the maximum point on the f(α) curve occurs at q = 0. This is most easily seen using calculus, specifically, that at the maximum of the f(α) curve, the derivative df/dα = 0. Because |
f(α) = q⋅α + β(q) |
the derivative condition becomes |
0 = df/dα = (d/dα)(q⋅α + β(q)) = q |
and so the maximum value of the f(α) curve occurs at q = 0. |
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Combining these three steps, we see the highest point on the f(α) curve is the dimension of the attractor of the IFS. |
Return to Multifractals from IFS.