| 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. |
| 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. |
| 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. |
| 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.