# Multifractals

## Different Probabilities and the f(α) Curve

The figures below are results of applying the random IFS alogrithm with the transformations
 T3(x, y) = (x/2, y/2) + (0, 1/2) T4(x, y) = (x/2, y/2) + (1/2, 1/2) T1(x, y) = (x/2, y/2) T2(x, y) = (x/2, y/2) + (1/2, 0)
with probabilities (p1, p2, p3, p4)
 (0.35,0.30,0.20,0.15) (0.50,0.25,0.20,0.05) (0.60,0.20,0.15,0.05) (0.80,0.10,0.06,0.04)
Here are the corresponding f(α) curves.

Note that because each of the log(pi)/log(ri) are distinct in all four examples, we have seen f(alphamin) = f(alphamax) = 0.
Note that the more uniform the probabilities (the red curve), the narrower the domain of the f(α) curve.
If probabilities are equal, the f(α) curve collapses to a vertical line segment.
Recalling the max and min values of α are given by the max and min values of log(pi)/log(ri), we see
 log(.35)/log(.5) ≈ 1.51,   log(.15)/log(.5) ≈ 2.74 log(.50)/log(.5) = 1.00,   log(.05)/log(.5) ≈ 4.32 log(.60)/log(.5) ≈ 0.74,   log(.05)/log(.5) ≈ 4.32 log(.80)/log(.5) ≈ 0.32,   log(.04)/log(.5) ≈ 4.64