Multifractals

Different Probabilities, Another Example

In this example, we introduce more variability in the probabilities:

p1 = 0.2, p2 = 0.25, p3 = 0.25, and p4 = 0.3.

Among other things, the number of values of the probabilities of regions increases more rapidly.

Smaller regions have smaller probabilities; if these graphs weren't rescalled vertically they would appear to become closer and closer to a flat surface of height 0. Click here for an animation of the first four iterates, all drawn to the same vertical scale.

For each region we expect that

prob scales as (side length)some power

So instead of letting the height of the graph represent the probability of the region, now we assign height Log(prob)/Log(side length) to the region.

Because the probability measures the fraction of the points that occupy a region, we think of this ratio as a dimension.

Being viewed at the resolution of the side length of the region, this is a coarse Holder exponent; it is also called the coarse dimension.

Compared with the pictures of the probabilities, perhaps the most noticeable feature is the graphs of the coarse Holder exonent curve in the opposite direction.

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