Multifractals

Local Holder Exponents

Taking limits as the side length of the regions go to zero, the coarse Holder exponent can be refined to the local Holder exponent (or roughness) at (x, y) is

dloc(x,y) = limn -> infinity Log(Prob(i1...in))/Log(2-n)

where Prob(i1...in) is the probability pr(i1)* ... *pr(in), if (x,y) lies in the square with address i1...in.

The value for a square of finite length address is called the coarse Holder exponent. So the local Holder exponent of a point (x, y) is the limit as N -> infinity of the coarse Holder exponents of the length N address squares containing (x, y).

Now define

Ealpha = {(x, y): dloc(x, y) = alpha},

the collection of all points of the fractal having local Holder exponent alpha.

As alpha takes on all values of the local Holder exponent, we decompose the fractal into these sets Ealpha.

Here are examples, Ealpha (alpha = column height) for the lowest value of alpha (on the left), two intermediate values, and the highest value.

Click here for an animation scanning through all the values of alpha, from lowest to highest, resolved to boxes have side length 1/24.

Because each local Holder exponent alpha is the exponent for a power law, a multifractal is a process exhibiting scaling for a range of different power laws.

The multifractal structure is revealed by plotting dim(Ealpha) as a function of alpha.

(In general, a dimension more subtle than the box-counting dimension must be used. We ignore this complication here.)

This graph is called the f(alpha) curve.

Here is the f(alpha) curve for the example with p1 = 0.2, p2 = p3 = 0.25, and p4 = 0.3.

At least in this example, sets Ealpha for the lowest and highest values of alpha reduce to points in the limit, hence have dimension f(alpha) = 0. This is represented in the left and right endpoints of the curve lying on the x-axis.

This result is derived under more general conditions in a later section.

Return to Multifractals.