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 |
dloc(x,y) = limn -> infinity Log(Prob(i1...in))/Log(2-n) |
where Prob(i1...in) is the probability pr(i1)* ... *pr(in), if |
The value for a square of finite length address is called the coarse Holder exponent.
So the local Holder exponent of a point |
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 |
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
|
(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
|
At least in this example, sets Ealpha for the lowest and highest values of alpha
reduce to points in the limit, hence have dimension |
![]() |
This result is derived under more general conditions in a later section. |
Return to Multifractals.