Random Fractals and the Stock Market

Unifractal Cartoons

Dimensions of Unifractal Graphs

Here we sketch the relation between the Holder exponent H of a function and the box-counting dimension of the graph of the function.

To simplify the calculation, suppose the function f(x) is defined for 0 ≤ x ≤ 1.

Here are the basic steps of the argument.

(1) Divide 0 ≤ x ≤ 1 into intervals of width r.

(2) Above each of these intervals, mark off a column of width r.

(3) In this situation the condition dY = (dt)^H means in each of these columns, the graph of f(x) passes through a height of about r^H.

(4) So the number of boxes needed to cover the part of the graph in that column is about (height of graph)/(height of box) = r^H/r = r^H-1.

(5) The number of these columns is about 1/r.

(6) The total number of boxes of side r needed to cover the graph is r^H-1⋅(1/r) = r^H-2.

(7) Then the box-counting dimension of the graph is about Log(r^H-2)/Log(1/r) = 2-H.

Return to Dimensions of UniFractal Cartoons.