FRACTAL MEASURES

Let us introduce the fractal dimension of an object. We will refer to sets in the plane (2-D space). When we talk about the fractal dimension of a curve we mean the fractal dimension of the set of points that belong to the curve.

An E-ball covering of a set S, is a collections of circles of radius E (balls) such that S is contained in the unions of the circles. For a given radius E let N(E) be the smallest number of circles that are necessary to cover the set, and let A(E)=N(E) E2 be the "area" of the covering (a factor Pi has been dropped). Compute the limit

limE→0 ( A(E) / E2 ) = const.   E-d

The exponent d is the fractal dimension of the set S.

For example a straight line has fractal dimension 1; a rectangle has fractal dimension 2. There are other interesting figures ("fractal curves") with fractional fractal dimension, for example the famous Mandelbrot sets.

The metric entropy is the limit, as the radius E goes to zero, of

HE = ln N(E)

The fractal dimension is the coefficient of ln(1/E), as E goes to zero. This definition is more general and applies to any compact set in a Banach space.

Fractal characteristic of a function

Given a function f(x) and fixed a spacing T on the x-axis, we define the Holder exponent by considering the expectation value

a(f,T) = E[ log(|f(x) - f(x-T)|) / log(T) ]

Taking the exponential, A(f,T) = exp(a(f,T)), we have

A(fk,T) = k A(f,kT)

where fk(x)=f(kx).

... TO CONTINUE.

Marco Corvi - 2000