From Osher Doctorow

Kenneth Falconer's (U. Bristol, U.K.) Fractal Geometry, Wiley:
Chichester 1990, p. 27, has for Hausdorff measures (which generalize
length, area, volume) H:

1) H^s(kF) = k^a H^s(F)

where kF = {kx: x in F}, th set F scaled by a factor k. He points out
that for ordinary length, area, volume, scaling properties are well
known, e.g., length when magnified by a factor k is multiplied by k for
a curve, area of plane region is multiplied by k^2m volume of
3-dimensional object by k^3, so (1) has s-dimensional Hausdorff measure
scaling with factor k^s. Note that a Holder condition with exponent
a, namely /f(x) - f(y)/ = c/x - y/^a, with constants c and a 0
implies H^(s/a)(f(F)) = c^(s/a)H^s(F) by Proposition 2.2 of Falconer
(p. 27) and scaling properties are fundamental to the theory of
fractals.

Note that if S1 to Sm are contractions, i.e., /Si(x) - Si(y)/ = c/x -
y/ for all x in a closed subset D of R^n, then a subset F of D is
called invariant for the transformations Si if F = U Si(F) where the
union goes from i 1 to m. Such invariant sets are often fractals, and
Falconer shows that families of contractions, also called iterated
function schemes, define unique non-empty compact invariant sets such
as the Cantor middle thirds set. In fact, Falconer describes a general
construction for fractals via the above apparatus and proves its main
theorem.

Osher Doctorow