CHAPTER 2
Dimension
Since the notion of a tangent plane is absent in the fractal setting, we need an
alternative way to explain why a curve has dimension 1 and a surface has dimension
2. We will now discuss a construction that works in any compact metric space X.
Let A be a closed subset of X. Fix some radius r 0. Let Nr(A) be the
least number of balls of radius r needed to cover A. If A is a rectifiable curve in
Rn,
it is not hard to show that Nr(A) is of order
r−1.
If A is a surface, Nr(A) is
approximately
r−2.
This suggests the idea to define the dimension of an arbitrary
set as the number d for which Nr(A)
r−d.
Definition. The limit lim
r→0+
log Nr (A)
log 1/r
, when it exists, is called the Minkowski
dimension of A and is denoted by M-dimA.
Even if the limit fails to exist, we still can talk about the upper and the lower
limits, which give us the upper and the lower Minkowski dimensions
M-dimA = lim sup
r→0+
log Nr(A)
log
1
r
and M-dimA = lim inf
r→0+
log Nr(A)
log
1
r
correspondingly.
Since Nr(A) is a non-increasing function of r and since log
1
r
changes very
slowly, instead of the (upper or lower) limit along all values of r we can take the
(upper or lower) limit along any sequence rj such that
log 1
rj+1
log 1
rj
1 as j ∞.
When X = Q is the unit cube in Rn, we can use cubic boxes instead of balls.
More precisely, take some integer p 0 and split Q into pn equal subcubes with
sidelength
1
p
. For A Q, let Hp(A) be the number of subcubes that intersect A.
Definition. The box dimension of A is
B-dimA = lim
p→+∞
log Hp(A)
log p
.
Again, in the cases when the limit fails to exist, we may talk about the upper
and the lower box dimensions instead.
Since each cube with sidelength
1
p
is contained in a ball of radius

n
p
, say, we
conclude that N

n
p
(A) Hp(A). Since each ball of radius
1
p
can intersect at most
some finite number C(n) cubes of size
1
p
, we have
Hp(A) C(n) N
1
p
(A).
Taking the logarithm, dividing by log p, and passing to the limit as p +∞, we
see that the box dimension we just defined coincides with the Minkowski dimension
and the same is true for the upper and lower versions.
11
http://dx.doi.org/10.1090/cbms/120/02
Previous Page Next Page