I left off the last post on An Introduction to Fractal Geometry discussing the issue of dimension for fractals. Namely, using the defined box dimension, we saw that the Cantor Set and the Sierpinski Carpet have non-integer dimension. We also saw some issues with the Box Dimension, specifically with density, and thus will seek different notions of dimension in this post, in the form of Minkowski Dimension and Hausdorff Dimension.
As promised, this discussion will be mathematically rigorous. In this post, we’ll first discuss Minkowski Dimension and Minkowski Content, and then Hausdorff Measure and Hausdorff Dimension.
I. Minkowski Dimension and Minkowski Content
Let us prove this, to gain some insight into the matter. In particular, let’s prove the upper and lower Box and Minkowski Dimensions are equal.
In the remainder of this post, let denote the -dimensional Lebesgue measure of some Lebesgue measurable set . Let denote the set
In a similar way to how we defined the box dimension, we define the upper Minkowski Dimension of , , as
and we define the lower Minkowski Dimension of ,, as
if these are equal, we call their common value the Minkowski Dimension of .
At first this definition may seem a bit, but not to worry, it’s actually equivalent to the Box Dimension.
Let us prove this, to gain some insight into the matter. In particular, let’s prove the upper and lower Box and Minkowski Dimensions are equal. Instead of letting denote the number of closed boxes needed to cover , we shall use open balls. These can easily be seen to give equivalent definitions of the Box Dimension.
Note first that , where is the volume of the -dimensional ball. Namely, if we have a cover of by balls, then surely, using the same balls with twice the radius, they will cover . Taking the log of both sides (log monotonicity) and dividing by yields
Taking the lim sup as , adding to the lhs, we have
For the reverse inequality, use the observation that where is the maximal number of disjoint open balls with centre in , known as a packing. Clearly these cover at most . Now, to compare and , observe that since given a maximal packing, if we double the radius of the balls, we will surely cover the entire set. To see this, we need only note that each point not in the packing can not be more than away than all members of the packing, for if it were this would surely contradict the maximality.
Ergo, to prove the equivalence of the dimensions, we just need to prove the reverse inequality, which is done in the same fashion as the first, since .
This proves the equivalence of the Box Dimension and the Minkowski Dimension.
With this notion established, we will now give ourselves some notion of size using this dimension. Namely, we’re going to essentially skew the Lebesgue Measure of some set by a small amount, depending on the Minkowski Dimension. Let denote the topological boundary of .
We define the lower Minkowski Content of a set $latexA&s=2$ to be
and define the upper Minkowski Content as
if these two values are equal, we say the set is Minkowski Measurable, and denote their common value by , called the Minkowski Content. So as I said, all we’ve essentially done is taken the Lebesgue measure and skewed it by a constant depending on the dimension of the set.
The two most important properties of the Minkowski Content and Minkowski Dimension are:
- If exists, .
- All other properties of are immediately inherited by by the proven equivalence.
Notice the particular importance of (1.). When we set out, we were looking for some way to assign size to a set that is non-zero and non-infinite. We have done so with this Minkowski content, provided we measure the set with respect to the Minkowski Dimension. Though, the Minkowski Dimension thus inherits the issues of the Box Dimension by (2.).
We’re going to define the Minkowski Dimension in another way in a bit. The reason we’re going to do this is quite intriguing from a theoretical construction. Recall how the above was constructed; namely, we defined the Minkowski Dimension and then said hey this is the thing that makes the Minkowski Content finite! Well after we define the Hausdorff Measure and Dimension, we’re going to see how we can get the Minkowski Dimension from the Minkowski Content.
II. Hausdorff Measure and Dimension
We’ll start off with the definition of the Hausdorff Measure (which truly is a measure most of the time!), and then will discuss its uses and implications. Again, let .
By a -cover of a set , we mean an at most countable collection of sets such that is covered by their union, and for all , the usual diameter of a set. Define
Note that as , is montonically non-decreasing. To see this, note that the collection of all -covers decreases in size as decreases, and so the infimum must be non-decreasing. Ergo, the limit as tends to exists, and we define the – dimensional Hausdorff Measure as
Let us present some properties of the Hausdorff Measure, most without proof, but whose proofs can be found in Fractal Geometry, Falconer.
- If , .
- If , .
- is countably subadditive.
- , is a measure.
We prove only (4.), (1.) and (2.) are easily shown, and (3.) appears in [Falc].
Let . Let . Let be covers such that
Now, is a -cover for . Thus,
Let and then to conclude countable subadditivity.
One can actually note that the value for which is unique. To see this, suppose . If , then if is a cover that, then, since all such members of the cover have diameter less than ,
and if , in the same manner,
so taking the limit as in both cases, we see that or
Ergo, the graph of with respect to looks like
Thus there is one unique value at which the measure seems to give a good notion of size for some set. But as we said before, the idea is to take a measure in the same dimension as the object. This motivates the Hausdorff Dimension to be defined as this unique value. Namely, the Hasudorff Dimension of a set is given as
We therefore have done the exact opposite as we did in the case of Minkowski Dimension. In the case of Minkowski Dimension, we defined the dimension, and showed it gave a finite measure in the case of Minkowski Content. In the case of Hausdorff Dimension, we defined a plausible measure and then gave the dimension as that value which let the measure be finite.
It should therefore be noted that we may do the same in the case of Minkowski Dimension. If we define the Minkowski Dimension as
we get an equivalent definition of the Minkowski Dimension, and this equivalence is proven on pages 78-79 of Geometry of Sets and Measures in Euclidean Spaces by Mattila.
The Hausdorff Measure behaves fairly well, as indicated by such propositions as
If is a Hölder Continuous function, meaning
for , then
whose proof is also in [Falc].
Let us finish with a final remark on the intuition on the Hausdorff Dimension compared to the Box/Minkowski Dimension.
In particular, we have that
which is the intuitive reason why the Hausdorff Dimension is smaller than the Box Dimension, because it can be more refined.
Originally I was going to discuss Iterated Function Systems and Moran’s Theorem in this post, but they’re extensive topics that deserve their own post, so that’ll be next. We’ve so far seen how to define dimension and size for fractals, but not where they come from or how to construct them. This will be the topic of the next post.