Minkowski-Bouligand dimension

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In fractal geometry, the Minkowski-Bouligand dimension or Minkowski dimension is a way of determining the fractal dimension of a set S in a Euclidean space $R n$ , or more generally of a metric space (X,d). This dimension is also known as the packing dimension or, less accurately, the box-counting dimension.

To calculate this dimension for a fractal S, imagine this fractal lying on an evenly-spaced grid, and count how many boxes are required to cover the set. The box-counting dimension is calculated by seeing how this number changes as we make the grid finer.

Suppose that N(ε) is the number of boxes of side length ε required to cover the set. Then the box-counting dimension is defined as:

$\dim_{\rm box}(S) := \lim_{\epsilon \to 0} \frac {\log N(\epsilon)}{\log (1/\epsilon)}$

If the limit does not exist then one must talk about the upper box dimension and the lower box dimension which correspond to the upper limit and lower limit respectively in the expression above. In other words, the box-counting dimension is well defined only if the upper and lower box dimensions are equal. The upper box dimension is sometimes called the entropy dimension, Kolmogorov dimension, Kolmogorov capacity or upper Minkowski dimension, while the lower box dimension is also called the lower Minkowski dimension.

Both are strongly related to the more popular Hausdorff dimension. Only in very specialized applications is it important to distinguish between the three. See below for more details. Also, another measure of fractal dimension is the correlation dimension.

1 Alternative definitions
2 Properties
3 Relations to the Hausdorff dimension
4 See also
5 References

[edit] Alternative definitions

It is possible to define the box dimensions using balls, with either the covering number or the packing number. The covering number $N covering (ε)$ is the minimal number of open balls of radius ε required to cover the fractal, or in other words, such that their union contains the fractal. We can also consider the intrinsic covering number $N' covering (ε)$ , which is defined the same way but with the additional requirement that the centers of the open balls lie inside the set S. The packing number $N packing (ε)$ is the maximal number of disjoint balls of radius ε one can situate such that their centers would be inside the fractal. While N, N_covering, N'_covering and N_packing are not exactly identical, they are closely related, and give rise to identical definitions of the upper and lower box dimensions. This is easy to prove once the following inequalities are proven:

$N'_{\rm covering}(2\epsilon) \leq N_{\rm covering}(\epsilon), N_{\rm packing}(\epsilon) \leq N'_{\rm covering}(\epsilon).$

These, in turn follow with a little effort from the triangle inequality.

The advantage of using balls rather than squares is that this definition generalizes to any metric space. In other words, the box definition is "external" — one needs to assume the fractal is contained in a Euclidean space, and define boxes according to the external structure "imposed" by the containing space. The ball definition is "internal". One can imagine the fractal disconnected from its environment, define balls using the distance between points on the fractal and calculate the dimension (to be more precise, the N_covering definition is also external, but the other two are internal).

The advantage of using boxes is that in many cases N(ε) may be easily calculated explicitly, and that for boxes the covering and packing numbers (defined in an equivalent way) are equal.

The logarithm of the packing and covering numbers are sometimes referred to as entropy numbers, and are somewhat analogous (though not identical) to the concepts of thermodynamic entropy and information-theoretic entropy, in that they measure the amount of "disorder" in the metric space or fractal at scale $ε$ , and also measure how many "bits" one would need to describe an element of the metric space or fractal to accuracy $ε$ .

Another equivalent definition for the box counting dimension, which is again "external", is given by the formula

$\dim_{\rm box}(S) = n - \lim_{r \to 0} \frac{\log {\rm vol}(S_r)}{\log r},$

where for each r>0, the set $S r$ is defined to be the r-neighborhood of S, i.e. the set of all points in $R n$ which are at distance less than r from S (or equivalently, $S r$ is the union of all the open balls of radius r which are centered at a point in S).

[edit] Properties

Both box dimensions are finitely additive, i.e. if { A₁, .... A_n } is a finite collection of sets then

$\dim (A_1 \cup \dotsb \cup A_n) = \max \dim A_1 ,\dots, \dim A_n$

However, they are not countably additive, i.e. this equality does not hold for an infinite sequence of sets. For example, the box dimension of a single point is 0, but the box dimension of the collection of rational numbers in the interval [0, 1] has dimension 1. The Hausdorff dimension by comparison, is countably additive.

An interesting property of the upper box dimension not shared with either the lower box dimension or the Hausdorff dimension is the connection to set addition. If A and B are two sets in a Euclidean space then A + B is formed by taking all the couples of points a,b where a is from A and b is from B and adding a+b. One has

$\dim_{\rm upper\; box}(A+B)\leq \dim_{\rm upper\; box}(A)+\dim_{\rm upper\; box}(B).$

[edit] Relations to the Hausdorff dimension

The box-counting dimension is one of a number of definitions for dimension that can be applied to fractals. For many well behaved fractals all these dimensions are equal. For example, the Hausdorff dimension, lower box dimension, and upper box dimension of the Cantor set are all equal to log(2)/log(3). However, the definitions are not equivalent.

The box dimensions and the Hausdorff dimension are related by the inequality

$\dim_\operatorname{Haus} \leq \dim_\operatorname{lower box} \leq \dim_\operatorname{upper box}$

In general both inequalities may be strict. The upper box dimension may be bigger than the lower box dimension if the fractal has different behavior in different scales. For example, examine the interval [0,1], and examine the set of numbers satisfying the condition

for any n, all the digits between the $2 2 n$ -th digit and the $2 2 n + 1 - 1$ -th digit are zero

The digits in the "odd places", i.e. between $2 2 n + 1$ and $2 2 n + 2 - 1$ are not restricted and may take any value. This fractal has upper box dimension 2/3 and lower box dimension 1/3, a fact which may be easily verified by calculating N(ε) for $\epsilon=10^{-2^n}$ and noting that their values behaves differently for n even and odd. To see that the Hausdorff dimension may be smaller than the lower box dimension, return to the example of the rational numbers in [0, 1] discussed above. The Hausdorff dimension of this set is 0.

Box counting dimension also lacks certain stability properties one would expect of a dimension. For instance, one might expect that adding a countable set would have no effect on the dimension of set. This property fails for box dimension. In fact

$\dim_\operatorname{box} \{0,1,\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots\} = \frac{1}{2}.$