In mathematics, the Hausdorff dimension (also known as the Hausdorff–Besicovitch dimension) is an extended non-negative real number associated to any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space. In particular, the Hausdorff dimension of a single point is zero, the Hausdoff dimension of a line is one, the Hausdoff dimension of the plane is two, etc. There are however many irregular sets that have noninteger Hausdorff dimension. The concept was introduced in 1918 by the mathematician Felix Hausdorff. Many of the technical developments used to compute the Hausdorff dimension for highly irregular sets were obtained by Abram Samoilovitch Besicovitch.
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Intuitively, the dimension of a set (for example, a subset of Euclidean space) is the number of independent parameters needed to describe a point in the set. One mathematical concept which closely models this naive idea is that of topological dimension of a set. For example a point in the plane is described by two independent parameters (the Cartesian coordinates of the point), so in this sense, the plane is two-dimensional. As one would expect, the topological dimension is always a natural number.
However, topological dimension behaves in quite unexpected ways on certain highly irregular sets such as fractals. For example, the Cantor set has topological dimension zero, but in some sense it behaves as a higher dimensional space. Hausdorff dimension gives another way to define dimension, which takes the metric into account.
To define the Hausdorff dimension for X as non-negative real number (that is a number in the half-closed infinite interval [0, ∞)), we first consider the number N(r) of balls of radius at most r required to cover X completely. Clearly, as r gets smaller N(r) gets larger. Very roughly, if N(r) grows in the same way as 1/rd as r is squeezed down towards zero, then we say X has dimension d. In fact the rigorous definition of Hausdorff dimension is somewhat roundabout, as it allows the covering of by balls of different sizes.
For many shapes that are often considered in mathematics, physics and other disciplines, the Hausdorff dimension is an integer. However, sets with non-integer Hausdorff dimension are important and prevalent. Benoît Mandelbrot, a popularizer of fractals, advocates that most shapes found in nature are fractals with non-integer dimension, explaining that "[c]louds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." [1]
There are various closely related notions of possibly fractional dimension. For example box-counting dimension, generalizes the idea of counting the squares of graph paper in which a point of X can be found, as the size of the squares is made smaller and smaller. (The box-counting dimension is also called the Minkowski-Bouligand dimension). The packing dimension is yet another notion of dimension admitting fractional values. These notions (packing dimension, Hausdorff dimension, Minkowski-Bouligand dimension) all give the same value for many shapes, but there are well documented exceptions.
Let be a metric space. If
and
, the
-dimensional Hausdorff content of
is defined by
In other words, is the infimum of the set of numbers
such that there is some (indexed) collection of balls
with
for each
which satisfies
. (One can assume, with no loss of generality, that the index set
is the natural numbers
.) Here, we use the standard convention that inf Ø =∞. The Hausdorff dimension of
is defined by
Equivalently, may be defined as the infimum of the set of
such that the
-dimensional Hausdorff measure of
is zero. This is the same as the supremum of the set of
such that the
-dimensional Hausdorff measure of
is infinite (except that when this latter set of numbers
is empty the Hausdorff dimension is zero).
Let X be an arbitrary separable metric space. There is a topological notion of inductive dimension for X which is defined recursively. It is always an integer (or +∞) and is denoted dimind(X).
Theorem. Suppose X is non-empty. Then
Moreover
where Y ranges over metric spaces homeomorphic to X. In other words, X and Y have the same underlying set of points and the metric dY of Y is topologically equivalent to dX.
These results were originally established by Edward Szpilrajn (1907-1976). The treatment in Chapter VII of the Hurewicz and Wallman reference is particularly recommended.
The Minkowski dimension is similar to the Hausdorff dimension, except that it is not associated with a measure. The Minkowski dimension of a set is at least as large as the Hausdorff dimension. In many situations, they are equal. However, the set of rational points in has Hausdorff dimension zero and Minkowski dimension one. There are also compact sets for which the Minkowski dimension is strictly larger than the Hausdorff dimension.
If there is a measure defined on Borel subsets of a metric space
such that
and
holds for some constant
and for every ball
in
, then
. A partial converse is provided by Frostman's lemma. That article also discusses another useful characterization of the Hausdorff dimension.
If is a finite or countable union, then
This can be verified directly from the definition.
If and
are metric spaces, then the Hausdorff dimension of their product satisfies
An example in which the inequality is strict has been constructed by J. M. Marstrand[3]. It is known that when and
are Borel subsets of
, the Hausdorff dimension of
is bounded from above by the Hausdorff dimension of
plus the upper packing dimension of
. These facts are discussed in Mattila (1995).
Many sets defined by a self-similarity condition have dimensions which can be determined explicitly. Roughly, a set E is self-similar if it is the fixed point of a set-valued transformation ψ, that is ψ(E) = E, although the exact definition is given below.
Theorem. Suppose
are contractive mappings on Rn with contraction constant rj < 1. Then there is a unique non-empty compact set A such that
The theorem follows from Stefan Banach's contractive mapping fixed point theorem applied to the complete metric space of non-empty compact subsets of Rn with the Hausdorff distance[4].
To determine the dimension of the self-similar set A (in certain cases), we need a technical condition called the open set condition on the sequence of contractions ψi which is stated as follows: There is a relatively compact open set V such that
where the sets in union on the left are pairwise disjoint.
Theorem. Suppose the open set condition holds and each ψi is a similitude, that is a composition of an isometry and a dilation around some point. Then the unique fixed point of ψ is a set whose Hausdorff dimension is s where s is the unique solution of
Note that the contraction coefficient of a similitude is the magnitude of the dilation.
We can use this theorem to compute the Hausdorff dimension of the Sierpinski triangle (or sometimes called Sierpinski gasket). Consider three non-collinear points a1, a2, a3 in the plane R² and let ψi be the dilation of ratio 1/2 around ai. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket and the dimension s is the unique solution of
Taking natural logarithms of both sides of the above equation, we can solve for s, that is:
The Sierpinski gasket is self-similar. In general a set E which is a fixed point of a mapping
is self-similar if and only if the intersections
where s is the Hausdorff dimension of E and denotes Hausdorff measure. This is clear in the case of the Sierpinski gasket (the intersections are just points), but is also true more generally:
Theorem. Under the same conditions as the previous theorem, the unique fixed point of ψ is self-similar.