Hausdorff dimension

Formal definition

The formal definition of the Hausdorff dimension is arrived at by defining first the d-dimensional Hausdorff measure, a fractional-dimension analogue of the Lebesgue measure. First, an outer measure is constructed: Let ${\displaystyle X}$ be a metric space. If ${\displaystyle S\subset X}$ and ${\displaystyle d\in [0,\mathrm{\infty })}$,

${\displaystyle H_{\delta }^d(S)=inf\left\{\sum _{i=1}^{\mathrm{\infty }}(\mathrm{diam}{U}_i{)}^d:\bigcup _{i=1}^{\mathrm{\infty }}{U}_i\supseteq S,\mathrm{diam}{U}_i<\delta \right\},}$

where the infimum is taken over all countable covers ${\displaystyle U}$ of ${\displaystyle S}$. The Hausdorff d-dimensional outer measure is then defined as ${\displaystyle {\mathcal{H}}^d(S)=\underset{\delta \to 0}{lim}{H}_{\delta }^d(S)}$, and the restriction of the mapping to measurable sets justifies it as a measure, called the ${\displaystyle d}$-dimensional Hausdorff Measure.

Hausdorff dimension

The Hausdorff dimension ${\displaystyle {\dim }_{\mathrm{H}}(X)}$ of ${\displaystyle X}$ is defined by

${\displaystyle {\dim }_{\mathrm{H}}(X):=inf\{d\ge 0:{\mathcal{H}}^d(X)=0\}.}$

This is the same as the supremum of the set of ${\displaystyle d\in [0,\mathrm{\infty })}$ such that the ${\displaystyle d}$-dimensional Hausdorff measure of ${\displaystyle X}$ is infinite (except that when this latter set of numbers ${\displaystyle d}$ is empty the Hausdorff dimension is zero).

Hausdorff content

The ${\displaystyle d}$-dimensional unlimited Hausdorff content of ${\displaystyle S}$ is defined by

${\displaystyle C_H^d(S):=H_{\mathrm{\infty }}^d(S)=inf\left\{\sum _{k=1}^{\mathrm{\infty }}(\mathrm{diam}{U}_k{)}^d:\bigcup _{k=1}^{\mathrm{\infty }}{U}_k\supseteq S\right\}}$

In other words, ${\displaystyle C_H^d(S)}$ has the construction of the Hausdorff measure where the covering sets are allowed to have arbitrarily large sizes. (Here we use the standard convention that ${\displaystyle inf\varnothing =\mathrm{\infty }}$.) The Hausdorff measure and the Hausdorff content can both be used to determine the dimension of a set, but if the measure of the set is non-zero, their actual values may disagree.

Examples

• Countable sets have Hausdorff dimension 0.
• The Euclidean space ${\displaystyle {\mathbb{R}}^n}$ has Hausdorff dimension ${\displaystyle n}$, and the circle ${\displaystyle S^1}$ has Hausdorff dimension 1.
• Fractals often are spaces whose Hausdorff dimension strictly exceeds the topological dimension. For example, the Cantor set, a zero-dimensional topological space, is a union of two copies of itself, each copy shrunk by a factor 1/3; hence, it can be shown that its Hausdorff dimension is ln(2)/ln(3) ≈ 0.63. The Sierpinski triangle is a union of three copies of itself, each copy shrunk by a factor of 1/2; this yields a Hausdorff dimension of ln(3)/ln(2) ≈ 1.58. These Hausdorff dimensions are related to the "critical exponent" of the Master theorem for solving recurrence relations in the analysis of algorithms.
• Space-filling curves like the Peano curve have the same Hausdorff dimension as the space they fill.
• Trajectories of Brownian motion through space of dimension 2 and above have Hausdorff dimension equal to 2 almost surely.

• Lewis Fry Richardson performed detailed experiments to measure the approximate Hausdorff dimension for various coastlines. His results have varied from 1.02 for the coastline of South Africa to 1.25 for the west coast of Great Britain.

Properties of Hausdorff dimension

Hausdorff dimension and inductive dimension

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 dim_ind(X).

Theorem. Suppose X is non-empty. Then

${\displaystyle {\dim }_{\mathrm{H}\mathrm{a}\mathrm{u}\mathrm{s}}(X)\ge {\dim }_{\mathrm{ind}}(X).}$

Moreover,

${\displaystyle \underset{Y}{inf}{\dim }_{\mathrm{Haus}}(Y)={\dim }_{\mathrm{ind}}(X),}$

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 d_Y of Y is topologically equivalent to d_X.

These results were originally established by Edward Szpilrajn (1907–1976), e.g., see Hurewicz and Wallman, Chapter VII.

Hausdorff dimension and Minkowski dimension

The Minkowski dimension is similar to, and at least as large as, the Hausdorff dimension, and they are equal in many situations. However, the set of rational points in [0, 1] 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.

Hausdorff dimensions and Frostman measures

If there is a measure μ defined on Borel subsets of a metric space X such that μ(X) > 0 and μ(B(x, r)) ≤ r^s holds for some constant s > 0 and for every ball B(x, r) in X, then dim_Haus(X) ≥ s. A partial converse is provided by Frostman's lemma.

Behaviour under unions and products

If ${\displaystyle X=\bigcup _{i\in I}{X}_i}$ is a finite or countable union, then

${\displaystyle {\dim }_{\mathrm{Haus}}(X)=\underset{i\in I}{sup}{\dim }_{\mathrm{Haus}}(X_i).}$

This can be verified directly from the definition.

If X and Y are non-empty metric spaces, then the Hausdorff dimension of their product satisfies

${\displaystyle {\dim }_{\mathrm{Haus}}(X\times Y)\ge {\dim }_{\mathrm{Haus}}(X)+{\dim }_{\mathrm{Haus}}(Y).}$

This inequality can be strict. It is possible to find two sets of dimension 0 whose product has dimension 1. In the opposite direction, it is known that when X and Y are Borel subsets of Rⁿ, the Hausdorff dimension of X × Y is bounded from above by the Hausdorff dimension of X plus the upper packing dimension of Y. These facts are discussed in Mattila (1995).

Self-similar sets

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

${\displaystyle {\psi }_i:{\mathbf{R}}^n\to {\mathbf{R}}^n,i=1,\dots ,m}$

are each a contraction mapping on Rⁿ with contraction constant r_i < 1. Then there is a unique non-empty compact set A such that

${\displaystyle A=\bigcup _{i=1}^m{\psi }_i(A).}$

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.

The open set condition

To determine the dimension of the self-similar set A (in certain cases), we need a technical condition called the open set condition (OSC) on the sequence of contractions ψ_i.

There is an open set V with compact closure, such that

${\displaystyle \bigcup _{i=1}^m{\psi }_i(V)\subseteq V,}$

where the sets in union on the left are pairwise disjoint.

The open set condition is a separation condition that ensures the images ψ_i(V) do not overlap "too much".

Theorem. Suppose the open set condition holds and each ψ_i 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

${\displaystyle \sum _{i=1}^m{r}_i^s=1.}$

The contraction coefficient of a similitude is the magnitude of the dilation.

In general, a set E which is carried onto itself by a mapping

${\displaystyle A\mapsto \psi (A)=\bigcup _{i=1}^m{\psi }_i(A)}$

is self-similar if and only if the intersections satisfy the following condition:

${\displaystyle H^s\left({\psi }_i(E)\cap {\psi }_j(E)\right)=0,}$

where s is the Hausdorff dimension of E and H^s denotes s-dimensional 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.

The Moran Equation

For self-similar sets satisfying the Open set condition, the Hausdorff dimension ${\displaystyle d}$ can be determined using the Moran equation. If a fractal is composed of ${\displaystyle m}$ self-similar copies with contraction ratios ${\displaystyle r_1,r_2,\dots ,r_m}$, the dimension ${\displaystyle d}$ is the unique positive solution to the equation:

${\displaystyle \sum _{i=1}^m{r}_i^d=1}$

This framework is particularly useful for calculating the dimensions of fractals generated by substitution tilings, such as those based on the metallic mean family.