Hausdorff dimension

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

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

This is the same as the supremum of the set of ${\displaystyle d\in [0,\infty )}$ 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).

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

${\displaystyle C_{H}^{d}(S):=H_{\infty }^{d}(S)=\inf \left\{\sum _{k=1}^{\infty }(\operatorname {diam} U_{k})^{d}:\bigcup _{k=1}^{\infty }U_{k}\supseteq S\right\}}$

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 ${\displaystyle \inf \varnothing =\infty }$).[7] 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.

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

${\displaystyle \dim _{\mathrm {Haus} }(X)\geq \dim _{\operatorname {ind} }(X).}$

Moreover,

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

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.

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.

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.[11]

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

${\displaystyle \dim _{\operatorname {Haus} }(X)=\sup _{i\in I}\dim _{\operatorname {Haus} }(X_{i}).}$

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[12]

${\displaystyle \dim _{\operatorname {Haus} }(X\times Y)\geq \dim _{\operatorname {Haus} }(X)+\dim _{\operatorname {Haus} }(Y).}$

This inequality can be strict. It is possible to find two sets of dimension 0 whose product has dimension 1.[13] 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).

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 dilatation 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[15]

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

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

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.