Hausdorff dimension

2020 Mathematics Subject Classification: Primary: 28A [MSN][ZBL]

A numerical invariant of a metric spaces, introduced by F. Hausdorff in [Ha].

Definition

Let $(X,d)$ be a metric space. In what follows, for any subset $E\subset X$, ${\rm diam}\, (E)$ will denote the diameter of $E$.

Definition 1 For any $E\subset X$, any $\delta \in ]0, \infty]$ and any $\alpha\in [0, \infty[$ we consider the outer measure \begin{equation}\label{e:hausdorff_m} \mathcal{H}^\alpha_\delta (E) := \inf \left\{ \sum_{i=1}^\infty ({\rm diam}\, E_i)^\alpha : E\subset \bigcup_i E_i \quad\mbox{and}\quad {\rm diam}\, (E_i) < \delta\right\}\, . \end{equation}

The map $\delta\mapsto \mathcal{H}^\alpha_\delta (E)$ is monotone nonincreasing and thus we can define the Hausdorff $\alpha$-dimensional measure of $E$ as \[ \mathcal{H}^\alpha (E) := \lim_{\delta\downarrow 0} \mathcal{H}^\alpha_\delta (E)\, . \]

Warning Several authors define $\mathcal{H}^\alpha_\delta$ in a way which differs from \ref{e:hausdorff_m} by a multiplicative positive factor $\omega_\alpha$. This factor ensures that $\mathcal{H}^n$ coincides with the Lebesgue (outer) measure when $X$ is the $n$-dimensional euclidean space. In any case the multiplicative factor does not make a difference in the definition of the Hausdorff dimension (see below).

Indeed $\mathcal{H}$ is an outer measure and the procedure above is a classical construction (sometimes called Caratheodory construction, see again Outer measure). The following is a simple consequence of the definition (cp. with Theorem 4.7 of [Ma]).

Theorem 2 For $0\leq s<t<\infty$ and $A\subset X$ we have

• $\mathcal{H}^s (A) < \infty \Rightarrow \mathcal{H}^t (A) = 0$;
• $\mathcal{H}^t (A)>0 \Rightarrow \mathcal{H}^s (A) = \infty$.

The Hausdorff dimension ${\rm dim}_H (A)$ of a subset $A\subset X$ is then defined as

Definition 3 \begin{align*} {\rm dim}_H (A) &= \sup \{s: \mathcal{H}^s (A)> 0\} = \sup \{s: \mathcal{H}^s (A) = \infty\}\\ &=\inf \{t: \mathcal{H}^t (A) = 0\} = \inf \{t: \mathcal{H}^t (A) < \infty\}\, . \end{align*}

Remarks

In the early developments of Geometric measure theory several seminal papers by Besicovitch played a fundamental role in clarifying the concepts of Hausdorff measure and Hausdorff dimension. Therefore the Hausdorff dimension is sometimes called Hausdorff-Besicovitch dimension.

Clearly the Hausdorff dimension is not necessarily an integer. Perhaps the most famous example of a set with non-integer ${\rm dim}_H$ is the Cantor set $C$, for which we have ${\rm dim}_H (C) = (\ln 2)/(\ln 3)$ (cp. with Section 4.10 of [Ma]). The construction in Section 4.13 of [Ma] leads easily to subsets of the euclidean space with arbitrary Hausdorff dimension.

If $(X,d)$ is a metric space and $Y\subset X$, we can then restrict the metric $d$ on $Y\times Y$, consider the resulting metric space and define the Hausdorff dimension of any $E\subset Y$ as a subset of $Y$. It is easy to see that this does not change the result: i.e. the Hausdorff dimension of $E$ as a subset of $Y$ or as a subset of $X$ is the same.

Properties

• If $\psi: X\to Y$ is a Lipschitz map, then the Hausdorff dimension of $\psi (A)$ is at most that of $A$.
• If $A$ is a countable union of sets $A_i$'s, the Hausdorff dimension of $A$ is the supremum of the Hausdorff dimensions of the $A_i$'s.
• The Hausdorff dimension of $A\times B$ is at least the sum of the Hausdorff dimensions of the spaces $A$ and $B$ and it is not necessarily equal to the sum.
• The Hausdorff dimension of a Riemannian manifold corresponds to its topological dimension.

For all these facts we refer to [Ma]. A useful tool to estimate the Hausdorff dimension of Borel subsets of the euclidean space is Frostman's Lemma.

Other definitions of dimension

For general metric spaces one can define the metric dimension (see [HW]), whereas for subsets of the euclidean space one can define the Minkowski dimension and the packing dimension (see [Ma]). For general sets these dimensions do not coincide.

References

[EG]	L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[Fa]	K.J. Falconer, "The geometry of fractal sets" , Cambridge Univ. Press (1985) MR0867284 Zbl 0587.28004
[Fe]	H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York Inc., New York, 1969. MR0257325 Zbl 0874.49001
[Ha]	F. Hausdorff, "Dimension and äusseres Mass" Math. Ann. , 79 (1918) pp. 157–179 MR1511917
[HW]	W. Hurevicz, G. Wallman, "Dimension theory" , Princeton Univ. Press (1948)
[Ma]	P. Mattila, "Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability". Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005