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A numerical invariant of metric spaces, introduced by F. Hausdorff in [[#References|[1]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h0466901.png" /> be a metric space. For real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h0466902.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h0466903.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h0466904.png" />, where the lower bound is taken over all countable coverings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h0466905.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h0466906.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h0466907.png" />. The Hausdorff dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h0466908.png" /> is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h0466909.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h04669010.png" />. The number thus defined depends on the metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h04669011.png" /> (on this, see also [[Metric dimension|Metric dimension]]) and is, generally speaking, not an integer (for example, the Hausdorff dimension of the [[Cantor set|Cantor set]] is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h04669012.png" />). A topological invariant is, for example, the lower bound of the Hausdorff dimensions over all metrics on a given topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h04669013.png" />; when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h04669014.png" /> is compact, this invariant is the same as the [[Lebesgue dimension|Lebesgue dimension]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h04669015.png" />.
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{{MSC|28A}}
  
====References====
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[[Category:Classical measure theory]]
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Hausdorff,  "Dimension and äusseres Mass"  ''Math. Ann.'' , '''79'''  (1918)  pp. 157–179  {{MR|1511917}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Hurevicz,  G. Wallman,  "Dimension theory" , Princeton Univ. Press  (1948)  {{MR|}} {{ZBL|}} </TD></TR></table>
 
  
 +
{{TEX|done}}
  
 +
A numerical invariant of a metric spaces, introduced by F. Hausdorff in {{Cite|Ha}}.
  
====Comments====
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===Defininition===
The limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h04669016.png" /> of the non-decreasing set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h04669017.png" /> is called the Hausdorff measure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h04669018.png" /> in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h04669019.png" />. There is then a unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h04669020.png" /> in the extended real line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h04669021.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h04669022.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h04669023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h04669024.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h04669025.png" />. This real number is the Hausdorff dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046690/h04669026.png" />. It is also called the Hausdorff–Besicovitch dimension.
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Let $(X,d)$ be a metric space. In what follows, for any subset $E\subset X$, ${\rm diam}\, (E)$ will denote the [[Diameter|diameter]] of $E$.  
  
See also (the editorial comments to) [[Hausdorff measure|Hausdorff measure]] for more material and references. The Hausdorff dimension is a basic notion in the theory of [[Fractals|fractals]], cf. also [[#References|[a1]]].
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'''Definition 1'''
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For any $E\subset X$, any $\delta \in ]0, \infty]$ and any $\alpha\in [0, \infty[$ we consider the [[Outer measure|outer measure]]  
 +
\begin{equation}\label{e:hausdorff_m}
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\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}
  
====References====
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The map $\delta\mapsto \mathcal{H}^\alpha_\delta (E)$ is monotone nonincreasing and thus we can define the [[Hausdorff measure|Hausdorff $\alpha$-dimensional measure]] of $E$ as
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"K.J. Falconer,  "The geometry of fractal sets" , Cambridge Univ. Press  (1985)  {{MR|0867284}} {{ZBL|0587.28004}} </TD></TR></table>
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\[
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\mathcal{H}^\alpha (E) := \lim_{\delta\downarrow 0} \mathcal{H}^\alpha_\delta (E)\, .
 +
\]
 +
 
 +
'''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 measure|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|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 {{Cite|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 ${\rm dim}_A$ is the [[Cantor set]] $C$, for which we have ${\rm dim}_H (C) = (\ln 3)/(\ln 2)$ (cp. with Section 4.10 of {{Cite|Ma}}). The construction in Section 4.13 lead 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 {{Cite|Ma}}. A useful tool to estimate the Hausdorff dimension of [[Borel set|Borel subsets]] of the euclidean space is Frostman's Lemma
 +
(see [[Hausdorff measure]]).
 +
 
 +
===Other definitions of dimension===
 +
For general metric spaces one can define the [[Metric dimension|metric dimension]] (see {{Cite|HW}}, whereas for subsets of the euclidean space one can define the Minkowski dimension and the packing dimension (see {{Cite|Ma}}). For general sets these dimensions do not coincide.
 +
 
 +
===References===
 +
{|
 +
|-
 +
|valign="top"|{{Ref|EG}}||    L.C. Evans, R.F. Gariepy, "Measure theory  and fine properties of    functions" Studies in Advanced Mathematics. CRC  Press, Boca Raton, FL,    1992. {{MR|1158660}} {{ZBL|0804.2800}}
 +
|-
 +
|valign="top"|{{Ref|Fa}}|| K.J. Falconer,  "The geometry of fractal sets" , Cambridge Univ. Press  (1985)  {{MR|0867284}} {{ZBL|0587.28004}}
 +
|-
 +
|valign="top"|{{Ref|Fe}}||    H. Federer, "Geometric measure  theory". Volume 153 of Die  Grundlehren  der mathematischen  Wissenschaften. Springer-Verlag New  York Inc., New  York, 1969.  {{MR|0257325}} {{ZBL|0874.49001}}
 +
|-
 +
|valign="top"|{{Ref|Ha}}|| F. Hausdorff,  "Dimension and äusseres Mass"  ''Math. Ann.'' , '''79'''  (1918)  pp. 157–179  {{MR|1511917}} {{ZBL|}}
 +
|-
 +
|valign="top"|{{Ref|HW}}|| W. Hurevicz,  G. Wallman,  "Dimension theory" , Princeton Univ. Press  (1948)  {{MR|}} {{ZBL|}}
 +
|-
 +
|valign="top"|{{Ref|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.  {{MR|1333890}} {{ZBL|0911.28005}}
 +
|-
 +
|}

Revision as of 17:53, 2 October 2012

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

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

Defininition

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 ${\rm dim}_A$ is the Cantor set $C$, for which we have ${\rm dim}_H (C) = (\ln 3)/(\ln 2)$ (cp. with Section 4.10 of [Ma]). The construction in Section 4.13 lead 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 (see Hausdorff measure).

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
How to Cite This Entry:
Hausdorff dimension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hausdorff_dimension&oldid=28299
This article was adapted from an original article by I.G. Koshevnikova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article