Assouad dimension

File:Assouad dimension.svg is equal to its Hausdorff dimension, $\alpha = \frac{\log(3)}{\log(2)}$ . In the illustration, we see that for a particular choice of {{mvar|r}}, {{mvar|R}}, and {{mvar|x}}, $N_{r}(B_{R}(x) \cap E) = 3 = 2^\alpha = \left( \frac{R}{r} \right)^{\alpha}.$ For other choices, the constant {{mvar|C}} may be greater than 1, but is still bounded.]]

In mathematics — specifically, in fractal geometry — the Assouad dimension is a definition of fractal dimension for subsets of a metric space. It was introduced by Patrice Assouad in his 1977 PhD thesis and later published in 1979,{{cite journal

| last = Assouad

| first = Patrice

| title = Étude d'une dimension métrique liée à la possibilité de plongements dans Rⁿ

| journal = Comptes Rendus de l'Académie des Sciences, Série A-B

| volume = 288

| year = 1979

| issue = 15

| pages = A731–A734

| language=fr

| issn = 0151-0509}} {{MathSciNet|id=532401}} although the same notion had been studied in 1928 by Georges Bouligand.{{cite journal

| last=Bouligand

| first=Georges

| author-link=Georges Bouligand

| year=1928

| title=Ensembles impropres et nombre dimensionnel

| journal=Bulletin des Sciences Mathématiques

| volume=52

| pages=320–344

| language=fr

| url=https://gallica.bnf.fr/ark:/12148/bpt6k486279t/f406.item}} As well as being used to study fractals, the Assouad dimension has also been used to study quasiconformal mappings and embeddability problems.

Definition

{{Quote|The Assouad dimension of $X, d_A(X)$ , is the infimum of all $s$ such that $(X, \varsigma)$ is $(M, s)$ -homogeneous for some $M \ge 1$ .{{cite book| last=Robinson |first=James C. |year=2010 |url=https://books.google.com/books?id=qFyXxiKfA9UC&dq=Assouad+dimension&pg=PA83 |title=Dimensions, Embeddings, and Attractors |page=85 |publisher=Cambridge University Press |isbn=9781139495189}}}}

Let $(X, d)$ be a metric space, and let {{mvar|E}} be a non-empty subset of {{mvar|X}}. For {{math|r > 0}}, let $N_{r}(E)$ denote the least number of metric open balls of radius less than or equal to {{mvar|r}} with which it is possible to cover the set {{mvar|E}}. The Assouad dimension of {{mvar|E}} is defined to be the infimal $\alpha \ge 0$ for which there exist positive constants {{mvar|C}} and $\rho$ so that, whenever

$0 < r < R \leq \rho,$

the following bound holds:

$\sup_{x \in E} N_{r}(B_{R}(x) \cap E) \leq C \left( \frac{R}{r} \right)^{\alpha}.$

The intuition underlying this definition is that, for a set {{mvar|E}} with "ordinary" integer dimension {{mvar|n}}, the number of small balls of radius {{mvar|r}} needed to cover the intersection of a larger ball of radius {{mvar|R}} with {{mvar|E}} will scale like {{math|(R/r)ⁿ}}.

Relationships to other notions of dimension

The Assouad dimension of a metric space is always greater than or equal to its Assouad–Nagata dimension.{{cite journal |last1=Le Donne |first1=Enrico |last2=Rajala |first2=Tapio |title=Assouad dimension, Nagata dimension, and uniformly close metric tangents |journal=Indiana University Mathematics Journal |date=2015 |volume=64 |issue=1 |pages=21–54 |doi=10.1512/iumj.2015.64.5469 |arxiv=1306.5859|s2cid=55039643 }}
The Assouad dimension of a metric space is always greater than or equal to its upper box dimension, which in turn is greater than or equal to the Hausdorff dimension.
The Lebesgue covering dimension of a metrizable space {{mvar|X}} is the minimal Assouad dimension of any metric on {{mvar|X}}. In particular, for every metrizable space there is a metric for which the Assouad dimension is equal to the Lebesgue covering dimension.{{cite journal |last1=Luukkainen |first1=Jouni |title=Assouad dimension: antifractal metrization, porous sets, and homogeneous measures |journal=Journal of the Korean Mathematical Society |date=1998 |volume=35 |issue=1 |pages=23–76 |url=https://koreascience.kr/article/JAKO199811919486026.page |issn=0304-9914}}

Assouad dimension

Definition

Relationships to other notions of dimension

References

Further reading