Assouad dimension explained

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,^[1] although the same notion had been studied in 1928 by Georges Bouligand.^[2] As well as being used to study fractals, the Assouad dimension has also been used to study quasiconformal mappings and embeddability problems.

Definition

Let

(X, d) be a metric space, and let be a non-empty subset of . For, let N _{r} (E) denote the least number of metric open balls of radius less than or equal to with which it is possible to cover the set . The Assouad dimension of is defined to be the infimal \alpha \ge 0 for which there exist positive constants and \rho so that, wheneverthe following bound holds:$$\backslash sup\_\; N\_(B\_(x)\; \backslash cap\; E)\; \backslash leq\; C\; \backslash left(\backslash frac\; \backslash right)^.$$

The intuition underlying this definition is that, for a set with "ordinary" integer dimension, the number of small balls of radius needed to cover the intersection of a larger ball of radius with will scale like .

Relationships to other notions of dimension

• The Assouad dimension of a metric space is always greater than or equal to its Assouad–Nagata dimension.^[3]
• 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 is the minimal Assouad dimension of any metric on . In particular, for every metrizable space there is a metric for which the Assouad dimension is equal to the Lebesgue covering dimension.^[4]

Further reading

• Book: Fraser , Jonathan M. . 2020. Assouad Dimension and Fractal Geometry. 10.1017/9781108778459. Cambridge University Press. 9781108478656. 218571013.

Notes and References

1. Assouad. Patrice. Étude d'une dimension métrique liée à la possibilité de plongements dans Rⁿ. Comptes Rendus de l'Académie des Sciences, Série A-B. 288. 1979. 15. A731 - A734. fr. 0151-0509.
2. Bouligand. Georges . Georges Bouligand. 1928. Ensembles impropres et nombre dimensionnel. Bulletin des Sciences Mathématiques. 52. 320–344. fr.
3. Le Donne . Enrico . Rajala . Tapio . Assouad dimension, Nagata dimension, and uniformly close metric tangents . . 2015 . 64 . 1 . 21–54 . 10.1512/iumj.2015.64.5469 . 1306.5859. 55039643 .
4. Luukkainen . Jouni . Assouad dimension: antifractal metrization, porous sets, and homogeneous measures . Journal of the Korean Mathematical Society . 1998 . 35 . 1 . 23–76 . 0304-9914.