Assouad dimension

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, although the same notion had been studied in 1928 by Georges Bouligand. 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 $N_(E)$ 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, whenever
$0 < r < R \leq \rho,$
the following bound holds:
$\sup_ N_(B_(x) \cap E) \leq C \left( \frac \right)^.$

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.
* 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.

References

Further reading

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{{Fractals

Dimension theory
Fractals
Metric geometry