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,[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.

The Assouad dimension of

be a metric space, and let E be a non-empty subset of X.

denote the least number of metric open balls of radius less than or equal to r with which it is possible to cover the set E. The Assouad dimension of E is defined to be the infimal

α ≥ 0

for which there exist positive constants C and

ρ

the following bound holds:

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

The Assouad dimension of the Sierpiński triangle is equal to its Hausdorff dimension , . In the illustration, we see that for a particular choice of r , R , and x , For other choices, the constant C may be greater than 1, but is still bounded.