Assouad dimension
Appearance
In
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
The Assouad dimension of , is the infimum of all such that is -homogeneous for some .[3]
Let be a metric space, and let E be a non-empty subset of X. For r > 0, let denote the least number of metric
infimal
for which there exist positive constants C and so that, whenever
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.
Relationships to other notions of dimension
- The Assouad dimension of a metric space is always greater than or equal to its Assouad–Nagata dimension.[4]
- 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.[5]
- The Lebesgue covering dimension of a metrizable space X is the minimal Assouad dimension of any metric on X. In particular, for every metrizable space there is a metric for which the Assouad dimension is equal to the Lebesgue covering dimension.[5]
References
- MR532401
- ^ Bouligand, Georges (1928). "Ensembles impropres et nombre dimensionnel". Bulletin des Sciences Mathématiques (in French). 52: 320–344.
- ISBN 9781139495189.
- S2CID 55039643.
- ^ ISSN 0304-9914.
Further reading
- Fraser, Jonathan M. (2020). Assouad Dimension and Fractal Geometry. Cambridge University Press. S2CID 218571013.