![Assouad dimension](https://upload.wikimedia.org/wikipedia/commons/3/3e/Assouad_dimension.svg)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 PHD or PhD may refer to:
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** Ph.D. (Ph.D. albu ...
thesis and later published in 1979, although the same notion had been studied in 1928 by
Georges Bouligand
Georges Louis Bouligand (13 October 1889 – 12 April 1979) was a French mathematician. He worked in Mathematical analysis, analysis, mechanics, Analytic geometry, analytical and differential geometry, topology, and mathematical physics. He is kno ...
. As well as being used to study fractals, the Assouad dimension has also been used to study
quasiconformal mapping
In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity.
Intuitively, let ''f'' : ''D' ...
s and
embeddability problems.
Definition
Let
be a
metric space, and let be a non-empty subset of . For , let
denote the least number of metric
open balls of radius less than or equal to with which it is possible to
cover
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** Book design
** Back cover copy, part of co ...
the set . The Assouad dimension of is defined to be the
infimal for which there exist positive constants and
so that, whenever
the following bound holds:
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 In mathematics, the Assouad–Nagata dimension (sometimes simply Nagata dimension) is a notion of dimension for metric spaces, introduced by Jun-iti Nagata in 1958 and reformulated by Patrice Assouad in 1982, who introduced the now-usual definition. ...
.
* The Assouad dimension of a metric space is always greater than or equal to its
upper box dimension
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, 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
*
{{Fractals
Dimension theory
Fractals
Metric geometry