Английская Википедия: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.
Definition
Let <math>(X, d)</math> be a metric space, and let Шаблон:Mvar be a non-empty subset of Шаблон:Mvar. For Шаблон:Math, let <math>N_{r}(E)</math> denote the least number of metric open balls of radius less than or equal to Шаблон:Mvar with which it is possible to cover the set Шаблон:Mvar. The Assouad dimension of Шаблон:Mvar is defined to be the infimal <math>\alpha \ge 0</math> for which there exist positive constants Шаблон:Mvar and <math>\rho</math> so that, whenever <math display="block">0 < r < R \leq \rho,</math> the following bound holds: <math display="block">\sup_{x \in E} N_{r}(B_{R}(x) \cap E) \leq C \left( \frac{R}{r} \right)^{\alpha}.</math>
The intuition underlying this definition is that, for a set Шаблон:Mvar with "ordinary" integer dimension Шаблон:Mvar, the number of small balls of radius Шаблон:Mvar needed to cover the intersection of a larger ball of radius Шаблон:Mvar with Шаблон:Mvar will scale like Шаблон:Math.
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.[4]
- The Lebesgue covering dimension of a metrizable space Шаблон:Mvar is the minimal Assouad dimension of any metric on Шаблон:Mvar. In particular, for every metrizable space there is a metric for which the Assouad dimension is equal to the Lebesgue covering dimension.[4]
References
Further reading