Assouad dimension

Assouad dimension on Sierpiński triangle. For R=2 and r=1 ${\displaystyle N_{r}(B_{R}(x)\cap E)=3=\left({\frac {2}{1}}\right)^{\alpha }}$, so the dimension can be ${\displaystyle {\frac {\log(3)}{\log(2)}}}$ like Hausdorff 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 in his 1977 PhD thesis and later published in 1979. It was defined earlier by Georges Bouligand (1928). 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 ${\displaystyle X,d_{A}(X)}$, is the infimum of all ${\displaystyle s}$ such that ${\displaystyle (X,\varsigma )}$ is ${\displaystyle (M,s)}$-homogeneous for some ${\displaystyle M\geq 1}$.^[1]

Let ${\displaystyle (X,d)}$ be a metric space, and let ${\displaystyle E}$ be a non-empty subset of ${\displaystyle X}$. For ${\displaystyle r>0}$, let ${\displaystyle N_{r}(E)}$ denote the least number of metric open balls of radius less than or equal to r with which it is possible to open cover the set ${\displaystyle E}$. The Assouad dimension of ${\displaystyle E}$ is defined to be the infimal ${\displaystyle \alpha \geq 0}$ for which there exist positive constants ${\displaystyle C}$ and ${\displaystyle \rho }$ so that, whenever

${\displaystyle 0<r<R\leq \rho ,}$

the following bound holds:

${\displaystyle \sup _{x\in E}N_{r}(B_{R}(x)\cap E)\leq C\left({\frac {R}{r}}\right)^{\alpha }.}$

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.

References[]

Further reading[]

• Assouad, Patrice (1979). "Étude d'une dimension métrique liée à la possibilité de plongements dans Rⁿ". Comptes Rendus de l'Académie des Sciences, Série A-B. 288 (15): A731–A734. ISSN 0151-0509. MR532401
• Bouligand, M.G. (1928). "Ensembles impropres et nombredimensionnel", Bulletin des Sciences Mathématiques 52, pp.320–344.
• Fraser, Jonathan M. (2020). Assouad Dimension and Fractal Geometry. Cambridge University Press. doi:10.1017/9781108778459. ISBN 9781108478656.