Assouad dimension

The Assouad dimension of the Sierpiński triangle is equal to its Hausdorff dimension, α = log ⁡ ( 3 ) log ⁡ ( 2 ) {\displaystyle \alpha ={\frac {\log(3)}{\log(2)}}} . In the illustration, we see that for a particular choice of r, R, and x,
N r ( B R ( x ) ∩ E ) = 3 = 2 α = ( R r ) α . {\displaystyle N_{r}(B_{R}(x)\cap E)=3=2^{\alpha }=\left({\frac {R}{r}}\right)^{\alpha }.}
For other choices, the constant C may be greater than 1, but is still bounded.

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

The Assouad dimension of X , d A ( X ) {\displaystyle X,d_{A}(X)} , is the infimum of all s {\displaystyle s} such that ( X , ς ) {\displaystyle (X,\varsigma )} is ( M , s ) {\displaystyle (M,s)} -homogeneous for some M ≥ 1 {\displaystyle M\geq 1} .[3]

Let ( X , d ) {\displaystyle (X,d)} be a metric space, and let E be a non-empty subset of X. For r > 0, let N r ( E ) {\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 cover the set E. The Assouad dimension of E is defined to be the infimal α ≥ 0 {\displaystyle \alpha \geq 0} for which there exist positive constants C and ρ {\displaystyle \rho } so that, whenever

0 < r < R ≤ ρ , {\displaystyle 0<r<R\leq \rho ,}
the following bound holds:
sup x ∈ E N r ( B R ( x ) ∩ E ) ≤ C ( R r ) α . {\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.

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

  1. ^ 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 (in French). 288 (15): A731–A734. ISSN 0151-0509. MR532401
  2. ^ Bouligand, Georges (1928). "Ensembles impropres et nombre dimensionnel". Bulletin des Sciences Mathématiques (in French). 52: 320–344.
  3. ^ Robinson, James C. (2010). Dimensions, Embeddings, and Attractors. Cambridge University Press. p. 85. ISBN 9781139495189.
  4. ^ Le Donne, Enrico; Rajala, Tapio (2015). "Assouad dimension, Nagata dimension, and uniformly close metric tangents". Indiana University Mathematics Journal. 64 (1): 21–54. arXiv:1306.5859. doi:10.1512/iumj.2015.64.5469. S2CID 55039643.
  5. ^ a b Luukkainen, Jouni (1998). "Assouad dimension: antifractal metrization, porous sets, and homogeneous measures". Journal of the Korean Mathematical Society. 35 (1): 23–76. ISSN 0304-9914.

Further reading

  • Fraser, Jonathan M. (2020). Assouad Dimension and Fractal Geometry. Cambridge University Press. doi:10.1017/9781108778459. ISBN 9781108478656. S2CID 218571013.
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