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Conformal dimension

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In mathematics, the conformal dimension of a metric space X is the infimum of the Hausdorff dimension over the conformal gauge of X, that is, the class of all metric spaces quasisymmetric to X.

Formal definition

Let X be a metric space and G {\displaystyle {\mathcal {G}}} be the collection of all metric spaces that are quasisymmetric to X. The conformal dimension of X is defined as such

C d i m X = inf Y ∈ G dim H ⁡ Y {\displaystyle \mathrm {Cdim} X=\inf _{Y\in {\mathcal {G}}}\dim _{H}Y}

Properties

We have the following inequalities, for a metric space X:

dim T ⁡ X ≤ C d i m X ≤ dim H ⁡ X {\displaystyle \dim _{T}X\leq \mathrm {Cdim} X\leq \dim _{H}X}

The second inequality is true by definition. The first one is deduced from the fact that the topological dimension T is invariant by homeomorphism, and thus can be defined as the infimum of the Hausdorff dimension over all spaces homeomorphic to X.

Examples

  • The conformal dimension of R N {\displaystyle \mathbf {R} ^{N}} is N, since the topological and Hausdorff dimensions of Euclidean spaces agree.
  • The Cantor set K is of null conformal dimension. However, there is no metric space quasisymmetric to K with a 0 Hausdorff dimension.

See also

References

  1. John M. Mackay, Jeremy T. Tyson, Conformal Dimension : Theory and Application, University Lecture Series, Vol. 54, 2010, Rhodes Island


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