Conformal Dimension

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.^[1]

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

${\displaystyle \mathrm{C}\mathrm{d}\mathrm{i}\mathrm{m}X=\underset{Y\in {𝒢}}{\inf }{\dim }_{H}Y}$

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

${\displaystyle {\dim }_{T}X\le \mathrm{C}\mathrm{d}\mathrm{i}\mathrm{m}X\le {\dim }_{H}X}$

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.

• The conformal dimension of ${\displaystyle {\mathbf{𝐑}}^N}$ 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.

• Anomalous scaling dimension

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