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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.

Contents

Formal definition

Let X be a metric space and 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

Properties

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

dim T ⁡ X ≤ C d i m X ≤ 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 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.

    • References

    Conformal dimension Wikipedia
    (Text) CC BY-SA


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