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]

Formal definition

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

Properties

We have the following inequalities, for a metric space 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 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.
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See also
  • Anomalous scaling dimension

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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