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.[1]

Formal definition[edit]

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

 \mathrm{Cdim} X = \inf_{Y \in \mathcal{G}} \dim_H Y

Properties[edit]

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

\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[edit]

  • The conformal dimension of \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[edit]

References[edit]

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