Conformal dimension

From Wikipedia, the free encyclopedia

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 {\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

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

  • 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

  • Anomalous scaling dimension

References

  1. ↑ John M. Mackay, Jeremy T. Tyson, Conformal Dimension : Theory and Application, University Lecture Series, Vol. 54, 2010, Rhodes Island
This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.