Conformal dimension explained

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

l{G} be the collection of all metric spaces that are quasisymmetric to X. The conformal dimension of X is defined as suchCdim X = inf _{Y} } \dim_H Y

Properties

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

\dim _{T} X \leq 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

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

Notes and References

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