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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 map|quasisymmetric]] to&nbsp;''X''.<ref>John M. Mackay, Jeremy T. Tyson, ''Conformal Dimension : Theory and Application'', University Lecture Series, Vol. 54, 2010, Rhodes Island</ref>
 
==Formal definition==
 
Let ''X'' be a metric space and <math>\mathcal{G}</math> be the collection of all metric spaces that are quasisymmetric to&nbsp;''X''. The conformal dimension of ''X'' is defined as such
 
: <math> \mathrm{Cdim} X = \inf_{Y \in \mathcal{G}} \dim_H Y</math>
 
==Properties==
 
We have the following [[Inequality (mathematics)|inequalities]], for a metric space&nbsp;''X'':
 
: <math>\dim_T X \leq \mathrm{Cdim} X \leq \dim_H X</math>
 
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&nbsp;''X''.
 
==Examples==
 
* The conformal dimension of <math>\mathbf{R}^N</math> is ''N'', since the topological and Hausdorff dimensions of [[Euclidean space]]s 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==
{{Reflist}}
[[Category:Fractals]]
[[Category:Metric geometry]]
[[Category:Dimension theory]]

Latest revision as of 00:30, 18 March 2013

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 Cantor set K is of null conformal dimension. However, there is no metric space quasisymmetric to K with a 0 Hausdorff dimension.

See also

References

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