Английская Википедия: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 <math>\mathcal{G}</math> be the collection of all metric spaces that are quasisymmetric to 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 inequalities, for a metric space 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 X.
Examples
- The conformal dimension of <math>\mathbf{R}^N</math> 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
References
Шаблон:Fractal-stub
Шаблон:Metric-geometry-stub
- ↑ John M. Mackay, Jeremy T. Tyson, Conformal Dimension : Theory and Application, University Lecture Series, Vol. 54, 2010, Rhodes Island