On the conformal gauge of a compact metric space

By Matias Carrasco Piaggio. 2012

60 citations recorded by the source corpus.

In this article we study the Ahlfors regular conformal gauge of a compact metric space $(X,d)$, and its conformal dimension $\mathrm{dim}_{AR}(X,d)$. Using a sequence of finite coverings of $(X,d)$, we construct distances in its Ahlfors regular conformal gauge of controlled Hausdorff dimension. We obtain in this way a combinatorial description, up to bi-Lipschitz homeomorphisms, of all the metrics in the gauge. We show how to compute $\mathrm{dim}_{AR}(X,d)$ using the critical exponent $Q_N$ associated to the combinatorial modulus.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bef8089cbec8a9c95aa45a71c648232731e11f0e