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On the conformal gauge of a compact metric space

Matias Carrasco Piaggio (2013)

Annales scientifiques de l'École Normale Supérieure

In this article we study the Ahlfors regular conformal gauge of a compact metric space ( X , d ) , and its conformal dimension dim A R ( 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 dim A R ( X , d ) using the critical exponent Q N associated to the combinatorial modulus.

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