On the conformal gauge of a compact metric space
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 46 (2013) no. 3, pp. 495-548.

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

Dans cet article, on étudie la jauge conforme Ahlfors régulière d’un espace métrique compact et sa dimension conforme dim AR (X,d). À l’aide d’une suite de recouvrements finis de (X,d), on construit des distances dans sa jauge Ahlfors régulière de dimension de Hausdorff contrôlée. On obtient ainsi une description combinatoire, à homéomorphismes bi-Lipschitz près, de toutes les métriques dans la jauge. On montre comment calculer dim AR X à partir de modules combinatoires en considérant un exposant critique Q N .

DOI: 10.24033/asens.2195
Classification: 30L10,  51F99,  20F67,  30C65,  28A78
Keywords: Ahlfors regular, conformal gauge, conformal dimension, combinatorial modulus, Gromov-hyperbolic
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Carrasco Piaggio, Matias. On the conformal gauge of a compact metric space. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 46 (2013) no. 3, pp. 495-548. doi : 10.24033/asens.2195. http://www.numdam.org/articles/10.24033/asens.2195/

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