On the conformal gauge of a compact metric space
[Sur la jauge conforme d'un espace métrique compact]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 3, pp. 495-548.

Dans cet article, on étudie la jauge conforme Ahlfors régulière d’un espace métrique compact et sa dimension conforme dimAR(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 dimARX à partir de modules combinatoires en considérant un exposant critique QN.

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

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

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