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, p. 495-548
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 .
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.
DOI : https://doi.org/10.24033/asens.2195
Classification:  30L10,  51F99,  20F67,  30C65,  28A78
Mots clés: Ahlfors régulier, jauge conforme, dimension conforme, module combinatoire, Gromov-hyperbolique
@article{ASENS_2013_4_46_3_495_0,
     author = {Carrasco Piaggio, Matias},
     title = {On the conformal gauge of a compact metric space},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 46},
     number = {3},
     year = {2013},
     pages = {495-548},
     doi = {10.24033/asens.2195},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2013_4_46_3_495_0}
}
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. http://www.numdam.org/item/ASENS_2013_4_46_3_495_0/

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