On the conformal gauge of a compact metric space

Carrasco Piaggio, Matias

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 ${dim}_{AR}(X,d)$. 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.

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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