On the conformal gauge of a compact metric space

Details On the conformal gauge of a compact metric space On the conformal gauge of a compact metric space

2012-04-15

arxiv.org/abs/1204.3309v1

Mathematics

Metric Geometry

49 pages, 9 figures

Scientific paper

In this article we study the Ahlfors regular conformal gauge of a compact metric space $(X,d)$, and its conformal dimension $\mathrm{dim}_{AR}(X,d)$. 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 $\mathrm{dim}_{AR}(X,d)$ using the critical exponent $Q_N$ associated to the combinatorial modulus.

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