On the conformal gauge of a compact metric space

TL;DR

This paper investigates the Ahlfors regular conformal gauge of compact metric spaces, providing a combinatorial method to describe and compute the conformal dimension using finite coverings and critical exponents.

Contribution

It introduces a new combinatorial approach to characterize the conformal gauge and compute the conformal dimension via the critical exponent associated with the combinatorial modulus.

Findings

Constructs distances with controlled Hausdorff dimension in the conformal gauge.

Provides a combinatorial description of metrics in the gauge up to bi-Lipschitz homeomorphisms.

Shows how to compute the conformal dimension using the critical exponent Q_N.

Abstract

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.