Conformal Assouad dimension as the critical exponent for combinatorial modulus

TL;DR

This paper establishes that the conformal Assouad dimension of a compact doubling metric space equals a critical exponent derived from the combinatorial modulus, extending previous results to a broader class of spaces.

Contribution

It generalizes the relation between conformal Assouad dimension and combinatorial modulus from Ahlfors regular spaces to all compact doubling metric spaces.

Findings

Conformal Assouad dimension equals a critical exponent for all compact doubling spaces.

Replacing quasisymmetry with power quasisymmetry does not change the conformal Assouad dimension.

The result extends previous work by Carrasco Piaggio to a larger class of metric spaces.

Abstract

The conformal Assouad dimension is the infimum of all possible values of Assouad dimension after a quasisymmetric change of metric. We show that the conformal Assouad dimension equals a critical exponent associated to the combinatorial modulus for any compact doubling metric space. This generalizes a similar result obtained by Carrasco Piaggio for the Ahlfors regular conformal dimension to a larger family of spaces. We also show that the value of conformal Assouad dimension is unaffected if we replace quasisymmetry with power quasisymmetry in its definition.