Rigidity of quasisymmetric mappings on self-affine carpets

TL;DR

This paper proves that quasisymmetric maps between certain self-affine carpets are highly restricted, existing only when the carpets have the same dimension, and these maps are quasi-Lipschitz, with implications for their conformal dimensions.

Contribution

It establishes the rigidity of quasisymmetric maps on self-affine carpets and characterizes their form and dimension-related properties.

Findings

Quasisymmetric maps only exist between carpets of the same dimension.

Such maps are necessarily quasi-Lipschitz.

Horizontal self-affine carpets are minimal for the conformal Assouad dimension.

Abstract

We show that the class of quasisymmetric maps between horizontal self-affine carpets is rigid. Such maps can only exist when the dimensions of the carpets coincide, and in this case, the quasisymmetric maps are quasi-Lipschitz. We also show that horizontal self-affine carpets are minimal for the conformal Assouad dimension.