Quasiregular distortion of dimensions

TL;DR

This paper studies how planar quasiregular maps affect the Assouad dimension and spectrum of sets, providing bounds and invariance results that extend understanding beyond classical dimensions like Hausdorff.

Contribution

It introduces bounds on the Assouad dimension and spectrum under quasiregular maps and proves porosity invariance, addressing gaps in the understanding of these dimensions' behavior.

Findings

Upper bounds on Assouad dimension and spectrum of images

Invariance of porosity under quasiregular maps

Extension of dimension distortion results beyond Hausdorff

Abstract

We investigate the distortion of the Assouad dimension and (regularized) spectrum of sets under planar quasiregular maps. While the respective results for the Hausdorff and upper box-counting dimension follow immediately from their quasiconformal counterparts by employing elementary properties of these dimension notions (e.g. countable stability and Lipschitz stability), the Assouad dimension and spectrum do not share such properties. We obtain upper bounds on the Assouad dimension and spectrum of images of compact sets under planar quasiregular maps by studying their behavior around their critical points. As an application the invariance of porosity of compact subsets of the plane under quasiregular maps is established.