Quasiconformal distortion of the Assouad spectrum and classification of polynomial spirals

TL;DR

This paper studies how quasiconformal maps distort the Assouad spectrum and dimension, and uses this to classify polynomial spirals up to quasiconformal equivalence based on their parameters.

Contribution

It provides new insights into the distortion of the Assouad spectrum under quasiconformal maps and classifies polynomial spirals according to their quasiconformal equivalence classes.

Findings

Assouad spectrum distortion under quasiconformal maps characterized.

Polynomial spirals classified by quasiconformal equivalence and dilatation bounds.

Abstract

We investigate the distortion of Assouad dimension and the Assouad spectrum under Euclidean quasiconformal maps. Our results complement existing conclusions for Hausdorff and box-counting dimension due to Gehring--V\"ais\"al\"a and others. As an application, we classify polynomial spirals $S_a:=\{x^{-a}e^{\mathbf{i} x}:x>0\}$ up to quasiconformal equivalence, up to the level of the dilatation. Specifically, for $a>b>0$ we show that there exists a quasiconformal map $f$ of $\mathbb{C}$ with dilatation $K_f$ and $f(S_a)=S_b$ if and only if $K_f \ge \tfrac{a}{b}$.