Quasiconformal distortion of the Assouad spectrum and classification of polynomial spirals

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--Väisälä 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 $K_f\ge \frac{a}{b}$.

arXiv comment21 pages, minor revisions and changes, accepted by Bulletin of the London Mathematical Society