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arXiv · 2112.02620

Quasiconformal distortion of the Assouad spectrum and classification of polynomial spirals

Efstathios Konstantinos Chrontsios Garitsis·Jeremy T. Tyson
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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 $K_f \ge \tfrac{a}{b}$.

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Efstathios Konstantinos Chrontsios Garitsis, Jeremy T. Tyson. 2021-12-05. Quasiconformal distortion of the Assouad spectrum and classification of polynomial spirals. https://arxiv.org/abs/2112.02620

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