Quasiconformal distortion of the Assouad spectrum and classification of polynomial spirals

We investigate the distortion of Assouad dimension and the Assouad spectrum under Euclidean quasicon-formal maps. Our results complement existing conclusions for Hausdorff and box-counting dimension due to Gehring-Vaisala and others. As an application, we classify polynomial spirals S-a := {x(-a)e(ix ): 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 C with dilatation K-f and f(S-a) = S-b if and only if K-f >= a/b.

Automated summary

This article investigates the distortion of Assouad dimension and the Assouad spectrum under Euclidean quasicon-formal maps. As an application, it provides a classification criterion for polynomial spirals up to quasiconformal equivalence.