Quasiconformal distortion of the Assouad spectrum and classification of polynomial spirals

2021/12/05 by Efstathios Konstantinos Chrontsios Garitsis, Garitsis, Efstathios Konstantinos Chrontsios, Jeremy T. Tyson +1 · 2 citations
Mathematics · #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2112.02620

openalex publication_date 2021/12/05 · openalex created_date 2021/12/31 · openalex updated_date 2026/07/28

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^-ae^{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 ℂ with dilatation K_f and f(S_a)=S_b if and only if K_f ≥ \tfracab.

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• On the dimension distortion under fractionally smooth mappings
• Recent Progress on Fractal Percolation

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