Quasiconformal distortion of the Assouad spectrum and classification of polynomial spirals

Efstathios K. Chrontsios Garitsis; Jeremy T. Tyson

doi:10.1112/blms.12727

Quasiconformal distortion of the Assouad spectrum and classification of polynomial spirals

Efstathios K. Chrontsios Garitsis, Jeremy T. Tyson

Mathematics

Research output: Contribution to journal › Article › peer-review

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 (Formula presented.) up to quasiconformal equivalence, up to the level of the dilatation. Specifically, for (Formula presented.) we show that there exists a quasiconformal map (Formula presented.) of (Formula presented.) with dilatation (Formula presented.) and (Formula presented.) if and only if (Formula presented.).

Original language	English (US)
Pages (from-to)	282-307
Number of pages	26
Journal	Bulletin of the London Mathematical Society
Volume	55
Issue number	1
Early online date	Sep 13 2022
DOIs	https://doi.org/10.1112/blms.12727
State	Published - Feb 2023

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General Mathematics

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10.1112/blms.12727License: CC BY-NC-ND

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title = "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–V{\"a}is{\"a}l{\"a} and others. As an application, we classify polynomial spirals (Formula presented.) up to quasiconformal equivalence, up to the level of the dilatation. Specifically, for (Formula presented.) we show that there exists a quasiconformal map (Formula presented.) of (Formula presented.) with dilatation (Formula presented.) and (Formula presented.) if and only if (Formula presented.).",

author = "{Chrontsios Garitsis}, {Efstathios K.} and Tyson, {Jeremy T.}",

note = "The second author acknowledges support from the National Science Foundation under grant DMS‐1600650, “Mappings and Measures in Sub‐Riemannian and Metric Spaces” and from the Simons Foundation under grant #852888, “Geometric mapping theory and geometric measure theory in sub‐Riemannian and metric spaces.” In addition, this material is based upon work conducted, while the second author is serving at the National Science Foundation. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.",

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AU - Tyson, Jeremy T.

N1 - The second author acknowledges support from the National Science Foundation under grant DMS‐1600650, “Mappings and Measures in Sub‐Riemannian and Metric Spaces” and from the Simons Foundation under grant #852888, “Geometric mapping theory and geometric measure theory in sub‐Riemannian and metric spaces.” In addition, this material is based upon work conducted, while the second author is serving at the National Science Foundation. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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N2 - 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 (Formula presented.) up to quasiconformal equivalence, up to the level of the dilatation. Specifically, for (Formula presented.) we show that there exists a quasiconformal map (Formula presented.) of (Formula presented.) with dilatation (Formula presented.) and (Formula presented.) if and only if (Formula presented.).

AB - 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 (Formula presented.) up to quasiconformal equivalence, up to the level of the dilatation. Specifically, for (Formula presented.) we show that there exists a quasiconformal map (Formula presented.) of (Formula presented.) with dilatation (Formula presented.) and (Formula presented.) if and only if (Formula presented.).

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Quasiconformal distortion of the Assouad spectrum and classification of polynomial spirals

Abstract

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