The fractal structure of elliptical polynomial spirals

Stuart Andrew Burrell; Kenneth John Falconer; Jonathan Fraser

doi:10.1007/s00605-022-01735-9

The fractal structure of elliptical polynomial spirals

Stuart Andrew Burrell^*, Kenneth John Falconer, Jonathan Fraser

^*Corresponding author for this work

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

Abstract

We investigate fractal aspects of elliptical polynomial spirals, that is planar spirals with differing polynomial rates of decay in the two axis directions. We give a full dimensional analysis of these spirals, computing explicitly their intermediate, box-counting and Assouad-type dimensions. An exciting feature is that these spirals exhibit two phase transitions within the Assouad spectrum, the first natural class of fractals known to have this property. We go on to use this dimensional information to obtain bounds for the Hölder regularity of maps that can deform one spiral into another, generalising the 'winding problem' of when spirals are bi-Lipschitz equivalent to a line segment. A novel feature is the use of fractional Brownian motion and dimension profiles to bound the Hölder exponents.

Original language	English
Pages (from-to)	1-22
Journal	Monatshefte für Mathematik
Volume	199
Early online date	3 Jul 2022
DOIs	https://doi.org/10.1007/s00605-022-01735-9
Publication status	Published - 1 Sept 2022

Keywords

Elliptical polynomial spiral
Generalised hyperbolic spiral
Box-counting dimension
Assouad dimension
Assouad spectrum
Intermediate dimensions
Hölder exponents
Fractional Brownian motion

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10.1007/s00605-022-01735-9Licence: CC BY

BurrellFalconerFraser2022_Article_TheFractalStructureOfElliptica
Copyright © The Author(s) 2022. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Final published version, 596 KBLicence: CC BY

New perspectives in the dimension: New perspectives in the dimension theory of fractals
Fraser, J. (PI)
The Leverhulme Trust
1/09/19 → 31/01/23
Project: Standard
Fourier analytic techniques: Fourier analytic techniques in geometry and analysis
Fraser, J. (PI) & Falconer, K. J. (CoI)
EPSRC
1/02/18 → 11/06/21
Project: Standard

Cite this

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title = "The fractal structure of elliptical polynomial spirals",

abstract = "We investigate fractal aspects of elliptical polynomial spirals, that is planar spirals with differing polynomial rates of decay in the two axis directions. We give a full dimensional analysis of these spirals, computing explicitly their intermediate, box-counting and Assouad-type dimensions. An exciting feature is that these spirals exhibit two phase transitions within the Assouad spectrum, the first natural class of fractals known to have this property. We go on to use this dimensional information to obtain bounds for the H{\"o}lder regularity of maps that can deform one spiral into another, generalising the 'winding problem' of when spirals are bi-Lipschitz equivalent to a line segment. A novel feature is the use of fractional Brownian motion and dimension profiles to bound the H{\"o}lder exponents.",

keywords = "Elliptical polynomial spiral, Generalised hyperbolic spiral, Box-counting dimension, Assouad dimension, Assouad spectrum, Intermediate dimensions, H{\"o}lder exponents, Fractional Brownian motion",

author = "Burrell, {Stuart Andrew} and Falconer, {Kenneth John} and Jonathan Fraser",

note = "Funding: SAB was supported by a Carnegie Trust PhD Scholarship (PHD060287) and would like to thank David Dritschel for helpful discussion on the physical applications of elliptical spirals. KJF and JMF were supported by an EPSRC Standard Grant (EP/R015104/1). JMF was also supported by a Leverhulme Trust Research Project Grant (RPG-2019-034).",

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doi = "10.1007/s00605-022-01735-9",

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AU - Burrell, Stuart Andrew

AU - Falconer, Kenneth John

AU - Fraser, Jonathan

N1 - Funding: SAB was supported by a Carnegie Trust PhD Scholarship (PHD060287) and would like to thank David Dritschel for helpful discussion on the physical applications of elliptical spirals. KJF and JMF were supported by an EPSRC Standard Grant (EP/R015104/1). JMF was also supported by a Leverhulme Trust Research Project Grant (RPG-2019-034).

PY - 2022/9/1

Y1 - 2022/9/1

N2 - We investigate fractal aspects of elliptical polynomial spirals, that is planar spirals with differing polynomial rates of decay in the two axis directions. We give a full dimensional analysis of these spirals, computing explicitly their intermediate, box-counting and Assouad-type dimensions. An exciting feature is that these spirals exhibit two phase transitions within the Assouad spectrum, the first natural class of fractals known to have this property. We go on to use this dimensional information to obtain bounds for the Hölder regularity of maps that can deform one spiral into another, generalising the 'winding problem' of when spirals are bi-Lipschitz equivalent to a line segment. A novel feature is the use of fractional Brownian motion and dimension profiles to bound the Hölder exponents.

AB - We investigate fractal aspects of elliptical polynomial spirals, that is planar spirals with differing polynomial rates of decay in the two axis directions. We give a full dimensional analysis of these spirals, computing explicitly their intermediate, box-counting and Assouad-type dimensions. An exciting feature is that these spirals exhibit two phase transitions within the Assouad spectrum, the first natural class of fractals known to have this property. We go on to use this dimensional information to obtain bounds for the Hölder regularity of maps that can deform one spiral into another, generalising the 'winding problem' of when spirals are bi-Lipschitz equivalent to a line segment. A novel feature is the use of fractional Brownian motion and dimension profiles to bound the Hölder exponents.

KW - Elliptical polynomial spiral

KW - Generalised hyperbolic spiral

KW - Box-counting dimension

KW - Assouad dimension

KW - Assouad spectrum

KW - Intermediate dimensions

KW - Hölder exponents

KW - Fractional Brownian motion

U2 - 10.1007/s00605-022-01735-9

DO - 10.1007/s00605-022-01735-9

M3 - Article

SN - 0026-9255

VL - 199

SP - 1

EP - 22

JO - Monatshefte für Mathematik

JF - Monatshefte für Mathematik

ER -

The fractal structure of elliptical polynomial spirals

Abstract

Keywords

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Projects

New perspectives in the dimension: New perspectives in the dimension theory of fractals

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

Cite this