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 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.
Citation
Burrell , S A , Falconer , K J & Fraser , J 2022 , ' The fractal structure of elliptical polynomial spirals ' , Monatshefte für Mathematik , vol. 199 , pp. 1-22 . https://doi.org/10.1007/s00605-022-01735-9
Publication
Monatshefte für Mathematik
Status
Peer reviewed
ISSN
0026-9255Type
Journal article
Description
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).Collections
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