The fractal structure of elliptical polynomial spirals

TL;DR

This paper analyzes the fractal dimensions of elliptical polynomial spirals, revealing phase transitions in their Assouad spectrum and applying this to bounds on H"older regularity of deformations, using fractional Brownian motion.

Contribution

It provides a comprehensive dimensional analysis of elliptical polynomial spirals and introduces phase transitions in the Assouad spectrum, extending the understanding of spiral deformations.

Findings

Explicit computation of fractal dimensions of the spirals

Identification of two phase transitions in the Assouad spectrum

Bounds on H"older regularity for spiral deformations

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\"older 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\"older exponents.