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.

Figures (7)

• Figure 1: An elliptical polynomial spiral $S_{p, q}$ with $p =0.7$ and $q=0.75$.
• Figure 2: A plot of $\dim_\theta S_{p, q}$ ($y$-axis) against $\theta$ ($x$-axis) for $\theta \in [0, 1]$ and $\dim_{\textup{A}}^{\theta-1} S_{p,q}$ against $\theta$ for $\theta \in [1,2]$. In this example, $p = 0.1$ and $q = 0.8$.
• Figure 3: A plot of $\dim_\theta S_{p, q}$ ($y$-axis) against $\theta$ ($x$-axis) for $p = 0.4$ and $q=0.7$, along with horizontal lines that indicate $\dim_{\textup{H}} S_{p, q} = 1$ and $\dim_{\textup{B}} S_{p, q} = (2+q-p)/(1+q)$.
• Figure 4: A plot of $\dim^\theta_{\mathrm{A}} S_{p, q}$ ($y$-axis) against $\theta$ ($x$-axis) for $p = 1.1$ and $q = 1.8$.
• Figure 5: A family of concentric ellipses $C_{p, q}$ dimensionally equivalent to $S_{p, q}$, where $p = 0.4$ and $q = 0.6$.

Theorems & Definitions (22)

• Theorem 2.1
• Lemma 2.2
• Corollary 2.3
• Corollary 2.4
• Corollary 2.5
• Theorem 2.6
• Corollary 2.7
• Corollary 2.8
• proof
• Theorem 2.9