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On concentric fractal spheres and spiral shells

Efstathios Konstantinos Chrontsios Garitsis

Abstract

We investigate dimension-theoretic properties of concentric topological spheres, which are fractal sets emerging both in pure and applied mathematics. We calculate the box dimension and Assouad spectrum of such collections, and use them to prove that fractal spheres cannot be shrunk into a point at a polynomial rate. We also apply these dimension estimates to quasiconformally classify certain spiral shells, a generalization of planar spirals in higher dimensions. This classification also provides a bi-Hölder map between shells, and constitutes an addition to a general programme of research proposed by J. Fraser.

On concentric fractal spheres and spiral shells

Abstract

We investigate dimension-theoretic properties of concentric topological spheres, which are fractal sets emerging both in pure and applied mathematics. We calculate the box dimension and Assouad spectrum of such collections, and use them to prove that fractal spheres cannot be shrunk into a point at a polynomial rate. We also apply these dimension estimates to quasiconformally classify certain spiral shells, a generalization of planar spirals in higher dimensions. This classification also provides a bi-Hölder map between shells, and constitutes an addition to a general programme of research proposed by J. Fraser.
Paper Structure (5 sections, 7 theorems, 83 equations, 2 figures)

This paper contains 5 sections, 7 theorems, 83 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Dimensions and impossible concentric spheres
  4. Quasiconformal classification of spiral shells
  5. Final Remarks

Key Result

Theorem 1.2

Let $S_0\subset \mathbb{R}^d$ be a topological sphere with ${\underline{\dim_B}} S_0\in (d-1,d]$. For any $p>0$, there is no $p$-concentric sphere collection generated by $S_0$.

Figures (2)

  • Figure 1: An example of a concentric collection of snowflakes.
  • Figure 2: An example of a polynomial spiral shell for $p=1/2$.

Theorems & Definitions (16)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Remark 3.4
  • ...and 6 more