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The Complex Dimensions of Every Sierpinski Carpet Modification of Dust Type

Jade Leathrum

Abstract

We investigate modified Sierpiński Carpet fractals, constructed by dividing a square into a square $n \times n$ grid, removing a subset of the squares at each step, and then repeating that process for each square remaining in that grid. If enough squares are removed and in the proper places, we get ``Dust Type'' carpets, which have a path-connected complement and are themselves not path-connected. We study these fractals using the Fractal Zeta Functions, first introduced by Michel Lapidus, Goran Radunović, and Darko \vZubrinić in their book \emph{Fractal Zeta Functions and Fractal Drums}, from which we devised an analytical and combinatorial algorithm to compute the complex dimensions of every Sierpiński Carpet modification of Dust Type.

The Complex Dimensions of Every Sierpinski Carpet Modification of Dust Type

Abstract

We investigate modified Sierpiński Carpet fractals, constructed by dividing a square into a square grid, removing a subset of the squares at each step, and then repeating that process for each square remaining in that grid. If enough squares are removed and in the proper places, we get ``Dust Type'' carpets, which have a path-connected complement and are themselves not path-connected. We study these fractals using the Fractal Zeta Functions, first introduced by Michel Lapidus, Goran Radunović, and Darko \vZubrinić in their book \emph{Fractal Zeta Functions and Fractal Drums}, from which we devised an analytical and combinatorial algorithm to compute the complex dimensions of every Sierpiński Carpet modification of Dust Type.
Paper Structure (21 sections, 11 theorems, 79 equations, 25 figures, 3 tables)

This paper contains 21 sections, 11 theorems, 79 equations, 25 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The Tubular Zeta Function
  3. Modified Sierpiński Carpets
  4. Motivating Example: Ternary Cantor Set
  5. Preview of Main Results
  6. Helper Theorems
  7. Scaling Theorem
  8. Entire Extension
  9. Threshold Theorems
  10. Simplest Case: Completely Dusty
  11. Worked Example: Cantor Dust
  12. Computing Dimensions of All the Dust Types
  13. Reduction to Combinatorics
  14. The Combinatorics
  15. General Dust Theorem
  16. ...and 6 more sections

Key Result

Theorem 2.1

If a measurable set $A \subseteq \mathbb{R}^N$ is uniformly scaled by a factor $\lambda > 0$, then for any $\delta > 0$, $\tilde{\zeta}_{\lambda A}(s, \lambda\delta) = \lambda^s \tilde{\zeta}_A(s, \delta)$. $\blacktriangleleft$$\blacktriangleleft$

Figures (25)

  • Figure 1.1: The first 5 steps of the construction of the classic Sierpiński Carpet.
  • Figure 1.2: The first 5 steps of the construction of a modified Sierpiński Carpet utilizing a $4\times4$ grid.
  • Figure 1.3: The first 5 steps of the construction of an example Sierpiński Carpet modification of dust type in the $4\times4$ grid.
  • Figure 1.4: The first 4 steps of the construction of all possible Sierpiński Carpet modifications in the $2\times2$ grid for $m = 2,3$.
  • Figure 1.5: The first 5 steps of the construction of a Sierpiński Carpet modification which produces the Ternary Cantor Set.
  • ...and 20 more figures

Theorems & Definitions (25)

  • Definition 1.1: Sierpiński Carpet Modification
  • Definition 1.2: Dust Type
  • Theorem 2.1: Scaling Theorem
  • proof
  • Lemma 2.1
  • Theorem 2.2: Entire Extension Theorem
  • proof
  • Corollary 2.1
  • proof
  • Theorem 2.3: Threshold Theorem
  • ...and 15 more