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General fractals represented by $\mathcal{F}$-limit sets of compression maps

Tynan Lazarus, Enrique G Alvarado, Qinglan Xia

Abstract

In this article, we provide a simple and systematic way to represent general (inhomogeneous) fractals that may look different at different scales and places. By using set-valued compression maps, we express these general fractals as $\mathcal{F}$-limit sets, which are represented as sequences of points in a fixed parameterization space $M$. By choosing different types of sequences in $M$, we get various types of fractals: from self-simlilar to non self-similar, and from deterministic to random. The computational complexity of producing a general fractal is independent of the sequence in $M$, and as a result, is the same as that of an iterated function system obtained from a constant sequence. In the metric space setting, we also estimate the Hausdorff dimension of limit sets for collections of sets that do not necessarily satisfy the Moran structure conditions. In particular, we introduce the concept ``uniform covering condition" for the study of the lower bound of the Hausdorff dimension of the limit set, and provide sufficient conditions for this condition. Specific examples (Cantor-like sets, Sierpiński-like Triangles, etc.) with the calculations of their corresponding Hausdorff dimensions are also studied.

General fractals represented by $\mathcal{F}$-limit sets of compression maps

Abstract

In this article, we provide a simple and systematic way to represent general (inhomogeneous) fractals that may look different at different scales and places. By using set-valued compression maps, we express these general fractals as -limit sets, which are represented as sequences of points in a fixed parameterization space . By choosing different types of sequences in , we get various types of fractals: from self-simlilar to non self-similar, and from deterministic to random. The computational complexity of producing a general fractal is independent of the sequence in , and as a result, is the same as that of an iterated function system obtained from a constant sequence. In the metric space setting, we also estimate the Hausdorff dimension of limit sets for collections of sets that do not necessarily satisfy the Moran structure conditions. In particular, we introduce the concept ``uniform covering condition" for the study of the lower bound of the Hausdorff dimension of the limit set, and provide sufficient conditions for this condition. Specific examples (Cantor-like sets, Sierpiński-like Triangles, etc.) with the calculations of their corresponding Hausdorff dimensions are also studied.

Paper Structure

This paper contains 9 sections, 10 theorems, 131 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Hausdorff Dimension of the Limit Sets
  3. General Setup of ${\mathcal{F}}$-Limit sets
  4. Hausdorff dimensions of ${\mathcal{F}}$-Limit sets
  5. Examples of ${\mathcal{F}}$-Limit sets
  6. Cantor-like sets
  7. Sierpiński Triangle
  8. Menger Sponge
  9. Sufficient conditions for the Uniform Covering Condition

Key Result

Proposition \oldthetheorem

Suppose $\mathcal{J} := \{J_{\sigma} : \sigma \in D\}$ is a collection of bounded subsets of a metric space $(X,d)$, and $s>0$. Let $E_{k} = \bigcup_{\sigma \in D_{k}} J_{\sigma}, \text{ and } F = \bigcap_{k \geq 0} E_{k}$ be defined as in (eqn: Limit_set). If there exists a sequence of positive num and for all $\sigma\in D_{k-1}$ and all $k=1,2,\cdots$, then $dim_H(F) \le s$.

Figures (19)

  • Figure 1: Generations $k = 0, 1, 2$, and $3$ of the tree structure of a $2$-ary tree $D$.
  • Figure 2: Comparison of classical Cantor set (blue) and new Cantor-like set (red)
  • Figure 3: A randomly generated Cantor-like set
  • Figure 4: Fractal of measure $\frac{1}{3}$ created by using $\sum_{n=0}^{\infty} \frac{1}{n!} = e$
  • Figure 5: Geometric illustration of $p = (p_1, p_2, p_3, p_4, p_5, p_6) \in M$
  • ...and 14 more figures

Theorems & Definitions (44)

  • Definition 1.1
  • Definition 1.2: Su
  • Proposition \oldthetheorem
  • proof
  • Definition 2.1: uniform covering condition
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Remark 3.1
  • ...and 34 more