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Dimension theory of Non-Autonomous iterated function systems

Yifei Gu, Jun Jie Miao

Abstract

In the paper, we define a class of new fractals named ``non-autonomous attractors", which are the generalization of classic Moran sets and attractors of iterated function systems. Simply to say, we replace the similarity mappings by contractive mappings and remove the separation assumption in Moran structure. We give the dimension estimate for non-autonomous attractors. Furthermore, we study a class of non-autonomous attractors, named `` non-autonomous affine sets or affine sets'', where the contractions are restricted to affine mappings. To study the dimension theory of such fractals, we define two critical values $s^*$ and $s_A$, and the upper box-counting dimensions and Hausdorff dimensions of non-autonomous affine sets are bounded above by $s^*$ and $s_A$, respectively. Unlike self-affine fractals where $s^*=s_A$, we always have that $s^*\geq s_A$, and the inequality may strictly hold. Under certain conditions, we obtain that the upper box-counting dimensions and Hausdorff dimensions of non-autonomous affine sets may equal to $s^*$ and $s_A$, respectively. In particular, we study non-autonomous affine sets with random translations, and the Hausdorff dimensions of such sets equal to $s_A$ almost surely.

Dimension theory of Non-Autonomous iterated function systems

Abstract

In the paper, we define a class of new fractals named ``non-autonomous attractors", which are the generalization of classic Moran sets and attractors of iterated function systems. Simply to say, we replace the similarity mappings by contractive mappings and remove the separation assumption in Moran structure. We give the dimension estimate for non-autonomous attractors. Furthermore, we study a class of non-autonomous attractors, named `` non-autonomous affine sets or affine sets'', where the contractions are restricted to affine mappings. To study the dimension theory of such fractals, we define two critical values and , and the upper box-counting dimensions and Hausdorff dimensions of non-autonomous affine sets are bounded above by and , respectively. Unlike self-affine fractals where , we always have that , and the inequality may strictly hold. Under certain conditions, we obtain that the upper box-counting dimensions and Hausdorff dimensions of non-autonomous affine sets may equal to and , respectively. In particular, we study non-autonomous affine sets with random translations, and the Hausdorff dimensions of such sets equal to almost surely.
Paper Structure (12 sections, 20 theorems, 194 equations, 3 figures)

This paper contains 12 sections, 20 theorems, 194 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Self-affine sets
  3. Moran sets
  4. Non-autonomous iterated function systems
  5. Definition of non-autonomous iterated function systems
  6. Non-autonomous affine iterated function systems
  7. Main conclusions for non-autonomous affine sets
  8. Dimension estimates of non-autonomous sets
  9. Box-counting dimensions of non-autonomous affine sets
  10. Hausdorff dimension of non-autonomous affine sets
  11. Affine Moran set with random translations
  12. Comparison of critical values and some examples

Key Result

Theorem 2.1

Let $E$ be the non-autonomous attractor given by attractor. For each integer $k\geq 1$, we assume that where $c_{k,i}<1$, for all $i=1,2,\ldots, n_k$. Let $d_*$ and $d^*$ be given by defd. Then Furthermore, if $c_*=\inf \{c_{k,j} : k\in \mathbb{N}, 1\leq j\leq n_k \}>0$, we have that

Figures (3)

  • Figure 1: non-autonomous affine set with positive finite Lebesgue measure
  • Figure 2: non-autonomous affine set contains countably many elements
  • Figure 3: Self-affine set contains only 3 elements, where $n=3$ and $\mathcal{D}=\{\binom{0}{0}, \binom{ 1 }{ 2 }, \binom{ 2 }{ 0 } \}$.

Theorems & Definitions (38)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Corollary 2.10
  • ...and 28 more