Dimension theory of Non-Autonomous iterated function systems

TL;DR

This paper introduces non-autonomous attractors, generalizing Moran sets and iterated function system attractors, and studies their dimension properties, especially for affine sets with random translations, revealing new inequalities and dimension bounds.

Contribution

It defines non-autonomous attractors and affine sets, establishes dimension estimates, and explores the relationships between critical values and fractal dimensions, including random translation cases.

Findings

Upper box-counting and Hausdorff dimensions are bounded by s* and s_A.

s* is always greater than or equal to s_A, with possible strict inequality.

Hausdorff dimensions of random translation affine sets equal s_A almost surely.

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.