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

TL;DR

This paper introduces a systematic method to represent and analyze general fractals, including inhomogeneous and non self-similar types, using set-valued compression maps and $\\mathcal{F}$-limit sets, with applications to Hausdorff dimension estimation.

Contribution

It develops a unified framework for representing diverse fractals via $\\mathcal{F}$-limit sets and introduces the uniform covering condition for Hausdorff dimension bounds.

Findings

Representation of various fractals as $\\mathcal{F}$-limit sets.

Computational complexity is independent of the sequence type.

Derived bounds for Hausdorff dimensions of complex fractals.

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, Sierpi\'nski-like Triangles, etc.) with the calculations of their corresponding Hausdorff dimensions are also studied.