Irreducible fractal structures for Moran's type theorems

TL;DR

This paper introduces a new separation property for IFS-attractors that is weaker than the SOSC, enabling the equality of similarity and Hausdorff dimensions, and explores the nature of overlaps and dimension equivalences.

Contribution

It characterizes a novel separation property for IFS-attractors, providing conditions for dimension equalities and analyzing overlaps in a new framework.

Findings

A new separation property weaker than SOSC is introduced.

Conditions for equality of similarity and Hausdorff dimensions are established.

Equivalent conditions for dimension equality using finite coverings are provided.

Abstract

In this paper, we characterize a novel separation property for IFS-attractors on complete metric spaces. Such a separation property is weaker than the strong open set condition (SOSC) and becomes necessary to reach the equality between the similarity and the Hausdorff dimensions of strict self-similar sets. We also investigate the size of the overlaps from the viewpoint of that separation property. In addition, we contribute some equivalent conditions to reach the equality between the similarity dimension and a new Hausdorff type dimension for IFS-attractors introduced by the authors in terms of finite coverings.