Generalizing Moran's Theorem

TL;DR

This paper extends Moran's Theorem to compute fractal dimensions of IFS attractors without needing the open set condition, broadening its applicability to more complex fractal structures.

Contribution

It introduces a generalized fractal dimension model that removes the open set condition requirement for IFS attractors.

Findings

Fractal dimensions can be calculated without OSC.

The model applies to a wider class of fractals.

It simplifies the computation of attractor dimensions.

Abstract

This work is aimed by the spirit of 1946 Moran's Theorem, which ensures that both the box and the Hausdorff dimensions for any attractor could be calculated as the solution of an equation involving only its similarity factors. To achieve such result, the open set condition (OSC, herein) is required to be satisfied by the pieces of the attractor in order to guarantee that they do not overlap too much. In this paper, we generalize the classical Moran's through a fractal dimension model which allows to obtain the fractal dimension of any IFS-attractor equipped with its natural fractal structure without requiring the OSC to be satisfied.