Fractal Dimension for Fractal Structures: A Hausdorff Approach

TL;DR

This paper introduces a new Hausdorff-based model for calculating fractal dimensions on generalized fractal spaces, enabling easier computation for self-similar sets without the open set condition.

Contribution

It generalizes existing fractal dimension definitions and provides a discretized Hausdorff approach applicable to a broader class of fractal structures.

Findings

New model for fractal dimension in generalized spaces

Connections established with classical fractal dimensions

Simplified calculation method for self-similar sets

Abstract

This paper provides a new model to compute the fractal dimension of a subset on a generalized-fractal space. Recall that fractal structures are a perfect place where a new definition of fractal dimension can be given, so we perform a suitable discretization of the Hausdorff theory of fractal dimension. We also find some connections between our definition and the classical ones and also with fractal dimensions I & II (see http://arxiv.org/submit/0080421/pdf). Therefore, we generalize them and obtain an easy method in order to calculate the fractal dimension of strict self-similar sets which are not required to verify the open set condition.