Counterexamples in theory of fractal dimension for fractal structures

TL;DR

This paper examines new models of fractal dimension within fractal structures, providing counterexamples to illustrate their mathematical behavior and potential applications, thereby highlighting their advantages over classical models.

Contribution

It introduces counterexamples to new fractal dimension models in fractal structures, demonstrating their properties and potential benefits over traditional models.

Findings

Counterexamples reveal limitations of new models

Illustrates how new models differ from classical ones

Highlights features useful for applications

Abstract

Fractal dimension constitutes the main tool to test for fractal patterns in Euclidean contexts. For this purpose, it is always used the box dimension, since it is easy to calculate, though the Hausdorff dimension, which is the oldest and also the most accurate fractal dimension, presents the best analytical properties. Additionally, fractal structures provide an appropriate topological context where new models of fractal dimension for a fractal structure could be developed in order to generalize the classical models of fractal dimension. In this paper, we provide some counterexamples regarding these new models of fractal dimension in order to show the reader how they behave mathematically with respect to the classical models, and also to point out which features of such models can be exploited to powerful effect in applications.