Fractal Dimension for Fractal Structures

TL;DR

This paper introduces a new, versatile definition of fractal dimension applicable to a broader class of spaces, including non-metrizable ones, with practical applications in language analysis and information streams.

Contribution

It proposes a novel fractal dimension definition compatible with any space supporting a fractal structure, extending classical box-counting methods and enabling new applications.

Findings

New fractal dimension definition applicable to non-metrizable spaces

Application to language structures and information streams

Demonstrated calculation of fractal dimension for languages generated by regular expressions

Abstract

The main goal of this paper has a double purpose. On the one hand, we propose a new definition in order to compute the fractal dimension of a subset respect to any fractal structure, which completes the theory of classical box-counting dimension. Indeed, if we select the so called natural fractal structure on each euclidean space, then we will get the box-counting dimension as a particular case. Recall that box-counting dimension could be calculated over any euclidean space, although it can be defined over any metrizable one. Nevertheless, the new definition we present can be computed on an easy way over any space admitting a fractal structure. Thus, since a space is metrizable if and only if it supports a starbase fractal structure, our model allows to classify and distinguish a much larger number of topological spaces than the classical definition. On the other hand, our aim consists also of studying some applications of effective calculation of the fractal dimension over a kind of contexts where the box-counting dimension has no sense, like the domain of words, which appears when modeling the streams of information in Kahn's parallel computation model. In this way, we show how to calculate and understand the fractal dimension value obtained for a language generated by means of a regular expression, and also we pay attention to an empirical and novel application of fractal dimension to natural languages.