On discrete models of fractal dimension to explore the complexity of discrete dynamical systems

Abstract

A fractal structure is a countable family of coverings which displays accurate information about the irregularities that a set presents when being explored with enough level of detail. It is worth noting that fractal structures become especially appropriate to provide new definitions of fractal dimension, which constitutes a valuable measure to test for chaos in dynamical systems. In this paper, we explore several approaches to calculate the fractal dimension of a subset with respect to a fractal structure. These models generalize the classical box dimension in the context of Euclidean subspaces from a discrete viewpoint. To illustrate the flexibility of the new models, we calculate the fractal dimension of a family of self-affine sets associated with certain discrete dynamical systems.

Keywords:

• Discrete dynamical system
• fractal structure
• fractal dimension
• box-counting dimension
• chaos measure

AMS Subject Classifications:

• 37F35
• 28A78
• 28A80
• 54E99

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Fundación Séneca – Agencia de Ciencia y Tecnología de la Región de Murcia [19219/PI/14]; Spanish Ministry of Economy and Competitiveness [MTM2014-51891-P].