Theoretical properties of fractal dimensions for fractal structures

Manuel  Fernández-Martínez

doi:10.3934/dcdss.2015.8.1113

Article Contents

2015, Volume 8, Issue 6: 1113-1128. Doi: 10.3934/dcdss.2015.8.1113

This issue Previous Article Research on operating mechanism for creative products supply chain based on game theory Next Article Generating pre-fractals to approach real IFS-attractors with a fixed Hausdorff dimension

Theoretical properties of fractal dimensions for fractal structures

Manuel Fernández-Martínez^1,

1.
University Centre of Defence at the Spanish Air Force Academy, MDE-UPCT, Coronel López Peña Street, w/n, 30720 Santiago de la Ribera, Murcia

Received: May 31, 2015

Revised: July 31, 2015

Published: November 2015

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Abstract

Hausdorff dimension, which is the oldest and also the most accurate model for fractal dimension, constitutes the main reference for any fractal dimension definition that could be provided. In fact, its definition is quite general, and is based on a measure, which makes the Hausdorff model pretty desirable from a theoretical point of view. On the other hand, it turns out that fractal structures provide a perfect context where a new definition of fractal dimension could be proposed. Further, it has been already shown that both Hausdorff and box dimensions can be generalized by some definitions of fractal dimension formulated in terms of fractal structures. Given this, and being mirrored in some of the properties satisfied by Hausdorff dimension, in this paper we explore which ones are satisfied by the fractal dimension definitions for a fractal structure, that are explored along this work.

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Mathematics Subject Classification: Primary: 28A80; Secondary: 37F35, 54E99.

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