# American Institute of Mathematical Sciences

December  2015, 8(6): 1113-1128. doi: 10.3934/dcdss.2015.8.1113

## Theoretical properties of fractal dimensions for fractal structures

 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, Spain

Received  June 2015 Revised  August 2015 Published  December 2015

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.
Citation: Manuel Fernández-Martínez. Theoretical properties of fractal dimensions for fractal structures. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1113-1128. doi: 10.3934/dcdss.2015.8.1113
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