<abstract xmlns="http://www.w3.org/1999/xhtml"><p>Along the years, the foundations of Fractal Geometry have received contributions starting from mathematicians like Cantor, Peano, Hilbert, Hausdorff, Carathéodory, Sierpiński, and Besicovitch, to quote some of them. They were some of the pioneers exploring objects having self-similar patterns or showing anomalous properties with respect to standard analytic attributes. Among the new tools developed to deal with this kind of objects, fractal dimension has become one of the most applied since it constitutes a single quantity which throws useful information concerning fractal patterns on sets. Several years later, fractal structures were introduced from Asymmetric Topology to characterize self-similar symbolic spaces. Our aim in this survey is to collect several results involving distinct definitions of fractal dimension we proved jointly with Prof.M.A. Sánchez-Granero in the context of fractal structures.</p></abstract>

Along the years, the foundations of Fractal Geometry have received contributions starting from mathematicians like Cantor, Peano, Hilbert, Hausdorff, Carathéodory, Sierpiński, and Besicovitch, to quote some of them. They were some of the pioneers exploring objects having self-similar patterns or showing anomalous properties with respect to standard analytic attributes. Among the new tools developed to deal with this kind of objects, fractal dimension has become one of the most applied since it constitutes a single quantity which throws useful information concerning fractal patterns on sets. Several years later, fractal structures were introduced from Asymmetric Topology to characterize self-similar symbolic spaces. Our aim in this survey is to collect several results involving distinct definitions of fractal dimension we proved jointly with Prof.M.A. Sánchez-Granero in the context of fractal structures.

A survey on fractal dimension for fractal structures

Centro Universitario de la Defensa, Academia General del Aire

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"A survey on fractal dimension for fractal structures"}

Along the years, the foundations of Fractal Geometry have received contributions starting from mathematicians like Cantor, Peano, Hilbert, Hausdorff,...

Theoretical properties	dim_B	${dim}_{Γ}^{1}$ $\begin{array}{} {\rm{dim}}_{\bf{\Gamma }}^1 \end{array}$	${dim}_{Γ}^{2}$ $\begin{array}{} {\rm{dim}}_{\bf{\Gamma }}^2 \end{array}$	${dim}_{Γ}^{3}$ $\begin{array}{} {\rm{dim}}_{\bf{\Gamma }}^3 \end{array}$	${dim}_{Γ}^{4}$ $\begin{array}{} {\rm{dim}}_{\bf{\Gamma }}^4 \end{array}$	${dim}_{Γ}^{5}$ $\begin{array}{} {\rm{dim}}_{\bf{\Gamma }}^5 \end{array}$	${dim}_{Γ}^{6}$ $\begin{array}{} {\rm{dim}}_{\bf{\Gamma }}^6 \end{array}$	dim_H
Monotonicity	1	1	1	1	1	1	1	1
Finite stability	1	1	0	1	1	1	1	1
Countable stability	0	0	0	0	0	1	1	1
Countable sets	0	0	0	0	0	1	1	1
Closure dimension	0	1	1	1	0	1	1	1

A survey on fractal dimension for fractal structures

Published Online: Sep 08, 2016

Page range: 437 - 472

Received: Apr 12, 2016

Accepted: Sep 08, 2016

DOI: https://doi.org/10.21042/AMNS.2016.2.00037

Keywords
Fractals, fractal structures, fractal dimension, Hausdorff dimension, box-counting dimension, fractal geometry, fractal stochastic processes, Hurst exponent, applications of fractal dimension

© 2016 M. Fernández-Martínez, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

The table above summarizes all the theoretical properties satisfied by each definition of fractal dimension explored throughout this paper. We set 1 to denote that the corresponding property is satisfied by each fractal dimension and 0 otherwise (c.f. [42]).

A survey on fractal dimension for fractal structures

Published Online: Sep 08, 2016

Page range: 437 - 472

Received: Apr 12, 2016

Accepted: Sep 08, 2016

DOI: https://doi.org/10.21042/AMNS.2016.2.00037

KeywordsFractals, fractal structures, fractal dimension, Hausdorff dimension, box-counting dimension, fractal geometry, fractal stochastic processes, Hurst exponent, applications of fractal dimension

© 2016 M. Fernández-Martínez, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

The table above summarizes all the theoretical properties satisfied by each definition of fractal dimension explored throughout this paper. We set 1 to denote that the corresponding property is satisfied by each fractal dimension and 0 otherwise (c.f. [42]).

Keywords
Fractals, fractal structures, fractal dimension, Hausdorff dimension, box-counting dimension, fractal geometry, fractal stochastic processes, Hurst exponent, applications of fractal dimension