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
References:
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F. G. Arenas and M. A. Sańchez-Granero, A characterization of self-similar symbolic spaces,, Mediterr. J. Math., 9 (2012), 709. doi: 10.1007/s00009-011-0146-4. Google Scholar

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A. S. Besicovitch and H. D. Ursell, Sets of fractional dimensions V: On dimensional numbers of some continuous curves,, J. Lond. Math. Soc., 12 (1937), 18. doi: 10.1112/jlms/s1-12.45.18. Google Scholar

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K. Falconer, Fractal Geometry. Mathematical Foundations and Applications,, John Wiley & Sons, (1990). Google Scholar

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K. Falconer, Fractal Geometry. Mathematical Foundations and Applications,, Third Edition, (2014). Google Scholar

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J. Feder, Fractals,, Plenum Press, (1988). doi: 10.1007/978-1-4899-2124-6. Google Scholar

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M. Fernández-Martínez and M. A. Sánchez-Granero, Fractal dimension for fractal structures,, Topology Appl., 163 (2014), 93. doi: 10.1016/j.topol.2013.10.010. Google Scholar

[13]

M. Fernández-Martínez and M. A. Sánchez-Granero, Fractal dimension for fractal structures: A Hausdorff approach,, Topology Appl., 159 (2012), 1825. doi: 10.1016/j.topol.2011.04.023. Google Scholar

[14]

M. Fernández-Martínez, M. A. Sánchez-Granero and J. E. Trinidad Segovia, Fractal dimension for fractal structures: Applications to the domain of words,, Appl. Math. Comput., 219 (2012), 1193. doi: 10.1016/j.amc.2012.07.029. Google Scholar

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M. Fernández-Martínez, M. A. Sánchez-Granero and J. E. Trinidad Segovia, Fractal Dimensions for Fractal Structures and Their Applications to Financial Markets,, Aracne, (2013). Google Scholar

[16]

M. Fernández-Martínez and M. A. Sánchez-Granero, Fractal dimension for fractal structures: A Hausdorff approach revisited,, Journal of Mathematical Analysis and Applications, 409 (2014), 321. doi: 10.1016/j.jmaa.2013.07.011. Google Scholar

[17]

M. Fernández-Martínez and M. A. Sánchez-Granero, How to calculate the Hausdorff dimension using fractal structures,, Appl. Math. Comput., 264 (2015), 116. doi: 10.1016/j.amc.2015.04.059. Google Scholar

[18]

F. Hausdorff, Dimension und äusseres mass,, Math. Ann., 79 (1919), 157. Google Scholar

[19]

L. Pontrjagin and L. Schnirelman, Sur une proprieté métrique de la dimension,, Ann. Math., 33 (1932), 156. doi: 10.2307/1968109. Google Scholar

[20]

M. A. Sánchez-Granero, Fractal structures,, in Asymmetric Topology and its Applications, (2011), 211. Google Scholar

show all references

References:
[1]

F. G. Arenas and M. A. Sánchez-Granero, A characterization of non-archimedeanly quasimetrizable spaces,, Rend. Istit. Mat. Univ. Trieste Suppl., 30 (1999), 21. Google Scholar

[2]

F. G. Arenas and M. A. Sánchez-Granero, A new approach to metrization,, Topology Appl., 123 (2002), 15. doi: 10.1016/S0166-8641(01)00165-1. Google Scholar

[3]

F. G. Arenas and M. A. Sánchez-Granero, A new metrization theorem,, Boll. Unione Mat. Ital. (8), 5 (2002), 109. Google Scholar

[4]

F. G. Arenas and M. A. Sańchez-Granero, A characterization of self-similar symbolic spaces,, Mediterr. J. Math., 9 (2012), 709. doi: 10.1007/s00009-011-0146-4. Google Scholar

[5]

A. S. Besicovitch, Sets of fractional dimensions IV: On rational approximation to real numbers,, J. Lond. Math. Soc., 9 (1934), 126. doi: 10.1112/jlms/s1-9.2.126. Google Scholar

[6]

A. S. Besicovitch and H. D. Ursell, Sets of fractional dimensions V: On dimensional numbers of some continuous curves,, J. Lond. Math. Soc., 12 (1937), 18. doi: 10.1112/jlms/s1-12.45.18. Google Scholar

[7]

C. Brown and L. Liebovitch, Fractal Analysis, in: Series 07-165: Quantitative Applications in the Social Sciences,, First ed., (2010). Google Scholar

[8]

C. Carathéodory, Über das lineare mass von punktmengen-eine verallgemeinerung das längenbegriffs,, Nach. Ges. Wiss. Göttingen, (1914), 406. Google Scholar

[9]

K. Falconer, Fractal Geometry. Mathematical Foundations and Applications,, John Wiley & Sons, (1990). Google Scholar

[10]

K. Falconer, Fractal Geometry. Mathematical Foundations and Applications,, Third Edition, (2014). Google Scholar

[11]

J. Feder, Fractals,, Plenum Press, (1988). doi: 10.1007/978-1-4899-2124-6. Google Scholar

[12]

M. Fernández-Martínez and M. A. Sánchez-Granero, Fractal dimension for fractal structures,, Topology Appl., 163 (2014), 93. doi: 10.1016/j.topol.2013.10.010. Google Scholar

[13]

M. Fernández-Martínez and M. A. Sánchez-Granero, Fractal dimension for fractal structures: A Hausdorff approach,, Topology Appl., 159 (2012), 1825. doi: 10.1016/j.topol.2011.04.023. Google Scholar

[14]

M. Fernández-Martínez, M. A. Sánchez-Granero and J. E. Trinidad Segovia, Fractal dimension for fractal structures: Applications to the domain of words,, Appl. Math. Comput., 219 (2012), 1193. doi: 10.1016/j.amc.2012.07.029. Google Scholar

[15]

M. Fernández-Martínez, M. A. Sánchez-Granero and J. E. Trinidad Segovia, Fractal Dimensions for Fractal Structures and Their Applications to Financial Markets,, Aracne, (2013). Google Scholar

[16]

M. Fernández-Martínez and M. A. Sánchez-Granero, Fractal dimension for fractal structures: A Hausdorff approach revisited,, Journal of Mathematical Analysis and Applications, 409 (2014), 321. doi: 10.1016/j.jmaa.2013.07.011. Google Scholar

[17]

M. Fernández-Martínez and M. A. Sánchez-Granero, How to calculate the Hausdorff dimension using fractal structures,, Appl. Math. Comput., 264 (2015), 116. doi: 10.1016/j.amc.2015.04.059. Google Scholar

[18]

F. Hausdorff, Dimension und äusseres mass,, Math. Ann., 79 (1919), 157. Google Scholar

[19]

L. Pontrjagin and L. Schnirelman, Sur une proprieté métrique de la dimension,, Ann. Math., 33 (1932), 156. doi: 10.2307/1968109. Google Scholar

[20]

M. A. Sánchez-Granero, Fractal structures,, in Asymmetric Topology and its Applications, (2011), 211. Google Scholar

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