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

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, 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.
Keywords: dimension properties., Fractal, fractal dimension, fractal structure, Hausdorff dimension, box-counting dimension.
Mathematics Subject Classification: Primary: 28A80; Secondary: 37F35, 54E9.
Citation: Manuel Fernández-Martínez. Theoretical properties of fractal dimensions for fractal structures. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1113-1128. doi: 10.3934/dcdss.2015.8.1113
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F. G. Arenas and M. A. Sánchez-Granero, A characterization of non-archimedeanly quasimetrizable spaces, Rend. Istit. Mat. Univ. Trieste Suppl., 30 (1999), 21-30.

[2]

F. G. Arenas and M. A. Sánchez-Granero, A new approach to metrization, Topology Appl., 123 (2002), 15-26. doi: 10.1016/S0166-8641(01)00165-1.

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F. G. Arenas and M. A. Sánchez-Granero, A new metrization theorem, Boll. Unione Mat. Ital. (8), 5 (2002), 109-122.

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F. G. Arenas and M. A. Sańchez-Granero, A characterization of self-similar symbolic spaces, Mediterr. J. Math., 9 (2012), 709-728. doi: 10.1007/s00009-011-0146-4.

[5]

A. S. Besicovitch, Sets of fractional dimensions IV: On rational approximation to real numbers, J. Lond. Math. Soc., 9 (1934), 126-131. doi: 10.1112/jlms/s1-9.2.126.

[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), 18-25. doi: 10.1112/jlms/s1-12.45.18.

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C. Brown and L. Liebovitch, Fractal Analysis, in: Series 07-165: Quantitative Applications in the Social Sciences, First ed., SAGE Publications Inc., New York, 2010.

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C. Carathéodory, Über das lineare mass von punktmengen-eine verallgemeinerung das längenbegriffs, Nach. Ges. Wiss. Göttingen, (1914), 406-426.

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K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, John Wiley & Sons, Chichester, 1990.

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K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, Third Edition, John Wiley & Sons, Chichester, 2014.

[11]

J. Feder, Fractals, Plenum Press, New York, 1988. doi: 10.1007/978-1-4899-2124-6.

[12]

M. Fernández-Martínez and M. A. Sánchez-Granero, Fractal dimension for fractal structures, Topology Appl., 163 (2014), 93-111. doi: 10.1016/j.topol.2013.10.010.

[13]

M. Fernández-Martínez and M. A. Sánchez-Granero, Fractal dimension for fractal structures: A Hausdorff approach, Topology Appl., 159 (2012), 1825-1837. doi: 10.1016/j.topol.2011.04.023.

[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), 1193-1199. doi: 10.1016/j.amc.2012.07.029.

[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, Roma, 2013.

[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), 321-330. doi: 10.1016/j.jmaa.2013.07.011.

[17]

M. Fernández-Martínez and M. A. Sánchez-Granero, How to calculate the Hausdorff dimension using fractal structures, Appl. Math. Comput., 264 (2015), 116-131. doi: 10.1016/j.amc.2015.04.059.

[18]

F. Hausdorff, Dimension und äusseres mass, Math. Ann., 79 (1919), 157-179.

[19]

L. Pontrjagin and L. Schnirelman, Sur une proprieté métrique de la dimension, Ann. Math., 33 (1932), 156-162. doi: 10.2307/1968109.

[20]

M. A. Sánchez-Granero, Fractal structures, in Asymmetric Topology and its Applications, Quaderni di Matematica, 26, Seconda Univ. Napoli, Caserta, 2011, 211-245.

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), 21-30.

[2]

F. G. Arenas and M. A. Sánchez-Granero, A new approach to metrization, Topology Appl., 123 (2002), 15-26. doi: 10.1016/S0166-8641(01)00165-1.

[3]

F. G. Arenas and M. A. Sánchez-Granero, A new metrization theorem, Boll. Unione Mat. Ital. (8), 5 (2002), 109-122.

[4]

F. G. Arenas and M. A. Sańchez-Granero, A characterization of self-similar symbolic spaces, Mediterr. J. Math., 9 (2012), 709-728. doi: 10.1007/s00009-011-0146-4.

[5]

A. S. Besicovitch, Sets of fractional dimensions IV: On rational approximation to real numbers, J. Lond. Math. Soc., 9 (1934), 126-131. doi: 10.1112/jlms/s1-9.2.126.

[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), 18-25. doi: 10.1112/jlms/s1-12.45.18.

[7]

C. Brown and L. Liebovitch, Fractal Analysis, in: Series 07-165: Quantitative Applications in the Social Sciences, First ed., SAGE Publications Inc., New York, 2010.

[8]

C. Carathéodory, Über das lineare mass von punktmengen-eine verallgemeinerung das längenbegriffs, Nach. Ges. Wiss. Göttingen, (1914), 406-426.

[9]

K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, John Wiley & Sons, Chichester, 1990.

[10]

K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, Third Edition, John Wiley & Sons, Chichester, 2014.

[11]

J. Feder, Fractals, Plenum Press, New York, 1988. doi: 10.1007/978-1-4899-2124-6.

[12]

M. Fernández-Martínez and M. A. Sánchez-Granero, Fractal dimension for fractal structures, Topology Appl., 163 (2014), 93-111. doi: 10.1016/j.topol.2013.10.010.

[13]

M. Fernández-Martínez and M. A. Sánchez-Granero, Fractal dimension for fractal structures: A Hausdorff approach, Topology Appl., 159 (2012), 1825-1837. doi: 10.1016/j.topol.2011.04.023.

[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), 1193-1199. doi: 10.1016/j.amc.2012.07.029.

[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, Roma, 2013.

[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), 321-330. doi: 10.1016/j.jmaa.2013.07.011.

[17]

M. Fernández-Martínez and M. A. Sánchez-Granero, How to calculate the Hausdorff dimension using fractal structures, Appl. Math. Comput., 264 (2015), 116-131. doi: 10.1016/j.amc.2015.04.059.

[18]

F. Hausdorff, Dimension und äusseres mass, Math. Ann., 79 (1919), 157-179.

[19]

L. Pontrjagin and L. Schnirelman, Sur une proprieté métrique de la dimension, Ann. Math., 33 (1932), 156-162. doi: 10.2307/1968109.

[20]

M. A. Sánchez-Granero, Fractal structures, in Asymmetric Topology and its Applications, Quaderni di Matematica, 26, Seconda Univ. Napoli, Caserta, 2011, 211-245.

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