Generating pre-fractals to approach real IFS-attractors with a fixed Hausdorff dimension

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December 2015, 8(6): 1129-1137. doi: 10.3934/dcdss.2015.8.1129

Generating pre-fractals to approach real IFS-attractors with a fixed Hausdorff dimension

Manuel Fernández-Martínez ^1, and Miguel Ángel López Guerrero ^2,

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
2.	Department of Mathematics at University of Castilla-La Mancha, Campus Universitario de Cuenca, 16071 Cuenca, Spain

Received May 2015 Revised September 2015 Published December 2015

Abstract
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In this paper, we explain how to generate adequate pre-fractals in order to properly approximate attractors of iterated function systems on the real line within a priori known Hausdorff dimension. To deal with, we have applied the classical Moran's Theorem, so we have been focused on non-overlapping strict self-similar sets. This involves a quite significant hypothesis: the so-called open set condition. The main theoretical result contributed in this paper becomes quite interesting from a computational point of view, since in such a context, there is always a maximum level (of the natural fractal structure we apply in this work) that may be achieved.

Keywords: IFS-attractor, fractal structure, fractal dimension, iterated function system, Hausdorff dimension., Fractal, pre-fractal.

Mathematics Subject Classification: Primary: 28A80; Secondary: 37F35, 54E9.

Citation: Manuel Fernández-Martínez, Miguel Ángel López Guerrero. Generating pre-fractals to approach real IFS-attractors with a fixed Hausdorff dimension. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1129-1137. doi: 10.3934/dcdss.2015.8.1129

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]	C. Bandt and T. Retta, Topological spaces admitting a unique fractal structure,, Fund. Math., 141* (1992), 257. Google Scholar*
[6]	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
[7]	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
[8]	C. Brown and L. Liebovitch, Fractal Analysis, in: Series 07-165: Quantitative Applications in the Social Sciences,, First ed., (2010). Google Scholar
[9]	C. Carathéodory, Über das lineare mass von punktmengen-eine verallgemeinerung das längenbegriffs,, Nach. Ges. Wiss. Göttingen, (1914), 406. Google Scholar
[10]	K. Falconer, Fractal Geometry. Mathematical Foundations and Applications,, John Wiley & Sons, (1990). 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: A Hausdorff approach,, Topology Appl., 159* (2012), 1825. doi: 10.1016/j.topol.2011.04.023. Google Scholar*
[13]	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*
[14]	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*
[15]	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*
[16]	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
[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]	J. Hutchinson, Fractals and self-similarity,, Indiana Univ. Math. J., 30 (1981), 713. doi: 10.1512/iumj.1981.30.30055. Google Scholar
[20]	B. B. Mandelbrot, Fractals: Form, Chance and Dimension,, W.H. Freeman & Company, (1977). Google Scholar
[21]	B. B. Mandelbrot, The Fractal Geometry of Nature,, W.H. Freeman & Company, (1982). Google Scholar
[22]	P. A. P. Moran, Additive functions of intervals and Hausdorff measure,, Proc. Camb. Phil. Soc., 42 (1946), 15. doi: 10.1017/S0305004100022684. Google Scholar
[23]	K. Morita, Completion of hyperspaces of compact subsets and topological completion of open-closed maps,, General Topology Appl., 4 (1974), 217. doi: 10.1016/0016-660X(74)90023-3. Google Scholar
[24]	L. Pontrjagin and L. Schnirelman, Sur une proprieté métrique de la dimension,, Ann. Math., 33 (1932), 156. doi: 10.2307/1968109. Google Scholar
[25]	M. A. Sánchez-Granero, Fractal structures,, in Asymmetric Topology and its Applications, (2011), 211. Google Scholar
[26]	A. Schief, Separation properties for self-similar sets,, Proc. Amer. Math. Soc., 122* (1994), 111. doi: 10.1090/S0002-9939-1994-1191872-1. Google Scholar*

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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]	C. Bandt and T. Retta, Topological spaces admitting a unique fractal structure,, Fund. Math., 141* (1992), 257. Google Scholar*
[6]	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
[7]	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
[8]	C. Brown and L. Liebovitch, Fractal Analysis, in: Series 07-165: Quantitative Applications in the Social Sciences,, First ed., (2010). Google Scholar
[9]	C. Carathéodory, Über das lineare mass von punktmengen-eine verallgemeinerung das längenbegriffs,, Nach. Ges. Wiss. Göttingen, (1914), 406. Google Scholar
[10]	K. Falconer, Fractal Geometry. Mathematical Foundations and Applications,, John Wiley & Sons, (1990). 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: A Hausdorff approach,, Topology Appl., 159* (2012), 1825. doi: 10.1016/j.topol.2011.04.023. Google Scholar*
[13]	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*
[14]	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*
[15]	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*
[16]	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
[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]	J. Hutchinson, Fractals and self-similarity,, Indiana Univ. Math. J., 30 (1981), 713. doi: 10.1512/iumj.1981.30.30055. Google Scholar
[20]	B. B. Mandelbrot, Fractals: Form, Chance and Dimension,, W.H. Freeman & Company, (1977). Google Scholar
[21]	B. B. Mandelbrot, The Fractal Geometry of Nature,, W.H. Freeman & Company, (1982). Google Scholar
[22]	P. A. P. Moran, Additive functions of intervals and Hausdorff measure,, Proc. Camb. Phil. Soc., 42 (1946), 15. doi: 10.1017/S0305004100022684. Google Scholar
[23]	K. Morita, Completion of hyperspaces of compact subsets and topological completion of open-closed maps,, General Topology Appl., 4 (1974), 217. doi: 10.1016/0016-660X(74)90023-3. Google Scholar
[24]	L. Pontrjagin and L. Schnirelman, Sur une proprieté métrique de la dimension,, Ann. Math., 33 (1932), 156. doi: 10.2307/1968109. Google Scholar
[25]	M. A. Sánchez-Granero, Fractal structures,, in Asymmetric Topology and its Applications, (2011), 211. Google Scholar
[26]	A. Schief, Separation properties for self-similar sets,, Proc. Amer. Math. Soc., 122* (1994), 111. doi: 10.1090/S0002-9939-1994-1191872-1. Google Scholar*

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