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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

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), 21-30.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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C. Bandt and T. Retta, Topological spaces admitting a unique fractal structure, Fund. Math., 141 (1992), 257-268.  Google Scholar

[6]

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

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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. Google Scholar

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

[10]

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

[11]

J. Feder, Fractals, Plenum Press, New York, 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), 1825-1837. 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), 93-111. 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), 321-330. 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), 1193-1199. 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, Roma, 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), 116-131. doi: 10.1016/j.amc.2015.04.059.  Google Scholar

[18]

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

[19]

J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055.  Google Scholar

[20]

B. B. Mandelbrot, Fractals: Form, Chance and Dimension, W.H. Freeman & Company, San Francisco, 1977.  Google Scholar

[21]

B. B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman & Company, New York, 1982.  Google Scholar

[22]

P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Camb. Phil. Soc., 42 (1946), 15-23. 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), 217-233. 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), 156-162. doi: 10.2307/1968109.  Google Scholar

[25]

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

[26]

A. Schief, Separation properties for self-similar sets, Proc. Amer. Math. Soc., 122 (1994), 111-115. doi: 10.1090/S0002-9939-1994-1191872-1.  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), 21-30.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

C. Bandt and T. Retta, Topological spaces admitting a unique fractal structure, Fund. Math., 141 (1992), 257-268.  Google Scholar

[6]

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.  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), 18-25. 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., SAGE Publications Inc., New York, 2010. Google Scholar

[9]

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

[10]

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

[11]

J. Feder, Fractals, Plenum Press, New York, 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), 1825-1837. 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), 93-111. 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), 321-330. 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), 1193-1199. 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, Roma, 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), 116-131. doi: 10.1016/j.amc.2015.04.059.  Google Scholar

[18]

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

[19]

J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055.  Google Scholar

[20]

B. B. Mandelbrot, Fractals: Form, Chance and Dimension, W.H. Freeman & Company, San Francisco, 1977.  Google Scholar

[21]

B. B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman & Company, New York, 1982.  Google Scholar

[22]

P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Camb. Phil. Soc., 42 (1946), 15-23. 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), 217-233. 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), 156-162. doi: 10.2307/1968109.  Google Scholar

[25]

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

[26]

A. Schief, Separation properties for self-similar sets, Proc. Amer. Math. Soc., 122 (1994), 111-115. doi: 10.1090/S0002-9939-1994-1191872-1.  Google Scholar

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