-
Previous Article
Dynamic systems based on preference graph and distance
- DCDS-S Home
- This Issue
-
Next Article
Theoretical properties of fractal dimensions for fractal structures
Generating pre-fractals to approach real IFS-attractors with a fixed Hausdorff dimension
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 |
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] |
C. Bandt and T. Retta, Topological spaces admitting a unique fractal structure, Fund. Math., 141 (1992), 257-268. |
[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. |
[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. |
[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. |
[9] |
C. Carathéodory, Über das lineare mass von punktmengen-eine verallgemeinerung das längenbegriffs, Nach. Ges. Wiss. Göttingen, (1914), 406-426. |
[10] |
K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, John Wiley & Sons, Chichester, 1990. |
[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: A Hausdorff approach, Topology Appl., 159 (2012), 1825-1837.
doi: 10.1016/j.topol.2011.04.023. |
[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. |
[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. |
[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. |
[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. |
[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] |
J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.
doi: 10.1512/iumj.1981.30.30055. |
[20] |
B. B. Mandelbrot, Fractals: Form, Chance and Dimension, W.H. Freeman & Company, San Francisco, 1977. |
[21] |
B. B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman & Company, New York, 1982. |
[22] |
P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Camb. Phil. Soc., 42 (1946), 15-23.
doi: 10.1017/S0305004100022684. |
[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. |
[24] |
L. Pontrjagin and L. Schnirelman, Sur une proprieté métrique de la dimension, Ann. Math., 33 (1932), 156-162.
doi: 10.2307/1968109. |
[25] |
M. A. Sánchez-Granero, Fractal structures, in Asymmetric Topology and its Applications, Quaderni di Matematica, 26 Seconda Univ. Napoli, Caserta, 2011, 211-245. |
[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. |
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] |
C. Bandt and T. Retta, Topological spaces admitting a unique fractal structure, Fund. Math., 141 (1992), 257-268. |
[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. |
[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. |
[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. |
[9] |
C. Carathéodory, Über das lineare mass von punktmengen-eine verallgemeinerung das längenbegriffs, Nach. Ges. Wiss. Göttingen, (1914), 406-426. |
[10] |
K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, John Wiley & Sons, Chichester, 1990. |
[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: A Hausdorff approach, Topology Appl., 159 (2012), 1825-1837.
doi: 10.1016/j.topol.2011.04.023. |
[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. |
[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. |
[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. |
[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. |
[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] |
J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.
doi: 10.1512/iumj.1981.30.30055. |
[20] |
B. B. Mandelbrot, Fractals: Form, Chance and Dimension, W.H. Freeman & Company, San Francisco, 1977. |
[21] |
B. B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman & Company, New York, 1982. |
[22] |
P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Camb. Phil. Soc., 42 (1946), 15-23.
doi: 10.1017/S0305004100022684. |
[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. |
[24] |
L. Pontrjagin and L. Schnirelman, Sur une proprieté métrique de la dimension, Ann. Math., 33 (1932), 156-162.
doi: 10.2307/1968109. |
[25] |
M. A. Sánchez-Granero, Fractal structures, in Asymmetric Topology and its Applications, Quaderni di Matematica, 26 Seconda Univ. Napoli, Caserta, 2011, 211-245. |
[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. |
[1] |
Raffaela Capitanelli, Maria Agostina Vivaldi. Uniform weighted estimates on pre-fractal domains. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 1969-1985. doi: 10.3934/dcdsb.2014.19.1969 |
[2] |
Michael L. Frankel, Victor Roytburd. Fractal dimension of attractors for a Stefan problem. Conference Publications, 2003, 2003 (Special) : 281-287. doi: 10.3934/proc.2003.2003.281 |
[3] |
Joseph Squillace. Estimating the fractal dimension of sets determined by nonergodic parameters. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5843-5859. doi: 10.3934/dcds.2017254 |
[4] |
Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887 |
[5] |
V. V. Chepyzhov, A. A. Ilyin. On the fractal dimension of invariant sets: Applications to Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 117-135. doi: 10.3934/dcds.2004.10.117 |
[6] |
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 |
[7] |
Uta Renata Freiberg. Einstein relation on fractal objects. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 509-525. doi: 10.3934/dcdsb.2012.17.509 |
[8] |
Umberto Mosco, Maria Agostina Vivaldi. Vanishing viscosity for fractal sets. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1207-1235. doi: 10.3934/dcds.2010.28.1207 |
[9] |
María Anguiano, Alain Haraux. The $\varepsilon$-entropy of some infinite dimensional compact ellipsoids and fractal dimension of attractors. Evolution Equations and Control Theory, 2017, 6 (3) : 345-356. doi: 10.3934/eect.2017018 |
[10] |
Hillel Furstenberg. From invariance to self-similarity: The work of Michael Hochman on fractal dimension and its aftermath. Journal of Modern Dynamics, 2019, 15: 437-449. doi: 10.3934/jmd.2019027 |
[11] |
Palle E. T. Jorgensen and Steen Pedersen. Orthogonal harmonic analysis of fractal measures. Electronic Research Announcements, 1998, 4: 35-42. |
[12] |
Abdon Atangana, Ali Akgül. On solutions of fractal fractional differential equations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3441-3457. doi: 10.3934/dcdss.2020421 |
[13] |
Seheon Ham, Hyerim Ko, Sanghyuk Lee. Circular average relative to fractal measures. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022100 |
[14] |
Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 235-246. doi: 10.3934/dcds.2008.22.235 |
[15] |
Michael Hochman. Lectures on dynamics, fractal geometry, and metric number theory. Journal of Modern Dynamics, 2014, 8 (3&4) : 437-497. doi: 10.3934/jmd.2014.8.437 |
[16] |
Alexander Plakhov, Vera Roshchina. Fractal bodies invisible in 2 and 3 directions. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1615-1631. doi: 10.3934/dcds.2013.33.1615 |
[17] |
Michael Barnsley, James Keesling, Mrinal Kanti Roychowdhury. Special issue on fractal geometry, dynamical systems, and their applications. Discrete and Continuous Dynamical Systems - S, 2019, 12 (8) : i-i. doi: 10.3934/dcdss.201908i |
[18] |
Eugen Mihailescu. Equilibrium measures, prehistories distributions and fractal dimensions for endomorphisms. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2485-2502. doi: 10.3934/dcds.2012.32.2485 |
[19] |
Thierry Coulbois. Fractal trees for irreducible automorphisms of free groups. Journal of Modern Dynamics, 2010, 4 (2) : 359-391. doi: 10.3934/jmd.2010.4.359 |
[20] |
Maria Rosaria Lancia, Paola Vernole. The Stokes problem in fractal domains: Asymptotic behaviour of the solutions. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1553-1565. doi: 10.3934/dcdss.2020088 |
2021 Impact Factor: 1.865
Tools
Metrics
Other articles
by authors
[Back to Top]