Fractal Dimension for IFS-Attractors Revisited
- Fernández-Martínez, M 1
- Guirao, J L G 2
- Vera López, Juan Antonio 1
- 1 University Centre of Defence at the Spanish Air Force Academy
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2
Universidad Politécnica de Cartagena
info
ISSN: 1575-5460
Datum der Publikation: 2018
Ausgabe: 17
Nummer: 3
Seiten: 709-722
Art: Artikel
Andere Publikationen in: Qualitative theory of dynamical systems
Zusammenfassung
One of the milestones in Fractal Geometry is the so-called Moran’s Theorem, which allows the calculation of the similarity dimension of any strict self-similar set under the open set condition. In this paper, we contribute a generalized version of the Moran’s theorem, which does not require the OSC to be satisfied by the similitudes that give rise to the corresponding attractor. To deal with, two generalized versions for the classical fractal dimensions, namely, the box and the Hausdorff dimensions, are explored in terms of fractal structures, a kind of uniform spaces.
Informationen zur Finanzierung
Geldgeber
-
Fundación Séneca
- 19219/PI/14
- 19219/PI/14
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Secretaría de Estado de Investigación, Desarrollo e Innovación
- MTM2014-51891-P
- MTM2014-51891-P
Bibliographische Referenzen
- 1. Arenas, F.G., Sánchez-Granero, M.A.: A characterization of self-similar symbolic spaces. Mediterr. J. Math. 9(4), 709–728 (2012)
- 2. Bandt, C., Hung, N.V., Rao, H.: On the open set condition for self-similar fractals. Proc. Am. Math. Soc. 134(5), 1369–1374 (2005)
- 3. Bandt, C., Retta, T.: Topological spaces admitting a unique fractal structure. Fundam. Math. 141(3), 257–268 (1992)
- 4. Deng, QiRong, Harding, John, TianYou, Hu: Hausdorff dimension of self-similar sets with overlaps. Sci. China Ser. A Math. 52(1), 119–128 (2009)
- 5. Falconer, K.: Fractal Geometry.Mathematical Foundations and Applications, 1st edn.Wiley, Chichester (1990)
- 6. Fernández-Martínez, M.: A survey on fractal dimension for fractal structures. Appl. Math. Nonlinear Sci. 1(2), 437–472 (2016)
- 7. Fernández-Martínez, M., Sánchez-Granero, M.A.: Fractal dimension for fractal structures: a Hausdorff approach. Topol. Appl. 159(7), 1825–1837 (2012)
- 8. Fernández-Martínez,M., Sánchez-Granero,M.A.: Fractal dimension for fractal structures. Topol. Appl. 163, 93–111 (2014)
- 9. Fernández-Martínez, M., Sánchez-Granero, M.A.: Fractal dimension for fractal structures: a Hausdorff approach revisited. J. Math. Anal. Appl. 409(1), 321–330 (2014)
- 10. Fernández-Martínez, M., Sánchez-Granero, M.A.: How to calculate the Hausdorff dimension using fractal structures. Appl. Math. Comput. 264, 116–131 (2015)
- 11. Fernández-Martínez,M., Sánchez-Granero,M.A., Segovia, J.E. Trinidad: Fractal dimensions for fractal structures and their applications to financial markets, Aracne Editrice, S.r.l., Roma (2013)
- 12. Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30(5), 713–747 (1981)
- 13. Lalley, S.P.: The packing and covering functions of some self-similar fractals. Indiana Univ. Math. J. 37(3), 699–710 (1988)
- 14. Moran, P.A.P.: Additive functions of intervals and Hausdorff measure. Math. Proc. Camb. Philos. Soc. 42(1), 15–23 (1946)
- 15. Ngai, S.-M., Wang, Yang: Hausdorff dimension of self-similar sets with overlaps. J. Lond. Math. Soc. 63(3), 655–672 (2001)
- 16. Schief, A.: Separation properties for self-similar sets. Proc. Am. Math. Soc. 122(1), 111–115 (1994)
- 17. Schief, A.: Self-similar sets in complete metric spaces. Proc. Am. Math. Soc. 124(2), 481–490 (1996)