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August & September  2019, 12(4&5): 723-725. doi: 10.3934/dcdss.2019046

A real attractor non admitting a connected feasible open set

Manuel Fernández-Martínez ,

University Centre of Defence at the Spanish Air Force Academy, MDE-UPCT, 30720 Santiago de la Ribera, Murcia, Spain

* Corresponding author: M. Fernández-Martínez

Received  September 2017 Revised  January 2018 Published  November 2018

Fund Project: The author has been partially supported by grants No. MTM2014-51891-P, No. MTM2015- 64373-P (both of them from Spanish Ministry of Economy and Competitiveness), and grant No. 19219/PI/14 from Fundación Séneca of Región de Murcia.

A self-similar set is described as the unique (nonempty) compact subset remaining invariant under the action of a finite collection of similitudes on a complete metric space. Among this kind of fractals, those satisfying the so-called Moran's open set condition are especially appropriate to deal with applications of Fractal Geometry since their Hausdorff dimensions can be easily computed. However, such a separation property depends on an external open set whose properties are not fully known. In this paper, we construct a self-similar set in the real line lying under the open set condition which does not admit a connected feasible open set. This answers an open question posed by Zhou and Li in 2009.

Keywords: Attractor, open set condition, feasible open set, connectedness, open question.
Mathematics Subject Classification: Primary: 28A80.
Citation: Manuel Fernández-Martínez. A real attractor non admitting a connected feasible open set. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 723-725. doi: 10.3934/dcdss.2019046
References:
[1]

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F. Hausdorff, Dimension und äußeres Maß, Mathematische Annalen, 79 (1918), 157-179.  doi: 10.1007/BF01457179.  Google Scholar

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D. Hilbert, Über die stetige Abbildung einer Line auf ein Flächenstück, Mathematische Annalen, 38 (1891), 459-460.  doi: 10.1007/BF01199431.  Google Scholar

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

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P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Mathematical Proceedings of the Cambridge Philosophical Society, 42 (1946), 15-23.  doi: 10.1017/S0305004100022684.  Google Scholar

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G. Peano, Sur une courbe, qui remplit toute une aire plane, Mathematische Annalen, 36 (1890), 157-160.  doi: 10.1007/BF01199438.  Google Scholar

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A. Schief, Separation properties for self-similar sets, Proceedings of the American Mathematical Society, 122 (1994), 111-115.  doi: 10.1090/S0002-9939-1994-1191872-1.  Google Scholar

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Z. Zhou and F. Li, Some problems on fractal geometry and topological dynamical systems, Analysis in Theory and Applications, 25 (2009), 5-15.  doi: 10.1007/s10496-009-0005-3.  Google Scholar

show all references

References:
[1]

G. Cantor, Ueber unendliche, lineare Punktmannichfaltigkeiten, Mathematische Annalen, 21 (1883), 545-591.  doi: 10.1007/BF01446819.  Google Scholar

[2]

F. Hausdorff, Dimension und äußeres Maß, Mathematische Annalen, 79 (1918), 157-179.  doi: 10.1007/BF01457179.  Google Scholar

[3]

D. Hilbert, Über die stetige Abbildung einer Line auf ein Flächenstück, Mathematische Annalen, 38 (1891), 459-460.  doi: 10.1007/BF01199431.  Google Scholar

[4]

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

[5]

P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Mathematical Proceedings of the Cambridge Philosophical Society, 42 (1946), 15-23.  doi: 10.1017/S0305004100022684.  Google Scholar

[6]

G. Peano, Sur une courbe, qui remplit toute une aire plane, Mathematische Annalen, 36 (1890), 157-160.  doi: 10.1007/BF01199438.  Google Scholar

[7]

A. Schief, Separation properties for self-similar sets, Proceedings of the American Mathematical Society, 122 (1994), 111-115.  doi: 10.1090/S0002-9939-1994-1191872-1.  Google Scholar

[8]

Z. Zhou and F. Li, Some problems on fractal geometry and topological dynamical systems, Analysis in Theory and Applications, 25 (2009), 5-15.  doi: 10.1007/s10496-009-0005-3.  Google Scholar

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