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A real attractor non admitting a connected feasible open set
University Centre of Defence at the Spanish Air Force Academy, MDE-UPCT, 30720 Santiago de la Ribera, Murcia, Spain |
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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.
References:
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G. Cantor,
Ueber unendliche, lineare Punktmannichfaltigkeiten, Mathematische Annalen, 21 (1883), 545-591.
doi: 10.1007/BF01446819. |
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F. Hausdorff,
Dimension und äußeres Maß, Mathematische Annalen, 79 (1918), 157-179.
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D. Hilbert,
Über die stetige Abbildung einer Line auf ein Flächenstück, Mathematische Annalen, 38 (1891), 459-460.
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Fractals and self-similarity, Indiana University Mathematics Journal, 30 (1981), 713-747.
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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.
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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. |
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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. |
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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. |
show all references
References:
| [1] |
G. Cantor,
Ueber unendliche, lineare Punktmannichfaltigkeiten, Mathematische Annalen, 21 (1883), 545-591.
doi: 10.1007/BF01446819. |
| [2] |
F. Hausdorff,
Dimension und äußeres Maß, Mathematische Annalen, 79 (1918), 157-179.
doi: 10.1007/BF01457179. |
| [3] |
D. Hilbert,
Über die stetige Abbildung einer Line auf ein Flächenstück, Mathematische Annalen, 38 (1891), 459-460.
doi: 10.1007/BF01199431. |
| [4] |
J. E. Hutchinson,
Fractals and self-similarity, Indiana University Mathematics Journal, 30 (1981), 713-747.
doi: 10.1512/iumj.1981.30.30055. |
| [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. |
| [6] |
G. Peano,
Sur une courbe, qui remplit toute une aire plane, Mathematische Annalen, 36 (1890), 157-160.
doi: 10.1007/BF01199438. |
| [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. |
| [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. |
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