Strong distributional chaos and minimal sets |
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Authors: | F. Balibrea |
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Affiliation: | a Departamento de Matemáticas, Universidad Murcia, Campus de Espinardo, 30100 Murcia, Spain b Mathematical Institute, Silesian University, 746 01 Opava, Czech Republic |
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Abstract: | In the class T of triangular maps of the square we consider the strongest notion of distributional chaos, DC1, originally introduced by Schweizer and Smítal [B. Schweizer, J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc. 344 (1994) 737-854] for continuous maps of the interval. We show that there is a DC1 homeomorphism F∈T such that any ω-limit set contains unique minimal set. This homeomorphism is constructed such that it is increasing on some fibres, and decreasing on the other ones. Consequently, F has zero topological entropy. Similar behavior is impossible when F is nondecreasing on the fibres, as shown by Paganoni and Smítal [L. Paganoni, J. Smítal, Strange distributionally chaotic triangular maps, Chaos Solitons Fractals 26 (2005) 581-589]. This result contributes to the solution of an old problem of Sharkovsky concerning classification of triangular maps but it is interesting by itself since it implies interesting open problems concerning relations between DC1 and minimality. |
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Keywords: | primary, 37D45, 54H20 secondary, 26A18 |
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