Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Specification properties and dense distributional chaos

Pages: 821 - 833, Volume 17, Issue 4, April 2007      doi:10.3934/dcds.2007.17.821

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Piotr Oprocha - Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland (email)

Abstract: The notion of distributional chaos was introduced by Schweizer and Smítal in [Trans. Amer. Math. Soc., 344 (1994) 737] for continuous maps of a compact interval. Further, this notion was generalized to three versions $d_1C$--$d_3C$ for maps acting on general compact metric spaces (see e.g. [Chaos Solitons Fractals, 23 (2005) 1581]). The main result of [ J. Math. Anal. Appl. , 241 (2000) 181] says that a weakened version of the specification property implies existence of a two points scrambled set which exhibits a $d_1 C$ version of distributional chaos. In this article we show that much more complicated behavior is present in that case. Strictly speaking, there exists an uncountable and dense scrambled set consisting of recurrent but not almost periodic points which exhibit uniform $d_1 C$ versions of distributional chaos.

Keywords:  distributional chaos, specification property, generalized specification property.
Mathematics Subject Classification:  Primary: 37D45; Secondary: 37B99.

Received: July 2006;      Revised: September 2006;      Available Online: January 2007.