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The three versions of distributional chaos. (English) Zbl 1069.37013
Summary: The notion of distributional chaos was introduced by {\it B. Schweizer} and {\it J. Smítal} [Trans. Am. Math. Soc. 344, 737--754 (1994; Zbl 0812.58062)] for continuous maps of the interval. However, it turns out that, for continuous maps of a compact metric space three mutually nonequivalent versions of distributional chaos, DC1--DC3, can be considered. Here, we consider the weakest one, DC3. We show that DC3 does not imply chaos in the sense of Li and Yorke. We also show that DC3 is not invariant with respect to topological conjugacy. In other words, there are lower and upper distribution functions $\Phi_{xy}$ and $\Phi^*_{xy}$ generated by a continuous map $f$ of a compact metric space $(M,\rho)$ such that $\Phi^*_{xy}(t)> \Phi_{xy}(t)$ for all $t$ in an interval. However, $f$ on the same space $M$, but with a metric $\rho'$ generating the same topology as $\rho$ is no more DC3. Recall that, contrary to this, either DC1 or DC2 is topological conjugacy invariant and implies Li and Yorke chaos (cf. {\it J. Smítal} and {\it M. Stefánková}, Chaos Solitons Fractals 21, 1125--1128 (2004; Zbl 1060.37037)]).

37B99Topological dynamics
54H20Topological dynamics
37D45Strange attractors, chaotic dynamics
Full Text: DOI
[1] Balibrea, F.; Reich, L.; Smı\acute{}tal, J.: Iteration theory: dynamical systems and functional equations. Int. J. Bifur. chaos 13, 1627-1648 (2003) · Zbl 1056.37003
[2] Forti, G. -L.; Paganoni, L.; Smı\acute{}tal, J.: Dynamics of homeomorphisms on minimal sets generated by triangular mappings. Bull. austral. Math. soc. 59, 1-20 (1999) · Zbl 0976.54043
[3] Bobok, J.: Chaos in countable dynamics. Topol. appl. 126, 207-216 (2002) · Zbl 1034.37005
[4] Schweizer, B.; Sklar, A.; Smı\acute{}tal, J.: Distributional (and other) chaos and its measurement. Real anal. Exchange 26, 495-524 (2000/2001) · Zbl 1012.37022
[5] Schweizer, B.; Smı\acute{}tal, J.: Measures of chaos and a spectral decomposition of dynamical systems on the interval. Trans. amer. Math. soc. 344, 737-854 (1994) · Zbl 0812.58062
[6] Smı\acute{}tal, J.; Štefánková, M.: Distributional chaos for triangular maps. Chaos, solitons & fractals 21, 1125-1128 (2004) · Zbl 1060.37037