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The three versions of distributional chaos. (English) Zbl 1069.37013

Summary: The notion of distributional chaos was introduced by B. Schweizer and 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 $\left(M,\rho \right)$ such that ${{\Phi }}_{xy}^{*}\left(t\right)>{{\Phi }}_{xy}\left(t\right)$ for all $t$ in an interval. However, $f$ on the same space $M$, but with a metric ${\rho }^{\text{'}}$ 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. J. Smítal and M. Stefánková, Chaos Solitons Fractals 21, 1125–1128 (2004; Zbl 1060.37037)]).

##### MSC:
 37B99 Topological dynamics 54H20 Topological dynamics 37D45 Strange attractors, chaotic dynamics