The three versions of distributional chaos

The notion of distributional chaos was introduced by Schweizer and Smital [Trans. Amer. Math. Soc. 344 (1994) 737] 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. In this paper 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. [Chaos, Solitons & Fractals 21 (2004) 1125]). (C) 2004 Elsevier Ltd. All rights reserved.