Strange distributionally chaotic triangular maps

The notion of distributional chaos was introduced by Schweizer, Smítal [Measures of chaos and a spectral decompostion of dynamical systems on the interval. Trans. Amer. Math. Soc. 344;1994:737–854] for continuous maps of the interval. For continuous maps of a compact metric space three mutually nonequivalent versions of distributional chaos, DC1–DC3, can be considered. In this paper we study distributional chaos in the class Tm of triangular maps of the square which are monotone on the fibres; such maps must have zero topological entropy. The main results: (i) There is an F in Tm such that F is not DC2 and F|Rec(F) is DC3. (ii) If no ω -limit set of an F in Tm contains two minimal subsets then F is not DC1. This completes recent results obtained by Forti et al. [Dynamics of homeomorphisms on minimal sets generated by triangular mappings. Bull Austral Math Soc 59;1999:1–20], Smítal, Štefánková [Distributional chaos for triangular maps, Chaos, Solitons & Fractals 21;2004:1125–8], and Balibrea et al. [The three versions of distributional chaos. Chaos, Solitons & Fractals 23;2005:1581–3]. The paper contributes to the solution of a long-standing open problem by Sharkovsky concerning classification of triangular maps.