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∈Tm such that F∉DC2 and F∣Rec(F) ∈ DC3. (ii) If no ω-limit set of an F∈Tm contains two minimal subsets then F∉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, Å tefánková [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.