Strange distributionally chaotic triangular maps II

The notion of distributional chaos was introduced by Schweizer and Smítal [Measures of chaos and a spectral decompostion of dynamical systems on the interval. Trans. Amer. Math. Soc. 1994; 344:737–854] 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. The main results: (i) If F in Tm has positive topological entropy then F is DC1, and hence, DC2 and DC3. This result is interesting since similar statement is not true for general triangular maps of the square [Smítal and Štefánková, Distributional chaos for triangular maps, Chaos, Solitons & Fractals 2004; 21:1125–8]. (ii) There are F1,F2 in Tm which are not DC3, and such that not every recurrent point of F1 is uniformly recurrent, while F2 is Li and Yorke chaotic on the set of uniformly recurrent points. This, along with recent results by Forti et al. [Dynamics of homeomorphisms on minimal sets generated by triangular mappings. Bull Austral Math Soc 59;1999:1–20], among others, make possible to compile complete list of the implications between dynamical properties of maps in Tm, solving a long-standing open problem by Sharkovsky.