Strange distributionally chaotic triangular maps II

The notion of distributional chaos was introduced by Schweizer and Smital [Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans Am Math Soc 1994;344:737-854] for continuous maps of the interval. For continuous maps of a compact metric space three mutually non-equivalent versions of distributional chaos, DC1-DC3, can be considered. In this paper we study distributional chaos in the class J(m) of triangular maps of the square which are monotone on the fibres. The main results: (i) If F is an element of J(m) 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 [Smital and stefankova, Distributional chaos for triangular maps, Chaos, Solitons & Fractals 2004;21:1125-8]. (ii) There are F-1,F-2 is an element of J(m) which are not DC3, and such that not every recurrent point of F, is uniformly recurrent, while F-2 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 1999;59:1-20], among others, make possible to compile complete list of the implications between dynamical properties of maps in J(m) solving a long-standing open problem by Sharkovsky. (c) 2005 Elsevier Ltd. All rights reserved.