Minimality and distributional chaos in triangular maps

Abstract

The result of this paper contributes to the classification of triangular maps of the square with zero topological entropy stated by A. N. Sharkovsky in the 1980s. The problem was if a triangular map of the square such that its any ω-limit set contains unique minimal set can be distributionally chaotic. So far such result was disproved only for the class of triangular maps non-decreasing on fibres [L. Paganoni, J. Smítal, Strange distributionally chaotic triangular maps, Chaos Solitons Fractals 26(2) (2005), pp. 581–589]. In this paper, we solve the problem in negative for all triangular maps of the square, correcting the original result from Balibrea and Smítal [Strong distributional chaos and minimal sets, Topology appl. 156 (2009), pp. 1673–1678].