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On solenoidal distribution of infinite ω-limit sets

The aim of this paper is to present a structure result on the infinite ω-limit sets of continuous triangular mappings defined on the unit square, i.e. maps of the form (x, y)→(f(x), g(x, y)) having zero topological entropy in the base map f and on the fibres defined over the periodic points of f. We show for this class of systems that the infinite ω-limit sets of the points of the form (x, y), where x is a periodic point of f, have a solenoidal distribution. It extends the results by Smítal [J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), pp. 269–282] on zero topological entropy of continuous interval maps. An application is presented.