On skew-product maps with the base having a closed set of periodic points

Sharkovskii˘ and Kolyada reported the problem of the characterization of skew-product maps having zero topological entropy. It is known that, even under additional assumptions, this aim has not been attained. Guirao and Chudziak partially solved this problem for the class of skew-product maps with the base map having a closed set of periodic points. The present paper has two aims for this class of map. On the one hand, to improve that solution showing the equivalence between the property ‘of having zero topological entropy’ and the fact of ‘not being Li–Yorke chaotic in the union of the ω-limit sets of recurrent points’. On the other hand, we show that the properties ‘of having a closed set of periodic points’ and ‘all non-wandering points are periodic’ are not mutually equivalent properties. In doing this we disprove a result of Efremova.