Stroboscopical property in topological dynamics

A dynamical system given by a metric space and its continuous selfmap is said to have the (Misiurewicz) stroboscopical property if, given any point z from the space and any increasing sequence of positive integers, there is a point whose ω-limit set relative to this sequence contains z. In the paper it is shown that some minimal skew product homeomorphisms on the torus do not have such property. heless, we show that some important classes of systems do have the stroboscopical property. On the other hand, there is a Devaney chaotic and topologically weakly mixing subshift without the stroboscopical property.