Semicocycle extensions and the stroboscopic property

We indicate a large class of almost 1–1 extensions over minimal systems, which do not possess the stroboscopic property, as defined by Misiurewicz and studied by Jimenez and Snoha [Topology Appl. 129 (2003) 301–316]. Sturmian flows and all Toeplitz flows belong to this class. This generalizes a theorem of [Topology Appl. 129 (2003) 301–316] for Sturmian flows. Our result allows to easily construct minimal weakly mixing systems without the stroboscopic property, which answers in the negative a question posed in [Topology Appl. 129 (2003) 301–316]. Finally we prove that even the strong stroboscopic property does not imply the stroboscopic property for induced (first return time) systems.