On topology of minimal sets: The case of fibre-preserving maps in graph bundles

We study topological structure of minimal sets in dynamical systems given by a continuous selfmap of a compact metric space X. First we consider the spaces X in which the full topological characterization of minimal sets is known. Then we discuss the alternative "nowhere dense or the whole space" which holds for minimal sets in some important classes of systems. Finally, we present results on the structure of minimal sets of fibre-preserving maps in graph bundles.