Topology of minimal sets

Discrete dynamical systems given by a continuous map on a topological (usually compact metrizable) space will be considered. Minimality of such a system/map can be defined as the density of all forward orbits. Every compact system contains at least one minimal set, i.e., a nonempty closed invariant subset such that the restriction of the map to this subset is minimal. A fundamental question in topological dynamics is the one on the topological structure of minimal sets of continuous maps in a given space (and, in particular, whether the space itself admits a minimal map or not). In the first part of the talk we will present a survey of some known facts on minimality (including those on topological properties of minimal maps and on noninvertible minimal maps). Then we will concentrate on the topological structure of minimal sets. Among others, recent results obtained jointly with Snoha and Trofimchuk will be presented. Minimal sets are studied for a compact dynamical system given by a fibre-preserving continuous map F in a graph bundle E (i.e., F is a skew product map). For minimal sets of such systems a kind of dichotomy holds, which can be described in terms of what we call end-points of a set: The set of end-points of a minimal set M is either dense in M (and then M is nowhere dense in E), or it is empty (and then M has nonempty interior in E).