A survey of some aspects of dynamical topology: Dynamical compactness and Slovak spaces

Abstract

The area of dynamical systems where one investigates dynamical properties that can be described in topological terms is Topological Dynamics. Investigating the topological properties of spaces and maps that can be described in dynamical terms is in a sense the opposite idea. This area has been recently called Topology. As an illustration, some topological properties of the space of all transitive interval maps are described. For (discrete) dynamical systems given by compact metric spaces and continuous (surjective) self-maps we survey some results on two new notions: Space and Compactness. A Slovak space, as a dynamical analogue of a rigid space, is a nontrivial compact metric space whose homeomorphism group is cyclic and generated by a minimal homeomorphism. Dynamical compactness is a new concept of chaotic dynamics. The omega-limit set of a point is a basic notion in the theory of dynamical systems and means the collection of states which attract this point while going forward in time. It is always nonempty when the phase space is compact. By changing the time we introduced the notion of the omega-limit set of a point with respect to a Furstenberg family. A dynamical system is called dynamically compact (with respect to a Furstenberg family) if for any point of the phase space this omega-limit set is nonempty. A nice property of dynamical compactness is that all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property.