Topological transitivity and mixing notions for group actions

This paper surveys six notions of dynamical transitivity and mixing, in the context of group actions on topological spaces. We discuss the relations between these notions, and the manner in which they are inherited by subgroups, by taking products, and when passing to the induced action on hyperspace, i.e., the space of compact subsets. The focus of the paper is on the fact that certain standard notions, which are equivalent in the classical theory of the dynamics of flows and the iteration of single maps, are distinct for general group actions. The paper examines how the notions coalesce (a) for actions of abelian groups and (b) for chaotic actions.