Asymptotic pairs, stable sets and chaos in positive entropy systems (Q2254806)

This paper studies the relationship between various topological dynamical properties of actions of various discrete groups by homeomorphisms of a compact metric space. Here the properties considered are the existence of asymptotic pairs, positive topological entropy of the action, and chaotic and stable sets. The properties make sense for countable discrete infinite amenable groups. In order to explore more subtle connections and to exhibit examples illustrating the relationships, additional hypotheses are needed. In order to exploit the dynamically significant notion of a past, the groups are assumed to be left-orderable throughout. Specific examples are constructed for actions of \(\mathbb{Z}^d\), the discrete Heisenberg group, and groups of integer unipotent upper triangular matrices.