Chain recurrence in $\beta$-compactifications of topological groups

Abstract

Let $G$ be a topological group. In this paper limit behavior in the Stone–Čech compactification $\beta G$ is studied. It depends on a family of translates of a reversible subsemigroup $S$. The notion of semitotal subsemigroup is introduced. It is shown that the semitotality property is equivalent to the existence of only two maximal chain transitive sets in $\beta G$ whenever $S$ is centric. This result links an algebraic property to a dynamical property. The concept of a chain recurrent function is also introduced and characterized via the compactification $\beta G$. Applications of chain recurrent function to linear differential systems and transformation groups are done.