Groups, Geometry, and Dynamics

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Volume 7, Issue 2, 2013, pp. 475–493
DOI: 10.4171/GGD/191

Published online: 2013-05-07

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

Josiney A. Souza[1]

(1) Universidade Estadual de Maringá, Maringá, Brazil

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.

Keywords: Transformation group, attractor, Morse decomposition, chain recurrence, Stone–Čech compactification

Souza Josiney: Chain recurrence in $\beta $-compactifications of topological groups. Groups Geom. Dyn. 7 (2013), 475-493. doi: 10.4171/GGD/191