Attractors and Chain Recurrence for Semigroup Actions

Let ${\mathcal{S}}$ be a semigroup acting on a topological space M. We define attractors for the action of ${\mathcal{S}}$ on M. This concept depends on a family ${\mathcal{F}}$ of subsets of ${\mathcal{S}}$ . For certain semigroups and families it recovers the concept of attractors for flows or semiflows. We define and study the complementary repeller of an attractor. We also characterize the set of chain recurrent points in terms of attractors.