M-systems and scattering systems of semigroup actions

We prove that: (1) an action of a semigroup S on a compact metric space X is an M -system if and only if N ( x , U ) is a piecewise syndetic set for every transitive point x in X and every neighborhood U of x ; (2) an action of a monoid S on a compact metric space X for which every s ∈ S is a surjective map from X onto itself is scattering if and only if N ( U , V ) is a set of topological recurrence for every pair of non-empty open subsets U , V in X . As applications, we show that: (1) if an action of a commutative semigroup S on a compact metric space X is an M -system then the system is finitely sensitive; (2) an action of a commutative semigroup S on a compact metric space X is a scattering system if and only if it is disjoint with any M -system.