Abstract
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 \(s \in S\) 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.
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