We introduce and study some concepts of sensitivity via Furstenberg families. A dynamical system(X,f)isℱ-sensitive if there exists a positiveεsuch that for everyx∈Xand every open neighborhoodUofxthere existsy∈Usuch that the pair(x,y)is notℱ-ε-asymptotic; that is, the time set{n:d(fn(x),fn(y))>ε}belongs toℱ, whereℱis a Furstenberg family. A dynamical system(X,f)is (ℱ1,ℱ2)-sensitive if there is a positiveεsuch that everyx∈Xis a limit of pointsy∈Xsuch that the pair(x,y)isℱ1-proximal but notℱ2-ε-asymptotic; that is, the time set{n:d(fn(x),fn(y))<δ}belongs toℱ1for any positiveδbut the time set{n:d(fn(x),fn(y))>ε}belongs toℱ2, whereℱ1andℱ2are Furstenberg families.
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.
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