Recurrence in the dynamical system ( X , 〈 T s 〉 s ∈ S ) and ideals of β S
Résumé
A dynamical system is a pair (X, ⟨T s ⟩ s∈S), where X is a compact Hausdorff space, S is a semigroup, for each s ∈ S, T s is a continuous function from X to X , and for all s, t ∈ S, T s • T t = T st. Given a point p ∈ β S, the Stone– ˇ Cech compactification of the discrete space S, T p : X → X is defined by, for x ∈ X , T p (x) = p −lim s∈S T s (x). We let β S have the operation extending the operation of S such that β S is a right topological semigroup and multiplication on the left by any point of S is continuous. Given p, q ∈ β S, T p • T q = T pq , but T p is usually not continuous. Given a dynamical system (X, ⟨T s ⟩ s∈S), and a point x ∈ X , we let U (x) = { p ∈ β S : T p (x) be uniformly recurrent}. We show that each U (x) is a left ideal of β S and for any semigroup we can get a dynamical system with respect to which K (β S) = ⋂ x∈X U (x) and cℓK (β S) = ⋂ {U (x) : x ∈ X and U (x) is closed}. And we show that weak cancellation assumptions guarantee that each such U (x) properly contains K (β S) and has U (x) \ cℓK (β S) ̸ = ∅.
Domaines
Systèmes dynamiques [math.DS]
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