Dynamical systems arising by iterated functions on arbitrary semigroups

• Published in: Semigroup forum 2024-10, Vol.109 (2), p.205-221
• Main Authors: Akbari Tootkaboni, M., Bagheri Salec, A. R., Abbas, S.
• Format: Article
• Language: English
• Subjects: Algebra; Dynamical systems; Homomorphisms; Mathematics; Mathematics and Statistics; Number theory; Research Article; Semigroups; Subsystems; Topology
• ISSN: 0037-1912; 1432-2137
• DOI: 10.1007/s00233-024-10465-3

Let S be a discrete semigroup and let S S denote the collection of all functions f : S → S . If ( P , ∘ ) is a subsemigroup of S S by composition operation, then P induces a natural topological dynamical system. In fact, ( β S , { T f } f ∈ P ) is a topological dynamical system, where β S is the Stone–Čech compactification of S , x ↦ T f ( x ) = f β ( x ) : β S → β S and f β is a unique continuous22 extension of f . In this paper, we concentrate on the dynamical system ( β S , { T f } f ∈ P ) , when S is an arbitrary discrete semigroup and P is a subsemigroup of S S and obtain some relations between subsets of S and subsystems of β S with respect to P . As a consequence, we prove that if ( S , + ) is an infinite commutative discrete semigroup and C is a finite partition of S , then for every finite number of arbitrary homomorphisms g 1 , ⋯ , g l : N → S , there exist an infinite subset B of the natural numbers and C ∈ C such that for every finite summations n 1 , ⋯ , n k of B there exists s ∈ S such that { s + g i ( n 1 ) , s + g i ( n 2 ) , ⋯ , s + g i ( n k ) } ⊆ C , ∀ i ∈ { 1 , ⋯ , l } .