Dynamical systems arising by iterated functions on arbitrary semigroups

Let S be a discrete semigroup and let \(^SS\) denote the collection of all functions \(f:S\rightarrow S\). If \((P,\circ )\) is a subsemigroup of \(^SS\) by composition operation, then P induces a natural topological dynamical system. In fact, \((\beta S,\{T_f\}_{f\in P})\) is a topological dynamical system, where \(\beta S\) is the Stone–Čech compactification of S, \(x\mapsto T_f(x)=f^\beta (x):\beta S\rightarrow \beta S\) and \(f^\beta \) is a unique continuous22 extension of f. In this paper, we concentrate on the dynamical system \((\beta S,\{T_f\}_{f\in P})\), when S is an arbitrary discrete semigroup and P is a subsemigroup of \(^SS\) and obtain some relations between subsets of S and subsystems of \(\beta S\) with respect to P. As a consequence, we prove that if \((S,+)\) is an infinite commutative discrete semigroup and \(\mathcal {C}\) is a finite partition of S, then for every finite number of arbitrary homomorphisms \(g_1,\dots ,g_l:\mathbb {N}\rightarrow S\), there exist an infinite subset B of the natural numbers and \(C\in \mathcal {C}\) such that for every finite summations \(n_1,\dots , n_k\) of B there exists \(s\in S\) such that

$$\begin{aligned} \{s+g_i(n_1),s+g_i(n_2),\dots , s+g_i(n_k)\}\subseteq C,\,\,\,\,\,\,\forall i\in \{1,\dots ,l\}. \end{aligned}$$