In this paper, we study abstract dynamical systems with discrete phase spaces. One example of such a system is induced by the \(3 x{+}1\) -map on the set of all natural numbers, also known as the Collatz map. Our main focus is on dynamical systems induced by maps on countable discrete sets that satisfy a bounded condition. When these maps satisfy the bounded and a separating conditions, a minimality of the induced dynamical systems is equivalent to the irreducibility of certain \(C^{*}\) -algebras on certain Hilbert spaces. For a map f on a general discrete phase space, we consider f-invariant sets and investigate their properties. When the phase space is countable and the map satisfies the bounded condition, we construct an order-preserving injection from the family of f-invariant sets to the family of reducing subspaces for the corresponding \(C^{*}\) -algebra. By introducing the totally uniqueness condition for f, we show that this injection is a bijection if f satisfies this condition. This condition is crucial in providing a symbolic representation of the dynamical system induced by f, and we discuss the relationship between this symbolic representation and that of a topological dynamical system.
