Dynamical Systems with Bounded Condition and $C^{*}$-algebras

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.