Dynamical Systems with Bounded Condition and C*-algebras

By introducing the totally uniqueness condition for maps, we establish a one-to-one correspondence between the family of invariant sets for the q x+d-function (e.g. the 3 x+1-map, also known as the Collatz map), where q and d are arbitrary positive odd numbers, and the family of reducing subspaces for the associated C*-algebra. This extends the connection between the Collatz conjecture (also known as the 3 n+1-problem) and the irreducibility of its associated C*-algebra. We also introduce homomorphisms between dynamical systems with bounded conditions that preserve the structures of these dynamical systems. We prove the existence of an isomorphism between their associated C*-algebras is proven for every isomorphism between dynamical systems with bounded condition.