Normal Behavior and Periodic Points of a Pseudo-Aliquot Map Associated with the Dedekind Psi Function

We study the iteration of (T(n)=ψ(n)-n), where (ψ) is the Dedekind psi function. We show that, for almost all integers, successive iterates satisfy strong asymptotic growth relations governed by (T(n)/n). As an application, we prove that for every fixed (\ell\ge1), the set of integers satisfying (T^{(\ell)}(n)=n) has asymptotic density zero, and hence so does the set of periodic points of any fixed period.