A Note about Iterated Arithmetic Functions

Let $f\text{\hspace{0.17em}⁣}:\mathbb{N}\to {\mathbb{N}}_0$ be a multiplicative arithmetic function such that for all primes $p$ and positive integers $\alpha$, $f(p^{\alpha })<p^{\alpha }$ and $f(p)\mathrm{\mid }f(p^{\alpha })$. Suppose also that any prime that divides $f(p^{\alpha })$ also divides $pf(p)$. Define $f(0)=0$, and let $H(n)={\displaystyle \underset{m\to \mathrm{\infty }}{\lim }{f}^m(n)}$, where $f^m$ denotes the $m^{th}$ iterate of $f$. We prove that the function $H$ is completely multiplicative.