  Given a number $r$, we consider the dynamical system generated by repeated
exponentiations modulo $r$, that is, by the map $u \mapsto f_g(u)$, where
$f_g(u) \equiv g^u \pmod r$ and $0 \le f_g(u) \le r-1$. The number of cycles of
the defined above dynamical system is considered for $r=p^n$.
