  This note will give an enumeration of $n$-cycles in the symmetric group
${\mathcal S}_n$ by their degree (also known as their cyclic descent number)
and studies similar counting problems for the conjugacy classes of $n$-cycles
under the action of the rotation subgroup of ${\mathcal S}_n$. This is achieved
by relating such cycles to periodic orbits of an associated dynamical system
acting on the circle. We also compute the mean and variance of the degree of a
random $n$-cycle and show that its distribution is asymptotically normal as $n
\to \infty$.
