  We study the set of periods of the Morse--Smale diffeomorphisms on the
$n$-dimensional sphere $\mathbb{S}^n$, on products of two spheres of arbitrary
dimension $\mathbb{S}^m \times \mathbb{S}^n$ with $m \neq n$, on the
$n$-dimensional complex projective space $\mathbb{C}\text{\textbf{P}}^n}$ and
on the $n$-dimensional quaternion projective space
$\mathbb{H}\text{\textbf{P}}^n}$. We classify the minimal sets of Lefschetz
periods for such Morse--Smale diffeomorphisms. This characterization is done
using the induced maps on the homology. The main tool used is the Lefschetz
zeta function.
