  The dynamics of a 1-parameter family of cluster maps $\varphi_r$ associated
to mutation-periodic quivers in dimension 4, is studied in detail. The use of
presymplectic reduction leads to a globally periodic symplectic map, and this
enables us to reduce the problem to the study of maps belonging to a group of
symplectic birational maps of the plane which is isomorphic to
$SL(2,\mathbb{Z})\ltimes\mathbb{R}^2$. We conclude that there are three
different types of dynamical behaviour for $\varphi_r$ characterized by the
integer parameter values $r=1$, $r=2$ and $r>2$. For each type, the periodic
points, the structure and the asymptotic behaviour of the orbits are completely
described. A finer description of the dynamics is provided by using first
integrals.
