  The dynamics of the maps associated to $F_0$ and $dP_3$ quivers is studied in
detail. We show that the corresponding reduced symplectic maps are conjugate to
globally periodic maps by providing explicit conjugations. The dynamics in
$\Rb^N_+$ of the original maps is obtained by lifting the dynamics of these
globally periodic maps and the solution of the discrete dynamical systems
generated by each map is given. A better understanding of the dynamics is
achieved by considering first integrals. The relationship between the complete
integrability of the globally periodic maps and the dynamics of the original
maps is explored.
