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The dynamics of the maps associated to $F_0$ and $dP_3$ quivers is studied indetail. We show that the corresponding reduced symplectic maps are conjugate toglobally periodic maps by providing explicit conjugations. The dynamics in$\Rb^N_+$ of the original maps is obtained by lifting the dynamics of theseglobally periodic maps and the solution of the discrete dynamical systemsgenerated by each map is given. A better understanding of the dynamics isachieved by considering first integrals. The relationship between the completeintegrability of the globally periodic maps and the dynamics of the originalmaps is explored.

Dynamics of the birational maps arising from $F_0$ and $dP_3$ quivers