Lagrangian reduction of discrete mechanical systems by stages

In this work we introduce a category of discrete Lagrange--Poincare systems $\mathfrak{L}\mathfrak{P}_d$ and study some of its properties. In particular, we show that the discrete mechanical systems and the discrete dynamical systems obtained by the Lagrangian reduction of symmetric discrete mechanical systems are objects in $\mathfrak{L}\mathfrak{P}_d$. We introduce a notion of symmetry group for objects of $\mathfrak{L}\mathfrak{P}_d$ as well as a reduction procedure that is closed in the category $\mathfrak{L}\mathfrak{P}_d$. Furthermore, under some conditions, we show that the reduction in two steps (first by a closed normal subgroup of the symmetry group and then by the residual symmetry group) is isomorphic in $\mathfrak{L}\mathfrak{P}_d$ to the reduction by the full symmetry group.