On discrete Routh reduction and structures on the reduced space

In this paper we work, first, with forced discrete-time mechanical systems on the configuration space $Q$ and construct closed $2$-forms $\omega^+$ and $\omega^-$ on $Q \times Q$, that are symplectic if and only if the system is regular. For a special type of discrete force, we prove that $\omega^+$ and $\omega^-$ are invariant by the flow of the system. We also consider the Lagrangian reduction of a discrete mechanical system by a symmetry group (using an affine discrete connection derived from the discrete momentum) and prove that, under some conditions on the action, the trajectories of the reduced system (with constant discrete momentum $\mu$) can be seen as trajectories of a forced discrete mechanical system, where the discrete force is of the type analyzed before. Therefore, we prove that there is a symplectic structure that is invariant by the flow of the forced reduced system; the symplectic structure can be seen as a pullback of a canonical cotangent structure plus a magnetic term. This discrete reduction process is the (discrete) Routh reduction and the behavior obtained runs parallel to the well known case for (continuous) Routh reduction.