DYNAMICAL R-MATRICES FOR INTEGRABLE MAPS

The integrability of two symplectic maps that can be considered as discrete-time analogs of the Gamier and Neumann systems is established in the framework of the r-matrix approach, starting from their Lax representation. In contrast with the continuous case, the r-matrix for such discrete systems turns out to be of dynamical type; remarkably, the induced Poisson structure appears as a linear combination of compatible ''more elementary'' Poisson structures. It is also shown that the Lax matrix naturally leads to define separation variables, whose discrete and continuous dynamics are investigated