A discrete-time relativistic Toda lattice

For each of the two simplest Hamiltonian flows from the relativistic Toda hierarchy we introduce two integrable symplectic discretizations. All four discrete-time systems are demonstrated to belong to the same hierarchy and to exemplify the general scheme for symplectic maps on groups equipped with quadratic Poisson brackets. The initial-value problem for the difference equations is solved in terms of a factorization problem in a group. Interpolating Hamiltonian flows are found for all maps.