An Integrable Symplectic Map of a Differential-Difference Hierarchy

By choosing a discrete matrix spectral problem, a hierarchy of integrable differential-difference equations is derived from the discrete zero curvature equation, and the Hamiltonian structures are built. Through a higher-order Bargmann symmetry constraint, the spatial part and the temporal part of the Lax pairs and adjoint Lax pairs, which we obtained are respectively nonlinearized into a new integrable symplectic map and a finite-dimensional integrable Hamiltonian system in Liouville sense.