A hierarchy of integrable lattice soliton equations and its Hamiltonian structure associated with a 3×3 matrix spectral problem are got and an integrable symplectic map is obtained by nonlinearization of Lax pairs and adjoint Lax pairs of the hierarchy. Moreover

the solutions of the prototype system in the lattice equations hierarchy are reduced to the solutions of a system of ordinary differential equations and a simple iterative process of the symplectic map.