Morse Index and Maslov-type Index of the Discrete Hamiltonian System

Abstract

In this paper, we give the definition of Maslov-type index of the discrete Hamiltonian system, and obtain the relation of Morse index and Maslov-type index of the discrete Hamiltonian system which is a generalization of the case ω = 1 to ω ∈ U degenerate case via direct method which is different from that of the known literatures. Moreover the well-posedness of the splitting numbers \(\cal{S}_{h,\omega}^{\pm}\) is proven, then the index iteration theories of Bott and Long are also valid for the discrete case, and those can be also applied to the study of the symplectic algorithm.