This paper is concerned with establishing the Weyl-Titchmarsh theory for a class of discrete linear Hamiltonian systems over a half-line. Fundamental properties of solutions, regular spectral problems, and the corresponding maximal and minimal operators are first studied. Matrix disks are constructed and proved to be nested and converge to a limiting set. Some precise relationships among the rank of the matrix radius of the limiting set, the number of linearly independent square summable solutions, and the defect indices of the minimal operator are established. Based on the above results, a classification of singular discrete linear Hamiltonian systems is given in terms of the defect indices of the minimal operator, and several equivalent conditions on the cases of limit point and limit circle are obtained, respectively. Especially, several problems in the limit point case are more carefully investigated, including fundamental properties of square summable solutions, properties of the Weyl function, which is the unique element in the limiting set in this case, and inhomogeneous boundary problems, self-adjointness of the corresponding Hamiltonian operator, relationship between the spectrum of the Hamiltonian operator and the analyticity of the Weyl function, as well as the dependence of the spectrum on the boundary data, in which some interesting separation results for the spectrum are obtained. Finally, another set of four equivalent conditions on the limit point case are established.

This paper is concerned with the limit circle and limit point criteria of second-order linear difference equations. A sufficient and necessary condition, a sufficient and necessary condition subject to a certain restriction, and several sufficient conditions are established. These results improve and extend some previous results.

In this paper, the Glazman-Krein-Naimark theory for a class of discrete Hamiltonian systems is developed. A minimal and a maximal operators, GKN-sets, and a boundary space for the system are introduced. Algebraic characterizations of the domains of self-adjoint extensions of the minimal operator are given. A close relationship between the domains of self-adjoint extensions and the GKN-sets is established. It is shown that there exist one-to-one correspondences among the set of all the self-adjoint extensions, the set of all the d-dimensional Lagrangian subspaces of the boundary space, and the set of all the complete Lagrangian subspaces of the boundary space.

This paper is concerned with the symplectic structure of discrete nonlinear Hamiltonian systems. The results are related to an open problem that was first proposed by C. D. Ahlbrandt [J. Math. Anal. Appl. 180 (1993), 498-517] discussed elsewhere in the literature. But we give a different statement and different proof. Under a solvable condition, we show that the solution operator of a discrete nonlinear Halmiltonian system is symplectic. Then its phase flow is a discrete one-parameter family of symplectic transformations and preserves the phase volume.

This paper is concerned with chaos induced by strictly turbulent maps in noncompact sets of complete metric spaces. Two criteria of chaos for such types of maps are established, and then a criterion of chaos, characterized by snap-back repellers in complete metric spaces, is obtained. All the maps presented in this paper are proved to be chaotic either in the sense of both Li-Yorke and Wiggins or in the sense of both Li-Yorke and Devaney. The results weaken the assumptions in some existing criteria of chaos. Several illustrative examples are provided with computer simulation.