Hyperbolic homoclinic points of Z(d)-actions in lattice dynamical systems

We study Z(d) action on a set of equilibrium solutions of a lattice dynamical system, i.e., a system with discrete spatial variables, and the stability and hyperbolicity of the equilibrium solutions. Complicated behavior of Z(d)-action corresponds to the existence of an infinite number of equilibrium solutions which are randomly situated along spatial coordinates. We prove that the existence of a homoclinic point of a Z(d)-action implies complicated behavior, provided the hyperbolicity of the homoclinic solution with respect to the lattice dynamical system (this is a generalization of the previous work of the first two authors). Similar result holds for hyperbolic partially homoclinic and heteroclinic points. We show the equivalence of stability for any equilibrium solutions and the equivalence of hyperbolicity for homoclinic points under various norms.