Energy Function for Direct Products of Discrete Dynamical Systems

This paper is devoted to the construction of an energy function, i.e. a smooth Lyapunov whose set critical points coincides with chain-recurrent dynamical system — for cascade that direct product two systems. One multipliers structurally stable diffeomorphism given on two-dimensional torus, non-wandering consists zero-dimensional non-trivial basic without pairs conjugated and fixed source sink, second one identical mapping real axis. It was previously proved if contains set, as under consideration has, then such does not have namely, any function will outside set. For mapping, constant entire line. In this paper, it shown absence sufficient condition systems, is, in some cases possible select way function.