  We investigate discrete-time dynamical systems generated by an
infinite-dimensional non-linear operator that maps the Banach space $l_1$ to
itself. It is demonstrated that this operator possesses up to seven fixed
points. By leveraging the specific form of our operator, we illustrate that
analyzing the operator can be simplified to a two-dimensional approach.
Subsequently, we provide a detailed description of all fixed points, invariant
sets for the two-dimensional operator and determine the set of limit points for
its trajectories. These results are then applied to find the set of limit
points for trajectories generated by the infinite-dimensional operator.
