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Trajectories of intervals in one-dimensional dynamical systems

The necessity of considering trajectories just of sets rather than of points occurs both in dynamical systems theory by itself and in many evolutionary problems reducible to dynamical systems. In the case of one-dimensional dynamical systems, we present a number of conditions for the trajectory of an interval to be asymptotically periodic. The obtained results find significant applications in the theory of continuous time difference equations and some classes of boundary value problems for partial differential equations.