Topological Limit of Trajectories of Intervals of Simplest One-Dimensional Dynamical Systems - Fedorenko, V. V. (Ukrains’kyi Matematychnyi Zhurnal, 2002)
We consider dynamical systems generated by continuous maps of an interval into itself. We investigate the asymptotic behavior of the trajectories of subsets of the interval. In particular, we prove that if the ω-limit set of an arbitrary trajectory is a fixed point, then the topological limit of the trajectory of any subinterval exists.