Dynamics of neighborhoods of points under a continuous mapping of an interval - Romanenko, Ye. Yu. (Ukrains’kyi Matematychnyi Zhurnal, 2005)
Let { I, f Z^+ } be a dynamical system induced by the continuous map f of a closed bounded interval I into itself. In order to describe the dynamics of neighborhoods of points unstable under f, we suggest a notion of ε ω - set ω_f, ε(x) of a point x as the ω-limit set of ε-neighborhood of x. We investigate the association between the ε ω - set and the domain of influence of a point. We also show that the domain of influence of an unstable point is always a cycle of intervals. The results obtained can be directly applied in the theory of continuous time difference equations and similar equations.