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 $ε ω - {\rm set} ω_{f, ε}(x)$ of a point $x$ as the $ω$-limit set of $ε$-neighborhood of $x$.We investigate the association between the $ε ω - {\rm set}$ 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.