# Dynamics of neighborhoods of points under a continuous mapping of an interval

### Abstract

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 $\varepsilon \omega - {\rm set} \omega_{f, \varepsilon}(x)$ of a point $x$ as the $\omega$-limit set of $\varepsilon$-neighborhood of $x$. We investigate the association between the $\varepsilon \omega - {\rm set}$ 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.
Published

25.11.2005

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 57, no. 11, Nov. 2005, pp. 1534–1547, https://umj.imath.kiev.ua/index.php/umj/article/view/3705.

Issue

Section

Research articles