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

  • Ye. Yu. Romanenko


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.
How to Cite
Romanenko, Y. Y. “Dynamics of Neighborhoods of Points under a Continuous Mapping of an Interval”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, no. 11, Nov. 2005, pp. 1534–1547, https://umj.imath.kiev.ua/index.php/umj/article/view/3705.
Research articles