2017
Том 69
№ 9

All Issues

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

Romanenko Ye. Yu.

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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.

English version (Springer): Ukrainian Mathematical Journal 57 (2005), no. 11, pp 1792-1808.

Citation Example: Romanenko Ye. Yu. Dynamics of neighborhoods of points under a continuous mapping of an interval // Ukr. Mat. Zh. - 2005. - 57, № 11. - pp. 1534–1547.

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