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

Let {I, f, Z ⁺} be a dynamical system induced by a continuous mapping f of a closed bounded interval I into itself. To describe the dynamics of neighborhoods of points unstable under the mapping f, we propose the concept of the εω-set ω ^{f, ε}(x) of a point x as the ω-limit set of the ε-neighborhood of the point x. We investigate the relationship between the εω-set and the domain of influence of a point. It is also shown that the domain of influence of an unstable point is always a cycle of intervals. The results obtained can be directly used in the theory of difference equations with continuous time and similar equations. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1534–1547, November, 2005.