On -limit sets of non-autonomous dynamical systems with a uniform limit of type

Abstract

This paper is devoted to the study of properties of -limit sets of non-autonomous dynamical systems on compact metric spaces given by sequences of maps which uniformly converge to a continuous map f. We show that, for systems defined on compact metric spaces, if an -limit set of the non-autonomous system is a subset of the set P(f) of periodic points of f then is necessarily the union of finitely many disjoint connected sets which are cyclically mapped to one another. Using this result, we answer a question posed by Cánovas in [3] [On -limit sets of non-autonomous systems. J. Difference Equ. Appl. 12 (2006), pp. 95–100] by proving that, if an interval map f has only finite -limit sets, then any -limit set of the non-autonomous system is a subset of the set of periodic points of f. We also show that a similar result applies to systems on trees but not on graphs with loops.