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Recurrence properties of a class of nonautonomous discrete systems. (English) Zbl 1373.37052

Summary: We study recurrence properties for the nonautonomous discrete system given by a sequence \((f_n)_{n=1}^\infty\) of continuous selfmaps on a compact metric space. In particular, our attention is paid to the case when the sequence \((f_n)_{n=1}^\infty\) converges uniformly to a map \(f\) or forms an equicontinuous family. In the first case we investigate the structure and behavior of an \(\omega\)-limit set of \((f_n)\) by a dynamical property of the limit map \(f\). We also present some examples of \((f_n)\) and \(f\) on the closed interval: (a) \(\omega(x, (f_n)) \smallsetminus \Omega(f)\neq \varnothing\) for some point \(x\); or (b) the set of periodic points of \(f\) is closed and for some point \(x, \omega(x, (f_n))\) is infinite. In the second case we create a perturbation of \((f_n)\) whose nonwandering set has small measure.

MSC:

37B55 Topological dynamics of nonautonomous systems
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
39A10 Additive difference equations
37E05 Dynamical systems involving maps of the interval
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Full Text: Euclid