## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 20, Number 4 (2013), 689-705.

### Recurrence properties of a class of nonautonomous discrete systems

#### Abstract

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) $\omega(x, (f_n)) \smallsetminus \Omega(f)\not=\emptyset$ 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.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 4 (2013), 689-705.

**Dates**

First available in Project Euclid: 22 October 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1382448189

**Digital Object Identifier**

doi:10.36045/bbms/1382448189

**Mathematical Reviews number (MathSciNet)**

MR3129068

**Zentralblatt MATH identifier**

1373.37052

**Subjects**

Primary: 37B55: Nonautonomous dynamical systems 37B20: Notions of recurrence

Secondary: 39A10: Difference equations, additive 37E05: Maps of the interval (piecewise continuous, continuous, smooth)

**Keywords**

$\omega$-limit set Nonautonomous Chain recurrent set

#### Citation

Yokoi, Katsuya. Recurrence properties of a class of nonautonomous discrete systems. Bull. Belg. Math. Soc. Simon Stevin 20 (2013), no. 4, 689--705. doi:10.36045/bbms/1382448189. https://projecteuclid.org/euclid.bbms/1382448189