Nonwandering sets and special \(\alpha\)-limit sets of monotone maps on regular curves. (English) Zbl 1540.37024
Elaydi, Saber (ed.) et al., Advances in discrete dynamical systems, difference equations and applications. ICDEA 26, Sarajevo, Bosnia and Herzegovina, July 26–30, 2021. Proceedings of the 26th international conference on difference equations and applications. Cham: Springer. Springer Proc. Math. Stat. 416, 339-362 (2023).
Summary: Let \(X\) be a regular curve and let \(f: X\rightarrow X\) be a monotone map. We show that \(\mathrm{AP} (f) = \mathrm{R}(f) = \varOmega (f)\), where \(\mathrm{AP}(f)\), \(\mathrm{R}(f)\) and \(\varOmega (f)\) are the sets of almost periodic points, recurrent points and nonwandering points of \(f\), respectively. On the other hand, we show that for every \(x\in X{\setminus} \mathrm{P}(f)\), the special \(\alpha\)-limit set \(s\alpha_f(x)\) is a minimal set, where \(P(f)\) is the set of periodic points of \(f\) and that \(s\alpha_f(x)\) is always closed, for every \(x\in X\). In addition, we prove that \(\mathrm{SA}(f) = \mathrm{R}(f)\), where \(\mathrm{SA}(f)\) denotes the special \(\alpha\)-limit set of \(f\). Further results related to the continuity of the limit maps are also obtained, we prove that the map \(\omega_f\) (resp. \(\alpha_f\), resp. \(\mathrm{s} \alpha_f)\) is continuous on \(X{\setminus} \mathrm{P}(f)\) (resp. \(X_{\infty}{\setminus} \mathrm{P}(f))\), where \(X_{\infty} = \underset{n\ge 0}{\cap}f^n(X)\).
For the entire collection see [Zbl 1531.37006].
For the entire collection see [Zbl 1531.37006].
MSC:
| 37B02 | Dynamics in general topological spaces |
| 37B45 | Continua theory in dynamics |
| 37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |
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