Nonwandering sets and special $\alpha$-limit sets of monotone maps on regular curves

Abstract

Let $X$ be a regular curve and let $f: X\to X$ be a monotone map. In this paper, nonwandering set of $f$ and the structure of special $\alpha$-limit sets for $f$ are investigated. We show that AP$(f)= \textrm{R}(f) =\Omega(f)$, where AP$(f)$, $\textrm{R}(f)$ and $\Omega(f)$ are the sets of almost periodic points, recurrent points and nonwandering of $f$, respectively. This result extends that of Naghmouchi established, whenever $f$ is a homeomorphism on a regular curve [J. Difference Equ. Appl., 23 (2017), 1485--1490] and [Colloquium Math., 162 (2020), 263--277], and that of Abdelli and Abdelli, Abouda and Marzougui, whenever $f$ is a monotone map on a local dendrite [Chaos, Solitons Fractals, 71 (2015), 66--72] and [Topology Appl., 250 (2018), 61--73], respectively. On the other hand, we show that for every $X\setminus \textrm{P}(f)$, 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 $\textrm{SA}(f) = \textrm{R}(f)$, where $\textrm{SA}(f)$ denotes the union of all special $\alpha$-limit sets of $f$; these results extend, for monotone case, recent results on interval and graph maps obtained respectively by Hant\'{a}kov\'{a} and Roth in [Preprint: arXiv 2007.10883.] and Fory\'{s}-Krawiec, Hant\'{a}kov\'{a} and Oprocha in [Preprint: arXiv:2106.05539.]. Further results related to the continuity of the limit maps are also obtained, we prove that the map $\omega_{f}$ (resp. $\alpha_{f}$, resp. s$\alpha_{f}$) is continuous on $X\setminus \textrm{P}(f)$ (resp. $X_{\infty}\setminus \textrm{P}(f)$). %In particular, it is continuous on $X$ (resp. $X_{\infty}$) whenever $\textrm{P}(f)=\emptyset$.