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

TL;DR

This paper investigates the structure of nonwandering and special alpha-limit sets for monotone maps on regular curves, extending previous results and establishing new properties about their minimality, closure, and continuity of limit maps.

Contribution

It extends existing theorems on nonwandering and alpha-limit sets from interval and graph maps to monotone maps on regular curves, providing new insights into their structure and continuity.

Findings

Nonwandering set equals almost periodic and recurrent sets.

Special alpha-limit sets are minimal and closed outside periodic points.

The limit maps are continuous outside the set of periodic points.

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 $\textrm{P}(f)=\emptyset$.