On Limit sets of Monotone maps on Regular curves

Abstract

We investigate the structure of $\omega$-limit (resp. $\alpha$-limit) sets for a monotone map $f$ on a regular curve $X$. %Let $X$ be a regular curve and let $f: X\longrightarrowX$ be a monotone map. We show that for any $x\in X$ (resp. for any negative orbit $(x_{n})_{n\geq 0}$ of $x$), the $\omega$-limit set $\omega_{f}(x)$ (resp. $\alpha$-limit set $\alpha_{f}((x_{n})_{n\geq 0})$) is a minimal set. This also hold for $\alpha$-limit set $\alpha_{f}(x)$ whenever $x$ is not a periodic point. These results extend those of Naghmouchi \cite{n} %[J. Difference Equ. Appl., 23 (2017), 1485--1490] established whenever $f$ is a homeomorphism on a regular curve and those of Abdelli \cite{a} %[Chaos, Solitons Fractals, 71 (2015), 66--72] , whenever $f$ is a monotone map on a local dendrite. Further results related to the basin of attraction of an infinite minimal set are also obtained.