On Limit sets of Monotone maps on Regular curves

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 $f:X\text{longrightarrowX}$ be a monotone map. We show that for any $x\in X$ (resp. for any negative orbit $(x_n{)}_{n\ge 0}$ of $x$), the $\omega$-limit set ${\omega }_f(x)$ (resp. $\alpha$-limit set ${\alpha }_f((x_n{)}_{n\ge 0})$) 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.