Dynamics of homeomorphisms of regular curves

In this paper, we prove first that the space of minimal sets of any homeomorphisms $f:X\to X$ of a regular curve $X$ is closed in the hyperspace $2^X$ of closed subsets of $X$ endowed with the Hausdorff metric, and the non-wandering set $\mathrm{\Omega }(f)$ is equal to the set of recurrent points of $f$. Second, we study the continuity of the map $\omega_f:X\to 2^X;x\mapsto \omega_f (x) $,weshowforinstancetheequivalencebetweenthecontinuityof$\omega_f $andtheequalitybetweenthe$\omega$-limitsetandthe$\alpha $-limit set of every point in $X$. Finally, we prove that there is only one (infinite) minimal set when there is no periodic point.