Structures of quasi-graphs and $\omega $-limit sets of quasi-graph maps

Abstract

An arcwise connected compact metric space $X$ is called a quasi-graph if there is a positive integer $N$ with the following property: for every arcwise connected subset $Y$ of $X$, the space $\overline {Y}-Y$ has at most $N$ arcwise connected components. If a quasi-graph $X$ contains no Jordan curve, then $X$ is called a quasi-tree. The structures of quasi-graphs and the dynamics of quasi-graph maps are investigated in this paper. More precisely, the structures of quasi-graphs are explicitly described; some criteria for $\omega$-limit points of quasi-graph maps are obtained; for every quasi-graph map $f$, it is shown that the pseudo-closure of $R(f)$ in the sense of arcwise connectivity is contained in $\omega (f)$; it is shown that $\overline {P(f)}=\overline {R(f)}$ for every quasi-tree map $f$. Here $P(f)$, $R(f)$ and $\omega (f)$ are the periodic point set, the recurrent point set and the $\omega$-limit set of $f$, respectively. These extend some well-known results for interval dynamics.