Structures of $R(f)-\overline{P(f)}$ for graph maps $f$

Abstract

Let $G$ be a graph and $f: G\rightarrow G$ be a continuous map. We establish a structure theorem which describes the structures of the set $R(f)-\overline{P(f)}$, where $R(f)$ and $P(f)$ are the recurrent point set and the periodic point set of $f$ respectively. Roughly speaking, the set $R(f)-\overline{P(f)}$ is covered by finitely many pairwise disjoint $f$-invariant open sets $U_{1\,},\,\cdots,\,U_{n\,}$; each $U_i$ contains a unique minimal set $W_i$ which absorbs each point of $U_i$; each $W_i$ lies in finitely many pairwise disjoint circles each of which is contained in a connected closed set; all of these connected closed sets are contained in $U_i$ and permutated cyclically by $f$. As applications of the structure theorem, several known results are improved or reproved.