Relatively pointwise recurrent graph map

Abstract:

Let $f$ be a self-continuous map of a graph $G$. Let $P(f)$ and $R(f)$ denote the sets of periodic points and recurrent points respectively. We say that the map $f$ is relatively recurrent if $\overline {R(f)} = G$. In this paper, it is shown that $f$ is relatively recurrent if and only if one of the following statements holds:

[(a)] $G$ is a circle and $f$ is a homeomorphism topologically conjugate to an irrational rotation of the unit circle $\mathbb {S}^1$;

[(b)] $\overline {P(f)} = G$.

Part (b) extends a result of Blokh.