Relatively pointwise recurrent graph map

Hattab Hawete
2010 Proceedings of the American Mathematical Society  
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 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 S 1 ; (b) P (f ) = G. Part (b) extends a result of Blokh.
doi:10.1090/s0002-9939-2010-10622-6 fatcat:we6dn6s5aza7lkomhee7ofvmlq