Abstract. This paper investigates the space C k (ω * , ω * ), the space of continuous self-maps on the Stone-Čech remainder of the integers, ω * , equipped with the compact-open topology. Our main results are that• Stone-Čech extensions of injective maps on ω form a dense set of weak P -points in C k (ω * , ω * ), • it is independent of ZFC whether C k (ω * , ω * ) contains P -points, and that • C k (ω * , ω * ) is not an F -space, but contains, as ω * , no non-trivial convergent sequences.