Non-expanding maps and Busemann functions

We give stronger versions and alternative simple proofs of some results of Beardon, [Be1] and [Be2]. These results concern contractions of locally com pact metric spaces and generalize the theorems of Wolff and Denjoy about th e iteration of a holomorphic map of the unit disk. In the case of unbounded orbits, there are two types of statements which can sometimes be proven; f irst, about invariant horoballs, and second, about the convergence of the i terates to a point on the boundary. A few further remarks of similar type a re made concerning certain random products of sernicontractions and also co ncerning semicontractions of Gromov hyperbolic spaces.