We give alternative simple proofs and stronger versions of some results of Beardon, (Be 90) and (Be 97). These results concern contractions of certain locally compact metric spaces and generalize the classical theorems of Wolff and Denjoy about the iteration of a holomorphic map of the unit disk. In the case of unbounded orbits, there are two types of statements that sometimes can be proven; first, about invariant horoballs, and second, about the convergence of the iterates to a point on the boundary. To establish statements of the latter type, some hyperbolicity or visibility property is assumed. A few further remarks of similar type are made concerning certain random products of semicontractions and concerning semicontractions of Gromov hyperbolic spaces.
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