Sections 8 and 9 in:

We provide an analysis of the dynamics of isometries and semicontractions of metric spaces. Certain subsets of the boundary at infinity play a fundamental role and are identified completely for the standard boundaries of CAT(0) spaces, Gromov hyperbolic spaces, Hilbert geometries, certain pseudoconvex domains, and partially for Thurston’s boundary of Teichmuller spaces. We present several rather general results concerning groups of isometries, as well as the proof of other more specific new theorems, for example concerning the existence of free nonabelian subgroups in CAT(0) geometry, iteration of holomorphic maps, a metric Furstenberg lemma, random walks on groups, noncompactness of automorphism groups of convex cones, and boundary behaviour of Kobayashi’s metric.

We give stronger versions and alternative simple proofs of some results of Beardon, [Be1] and [Be2]. These results concern contractions of locally compact metric spaces and generalize the 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 which can sometimes be proven; first, about invariant horoballs, and second, about the convergence of the iterates to a point on the boundary. A few further remarks of similar type are made concerning certain random products of semicontractions and also concerning semicontractions of Gromov hyperbolic spaces.