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On the dynamics of isometries. (English) Zbl 1120.53026
Summary: 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 Teichmüller 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.

##### MSC:
 53C24 Rigidity results 37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) 22F50 Groups as automorphisms of other structures 32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
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