Geometry & Topology, Vol. 9 (2005) Paper no. 55, pages 2359--2394.

On the dynamics of isometries

Anders Karlsson

Abstract. 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 Teichmueller 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.

Keywords. Metric spaces, isometries, nonpositive curvature, Kobayashi metric, random walk

AMS subject classification. Primary: 37B05, 53C24, . Secondary: 22F50, 32H50.

E-print: arXiv:math.MG/0512638

DOI: 10.2140/gt.2005.9.2359

Submitted to G&T on 12 March 2005. Paper accepted 16 December 2005. Paper published 26 December 2005.

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Anders Karlsson
Mathematics Department, Royal Institute of Technology
100 44 Stockholm, Sweden

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