Geometry & Topology, Vol. 9 (2005)
Paper no. 55, pages 2359--2394.
On the dynamics of isometries
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.
Metric spaces, isometries, nonpositive curvature, Kobayashi metric, random walk
AMS subject classification.
Primary: 37B05, 53C24, .
Secondary: 22F50, 32H50.
Submitted to G&T on 12 March 2005.
Paper accepted 16 December 2005.
Paper published 26 December 2005.
Notes on file formats
Mathematics Department, Royal Institute of Technology
100 44 Stockholm, Sweden
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