Geometry & Topology

On the dynamics of isometries

Anders Karlsson

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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.

Article information

Geom. Topol., Volume 9, Number 4 (2005), 2359-2394.

Received: 12 March 2005
Accepted: 16 December 2005
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.) 53C24: Rigidity results
Secondary: 22F50: Groups as automorphisms of other structures 32H50: Iteration problems

metric spaces isometries nonpositive curvature Kobayashi metric random walk


Karlsson, Anders. On the dynamics of isometries. Geom. Topol. 9 (2005), no. 4, 2359--2394. doi:10.2140/gt.2005.9.2359.

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