Volume 9, issue 4 (2005)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
On the dynamics of isometries

Anders Karlsson

Geometry & Topology 9 (2005) 2359–2394

arXiv: math.MG/0512638

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 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.

Keywords
metric spaces, isometries, nonpositive curvature, Kobayashi metric, random walk
Mathematical Subject Classification 2000
Primary: 37B05, 53C24
Secondary: 22F50, 32H50
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Publication
Received: 12 March 2005
Accepted: 16 December 2005
Published: 26 December 2005
Proposed: Benson Farb
Seconded: Jean-Pierre Otal, Steven Ferry
Authors
Anders Karlsson
Mathematics Department
Royal Institute of Technology
100 44 Stockholm
Sweden