## Geometry & Topology

### 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 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

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

Dates
Accepted: 16 December 2005
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513799683

Digital Object Identifier
doi:10.2140/gt.2005.9.2359

Mathematical Reviews number (MathSciNet)
MR2209375

Zentralblatt MATH identifier
1120.53026

#### Citation

Karlsson, Anders. On the dynamics of isometries. Geom. Topol. 9 (2005), no. 4, 2359--2394. doi:10.2140/gt.2005.9.2359. https://projecteuclid.org/euclid.gt/1513799683

#### References

• M Abate, Iteration theory on weakly convex domains, from: “Seminars in complex analysis and geometry (Arcavacata, 1988)” Sem. Conf. 4, EditEl, Rende (1990) 1–16
• M Abate, Iteration Theory of Holomorphic Maps on Taut Manifolds, Mediterranean Press, Rende (1989)
• R C Alperin, B Farb, G A Noskov, A strong Schottky lemma for nonpositively curved singular spaces, Geom. Dedicata 92 (2002) 235–243
• W Ballmann, Lectures on spaces of nonpositive curvature, DMV Seminar 25, Birkhäuser Verlag, Basel (1995)
• W Ballmann, M Brin, Orbihedra of nonpositive curvature, Inst. Hautes Études Sci. Publ. Math. (1995) 169–209 (1996)
• W Ballmann, M Gromov, V Schroeder, Manifolds of Nonpositive Curvature, Progress in Math. 61, Birkhäuser, Boston (1985)
• Y Benoist, Convexes divisibles, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 387–390
• M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundlehren series 319, Springer–Verlag, Berlin (1999)
• A Całka, On conditions under which isometries have bounded orbits, Colloq. Math. 48 (1984) 219–227
• K Diederich, J E Fornaess, Proper holomorphic maps onto pseudoconvex domains with real analytic boundary, Ann. of Math. 110 (1979) 575–592
• B Farb, L Mosher, Quasi-isometric rigidity for the solvable Baumslag–Solitar groups. II, Invent. Math. 137 (1999) 613–649
• F Forstneric, J-P Rosay, Localization of the Kobayashi metric and the boundary continuity of proper holomorphic mappings, Math. Ann. 279 (1987) 239–252
• H Furstenberg, A Poisson formula for semi-simple Lie groups, Ann. of Math. (2) 77 (1963) 335–386
• M Gromov, Hyperbolic groups, from: “Essays in group theory”, Math. Sci. Res. Inst. Publ. 8, Springer, New York (1987) 75–263
• M Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics 152, Birkhäuser Boston Inc., Boston, MA (1999)
• P de la Harpe, On Hilbert's metric for simplices, from: “Geometric Group Theory, vol. 1 (Sussex, 1991)”, London Math. Soc. Lecture Note Ser. 181, Cambridge University Press (1993) 97–119
• P de la Harpe, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL (2000)
• N V Ivanov, Subgroups of Teichmüller modular groups. Transl. of Math. Monographs. 115, American Mathematical Society, Providence, RI (1992)
• N V Ivanov, Mapping class groups, from: “Handbook of geometric topology”, North-Holland, Amsterdam (2002) 523–633
• M Jarnicki, P Pflug, Invariant Distances and Metrics in Complex Analysis, Expositions in Math. 9, Walter de Gruyter, Berlin–New York (1993)
• V A Kaimanovich, The Poisson formula for groups with hyperbolic properties, Ann. of Math. (2) 152 (2000) 659–692
• V A Kaimanovich, H Masur, The Poisson boundary of the mapping class group, Invent. Math. 125 (1996) 221–264
• A Karlsson, Non-expanding maps and Busemann functions, Ergodic Theory Dynam. Systems 21 (2001) 1447–1457
• A Karlsson, G A Margulis, A multiplicative ergodic theorem and nonpositively curved spaces, Comm. Math. Phys. 208 (1999) 107–123
• A Karlsson, G A Noskov, The Hilbert metric and Gromov hyperbolicity, Enseign. Math. (2) 48 (2002) 73–89
• A Karlsson, G A Noskov, Some groups having only elementary actions on metric spaces with hyperbolic boundaries, Geom. Dedicata 104 (2004) 119–137
• S Kobayashi, Hyperbolic complex spaces, Grundlehren series 318, Springer–Verlag, Berlin (1998)
• H Masur, Two boundaries of Teichmüller space, Duke Math. J. 49 (1982) 183–190
• J McCarthy, A Papadopoulos, Dynamics on Thurston's sphere of projective measured foliations, Comment. Math. Helv. 64 (1989) 133–166
• C T McMullen, Coxeter groups, Salem numbers and the Hilbert metric, Publ. Math. Inst. Hautes Études Sci. (2002) 151–183
• R D Nussbaum, Hilbert's projective metric and iterated nonlinear maps, Mem. Amer. Math. Soc. 75 (1988) iv+137
• J G Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics 149, Springer–Verlag, New York (1994)
• K E Ruane, Dynamics of the action of a ${\rm CAT}(0)$ group on the boundary, Geom. Dedicata 84 (2001) 81–99
• E L Swenson, A cut point theorem for ${\rm CAT}(0)$ groups, J. Differential Geom. 53 (1999) 327–358
• J Tits, Buildings of spherical type and finite BN–pairs, Springer–Verlag, Berlin (1974)
• W Zhang, F Ren, Dynamics on weakly pseudoconvex domains, Chinese Ann. Math. Ser. B 16 (1995) 467–476