On the dynamics of isometries

Karlsson, Anders¹

¹ Mathematics Department, Royal Institute of Technology, 100 44 Stockholm, Sweden

Geometry & topology, Tome 9 (2005) no. 4, pp. 2359-2394.

Résumé

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.

Détail

DOI : 10.2140/gt.2005.9.2359

Keywords: metric spaces, isometries, nonpositive curvature, Kobayashi metric, random walk

Bibliographie

[1] M Abate, Iteration theory on weakly convex domains, from: "Seminars in Complex Analysis and Geometry, 1988 (Arcavacata, 1988)", Sem. Conf. 4, EditEl (1990) 1

[2] M Abate, Iteration theory of holomorphic maps on taut manifolds, Research and Lecture Notes in Mathematics. Complex Analysis and Geometry, Mediterranean Press (1989)

[3] R C Alperin, B Farb, G A Noskov, A strong Schottky lemma for nonpositively curved singular spaces, Geom. Dedicata 92 (2002) 235

[4] W Ballmann, Lectures on spaces of nonpositive curvature, DMV Seminar 25, Birkhäuser Verlag (1995)

[5] W Ballmann, M Brin, Orbihedra of nonpositive curvature, Inst. Hautes Études Sci. Publ. Math. (1995)

[6] W Ballmann, M Gromov, V Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics 61, Birkhäuser (1985)

[7] Y Benoist, Convexes divisibles, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 387

[8] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, Springer (1999)

[9] A Całka, On conditions under which isometries have bounded orbits, Colloq. Math. 48 (1984) 219

[10] K Diederich, J E Fornæss, Proper holomorphic maps onto pseudoconvex domains with real-analytic boundary, Ann. of Math. $(2)$ 110 (1979) 575

[11] B Farb, L Mosher, Quasi-isometric rigidity for the solvable Baumslag–Solitar groups II, Invent. Math. 137 (1999) 613

[12] F Forstnerič, J P Rosay, Localization of the Kobayashi metric and the boundary continuity of proper holomorphic mappings, Math. Ann. 279 (1987) 239

[13] H Furstenberg, A Poisson formula for semi-simple Lie groups, Ann. of Math. $(2)$ 77 (1963) 335

[14] M Gromov, Hyperbolic groups, from: "Essays in group theory", Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75

[15] M Gromov, Metric structures for Riemannian and non–Riemannian spaces, Progress in Mathematics 152, Birkhäuser (1999)

[16] 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 Univ. Press (1993) 97

[17] P De La Harpe, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press (2000)

[18] N V Ivanov, Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs 115, American Mathematical Society (1992)

[19] N V Ivanov, Mapping class groups, from: "Handbook of geometric topology", North-Holland (2002) 523

[20] M Jarnicki, P Pflug, Invariant distances and metrics in complex analysis, de Gruyter Expositions in Mathematics 9, Walter de Gruyter Co. (1993)

[21] V A Kaimanovich, The Poisson formula for groups with hyperbolic properties, Ann. of Math. $(2)$ 152 (2000) 659

[22] V A Kaimanovich, H Masur, The Poisson boundary of the mapping class group, Invent. Math. 125 (1996) 221

[23] A Karlsson, Non-expanding maps and Busemann functions, Ergodic Theory Dynam. Systems 21 (2001) 1447

[24] A Karlsson, G A Margulis, A multiplicative ergodic theorem and nonpositively curved spaces, Comm. Math. Phys. 208 (1999) 107

[25] A Karlsson, G A Noskov, The Hilbert metric and Gromov hyperbolicity, Enseign. Math. $(2)$ 48 (2002) 73

[26] A Karlsson, G A Noskov, Some groups having only elementary actions on metric spaces with hyperbolic boundaries, Geom. Dedicata 104 (2004) 119

[27] S Kobayashi, Hyperbolic complex spaces, Grundlehren der Mathematischen Wissenschaften, Springer (1998)

[28] H Masur, Two boundaries of Teichmüller space, Duke Math. J. 49 (1982) 183

[29] J Mccarthy, A Papadopoulos, Dynamics on Thurston's sphere of projective measured foliations, Comment. Math. Helv. 64 (1989) 133

[30] C T Mcmullen, Coxeter groups, Salem numbers and the Hilbert metric, Publ. Math. Inst. Hautes Études Sci. (2002) 151

[31] R D Nussbaum, Hilbert's projective metric and iterated nonlinear maps, Mem. Amer. Math. Soc. 75 (1988)

[32] J G Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics 149, Springer (1994)

[33] K E Ruane, Dynamics of the action of a $\mathrm{CAT}(0)$ group on the boundary, Geom. Dedicata 84 (2001) 81

[34] E L Swenson, A cut point theorem for $\mathrm{CAT}(0)$ groups, J. Differential Geom. 53 (1999) 327

[35] J Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics 386, Springer (1974)

[36] W Zhang, F Ren, Dynamics on weakly pseudoconvex domains, Chinese Ann. Math. Ser. B 16 (1995) 467