Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Group actions on one-manifolds, II: Extensions of Hölder's Theorem

Authors: Benson Farb and John Franks
Journal: Trans. Amer. Math. Soc. 355 (2003), 4385-4396
MSC (2000): Primary 37E10
Published electronically: July 8, 2003
MathSciNet review: 1986507
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This self-contained paper is part of a series seeking to understand groups of homeomorphisms of manifolds in analogy with the theory of Lie groups and their discrete subgroups. In this paper we consider groups which act on $\mathbf R$ with restrictions on the fixed point set of each element. One result is a topological characterization of affine groups in $\mathrm{Diff}^2(\mathbf R)$ as those groups whose elements have at most one fixed point.

References [Enhancements On Off] (What's this?)

  • [B] T. Barbot, Caracterisation des flots d'Anosov en dimension 3 par leurs feuilletages faibles, Ergodic Theory Dynam. Systems 15 (1995), no. 2, 247-270. MR 96d:58100
  • [dMvS] W. de Melo and S. van Strien One-dimensional Dynamics, Springer-Verlag, Berlin, (1993). MR 95a:58035
  • [FF1] B. Farb and J. Franks, Groups of homeomorphisms of one-manifolds, I: Nonlinear group actions, June 2001 preprint.
  • [FF2] B. Farb and J. Franks, Groups of homeomorphisms of one-manifolds, III: Nilpotent subgroups, Ergodic Theory and Dynam. Syst., to appear.
  • [FS] B. Farb and P. Shalen, Groups of real-analytic diffeomorphisms of the circle, Ergodic Theory and Dynam. Syst. 22 (2002), 835-844. MR 2003e:37030
  • [Gh] E. Ghys, Groups acting on the circle, IMCA, Lima, June 1999.
  • [H] M. Hirsch, A stable analytic foliation with only exceptional minimal set, in Lecture Notes in Math., Vol. 468. Springer-Verlag, 1975. MR 52:13094
  • [Ho] O. Hölder, Die Axiome der Quantität und die Lehre vom Mass. Ber. Verh. Sachs. Ges. Wiss. Leipzig, Math. Phys. C1. 53, 1-64 (1901).
  • [K] N. Kovacevic, Möbius-like groups of homeomorphisms of the circle. Trans. Amer. Math. Soc. 351 (1999), no. 12, 4791-4822. MR 2000c:20074
  • [P] J. Plante, Solvable Groups acting on the line, Trans. Amer. Math. Soc. 278 (1983), 401-414. MR 85b:57048
  • [P2] J. Plante, Subgroups of continuous groups acting differentiably on the half-line, Ann. Inst. Fourier, Grenoble 34 1 (1984), 47-56. MR 86j:58020
  • [S] V. V. Solodov, Topological problems in the theory of dynamical systems. (Russian) Uspekhi Mat. Nauk 46 (1991), no. 4(280),93-114, 192 translation in Russian Math. Surveys 46 (1991), no. 4, 107-134. MR 92k:58203

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37E10

Retrieve articles in all journals with MSC (2000): 37E10

Additional Information

Benson Farb
Affiliation: Department of Mathematics, University of Chicago, 5734 University Ave., Chicago, Illinois 60637

John Franks
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208

Received by editor(s): September 6, 2001
Received by editor(s) in revised form: November 29, 2001
Published electronically: July 8, 2003
Additional Notes: The first author was supported in part by NSF grant DMS9704640
The second author was supported in part by NSF grant DMS9803346
Article copyright: © Copyright 2003 American Mathematical Society