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
DOI:
https://doi.org/10.1090/S0002-9947-03-03376-2
Published electronically:
July 8, 2003
MathSciNet review:
1986507
Full-text PDF Free Access
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 with restrictions on the fixed point set of each element. One result is a topological characterization of affine groups in
as those groups whose elements have at most one fixed point.
- [B] Thierry Barbot, Caractérisation des flots d’Anosov en dimension 3 par leurs feuilletages faibles, Ergodic Theory Dynam. Systems 15 (1995), no. 2, 247–270 (French, with English summary). MR 1332403, https://doi.org/10.1017/S0143385700008361
- [dMvS] Welington de Melo and Sebastian van Strien, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. MR 1239171
- [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] Benson Farb and Peter Shalen, Groups of real-analytic diffeomorphisms of the circle, Ergodic Theory Dynam. Systems 22 (2002), no. 3, 835–844. MR 1908556, https://doi.org/10.1017/S014338570200041X
- [Gh] E. Ghys, Groups acting on the circle, IMCA, Lima, June 1999.
- [H] An index to volumes 1-500 of Lecture Notes in Mathematics and other useful information, Lecture Notes in Mathematics, Vol. 1-500, Springer-Verlag, Berlin-New York, 1975. MR 0392277
- [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] Nataša Kovačević, Möbius-like groups of homeomorphisms of the circle, Trans. Amer. Math. Soc. 351 (1999), no. 12, 4791–4822. MR 1473447, https://doi.org/10.1090/S0002-9947-99-02189-3
- [P] J. F. Plante, Solvable groups acting on the line, Trans. Amer. Math. Soc. 278 (1983), no. 1, 401–414. MR 697084, https://doi.org/10.1090/S0002-9947-1983-0697084-7
- [P2] J. F. Plante, Subgroups of continuous groups acting differentiably on the half-line, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 1, 47–56 (English, with French summary). MR 743621
- [S] V. V. Solodov, Topological problems in the theory of dynamical systems, Uspekhi Mat. Nauk 46 (1991), no. 4(280), 93–114, 192 (Russian); English transl., Russian Math. Surveys 46 (1991), no. 4, 107–134. MR 1138953, https://doi.org/10.1070/RM1991v046n04ABEH002819
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
Email:
farb@math.uchicago.edu
John Franks
Affiliation:
Department of Mathematics, Northwestern University, Evanston, Illinois 60208
Email:
john@math.northwestern.edu
DOI:
https://doi.org/10.1090/S0002-9947-03-03376-2
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