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 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**, 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**, 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**, 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**, 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**, 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:
http://dx.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