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Subgroups of $\operatorname{Diff}^{\infty}_+ (\mathbb S^1)$ acting transitively on $4$-tuples


Author: Julio C. Rebelo
Journal: Trans. Amer. Math. Soc. 356 (2004), 4543-4557
MSC (2000): Primary 37B05, 22E65
Published electronically: March 12, 2004
MathSciNet review: 2067133
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Abstract: We consider subgroups of $C^{\infty}$-diffeomorphisms of the circle $\mathbb S^1$which act transitively on $4$-tuples of points. We show, in particular, that these subgroups are dense in the group of homeomorphisms of $\mathbb S^1$. A stronger result concerning $C^{\infty}$-approximations is obtained as well. The techniques employed in this paper rely on Lie algebra ideas and they also provide partial generalizations to the differentiable case of some results previously established in the analytic category.


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  • [Be] Michel Belliart, On the dynamics of certain actions of free groups on closed real analytic manifolds, Comment. Math. Helv. 77 (2002), no. 3, 524–548. MR 1933788, 10.1007/s00014-002-8350-2
  • [De] M. DEMAZURE, Classification des algèbres de Lie filtrées, Sém. Bourbaki 326 (1967).
  • [E-T] Yakov M. Eliashberg and William P. Thurston, Confoliations, University Lecture Series, vol. 13, American Mathematical Society, Providence, RI, 1998. MR 1483314
  • [H-H] Gilbert Hector and Ulrich Hirsch, Introduction to the geometry of foliations. Part B, 2nd ed., Aspects of Mathematics, E3, Friedr. Vieweg & Sohn, Braunschweig, 1987. Foliations of codimension one. MR 1110794
  • [He] Michael-Robert Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math. 49 (1979), 5–233 (French). MR 538680
  • [Ka] Irving Kaplansky, Lie algebras and locally compact groups, The University of Chicago Press, Chicago, Ill.-London, 1971. MR 0276398
  • [Ko] Nancy Kopell, Commuting diffeomorphisms, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 165–184. MR 0270396
  • [Lie] Sophus Lie’s 1880 transformation group paper, Math Sci Press, Brookline, Mass., 1975. In part a translation of “Theorie der Transformations-gruppen” by S. Lie [Math. Ann. 16 (1880), 441–528]; Translated by Michael Ackerman; Comments by Robert Hermann; Lie Groups: History, Frontiers and Applications, Vol. I. MR 0460053
  • [L-R] F. LORAY & J.C. REBELO, Minimal, rigid foliations by curves on ${\mathbb C} P^n$, to appear in Journal of the European Mathematical Society.
  • [Mo] Edwin E. Moise, Geometric topology in dimensions 2 and 3, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, Vol. 47. MR 0488059
  • [Na] Isao Nakai, Separatrices for nonsolvable dynamics on 𝐶,0, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 2, 569–599 (English, with English and French summaries). MR 1296744
  • [Pe] M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology 1 (1962), 101–120. MR 0142859
  • [Re] Julio C. Rebelo, Ergodicity and rigidity for certain subgroups of 𝐷𝑖𝑓𝑓^{𝜔}(𝑆¹), Ann. Sci. École Norm. Sup. (4) 32 (1999), no. 4, 433–453 (English, with English and French summaries). MR 1693579, 10.1016/S0012-9593(99)80019-6
  • [R-S] J.C. REBELO & R.R. SILVA, The multiple ergodicity of non-discrete subgroups of $\operatorname{Diff}^{\omega} (\mathbb{S}^1)$, to appear in the Moscow Mathematical Journal.
  • [Sh] A. A. Shcherbakov, Density of the orbit of a pseudogroup of conformal mappings and generalization of the Khudaĭ-Verenov theorem, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 4 (1982), 10–15, 84 (Russian, with English summary). MR 671879
  • [St] Shlomo Sternberg, Local 𝐶ⁿ transformations of the real line, Duke Math. J. 24 (1957), 97–102. MR 0102581

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Additional Information

Julio C. Rebelo
Affiliation: Pontificia Universidade Catolica do Rio de Janeiro PUC-Rio, Rua Marques de São Vicente 225 - Gavea, Rio de Janeiro, RJ CEP 22453-900, Brazil
Address at time of publication: Institute for Mathematical Sciences, State University of New York at Stony Brook, Stony Brook, New York 11794-3660
Email: jrebelo@mat.puc-rio.br, jrebelo@math.sunysb.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03466-X
Keywords: Groups, vector fields, Lie algebras
Received by editor(s): July 3, 2002
Received by editor(s) in revised form: July 1, 2003
Published electronically: March 12, 2004
Article copyright: © Copyright 2004 American Mathematical Society