Subgroups of 𝐷𝑖𝑓𝑓^{∞}₊(𝕊¹) acting transitively on 4-tuples

Rebelo, Julio

doi:10.1090/S0002-9947-04-03466-X

Subgroups of $\operatorname {Diff}^{\infty }_+ (\mathbb S^1)$ acting transitively on $4$-tuples
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Trans. Amer. Math. Soc. 356 (2004), 4543-4557 Request permission

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.

References

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, DOI 10.1007/s00014-002-8350-2
M. Demazure, Classification des algèbres de Lie filtrées, Sém. Bourbaki 326 (1967).
Yakov M. Eliashberg and William P. Thurston, Confoliations, University Lecture Series, vol. 13, American Mathematical Society, Providence, RI, 1998. MR 1483314, DOI 10.1090/ulect/013
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, DOI 10.1007/978-3-322-90161-3
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
Irving Kaplansky, Lie algebras and locally compact groups, University of Chicago Press, Chicago, Ill.-London, 1971. MR 0276398
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
Sophus Lie’s 1880 transformation group paper, Lie Groups: History, Frontiers and Applications, Vol. I, 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. MR 0460053
F. Loray & J.C. Rebelo, Minimal, rigid foliations by curves on ${\mathbb C} P^n$, to appear in Journal of the European Mathematical Society.
Edwin E. Moise, Geometric topology in dimensions $2$ and $3$, Graduate Texts in Mathematics, Vol. 47, Springer-Verlag, New York-Heidelberg, 1977. MR 0488059
Isao Nakai, Separatrices for nonsolvable dynamics on $\textbf {C},0$, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 2, 569–599 (English, with English and French summaries). MR 1296744
M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology 1 (1962), 101–120. MR 142859, DOI 10.1016/0040-9383(65)90018-2
Julio C. Rebelo, Ergodicity and rigidity for certain subgroups of $\textrm {Diff}^\omega (S^1)$, Ann. Sci. École Norm. Sup. (4) 32 (1999), no. 4, 433–453 (English, with English and French summaries). MR 1693579, DOI 10.1016/S0012-9593(99)80019-6
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.
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
Shlomo Sternberg, Local $C^{n}$ transformations of the real line, Duke Math. J. 24 (1957), 97–102. MR 102581