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On transverse rigidity for singular foliations in (ℂ2,0)

Published online by Cambridge University Press:  22 April 2010

JULIO C. REBELO
JULIO C. REBELO*
Affiliation:
Université de Toulouse, UPS, INSA, UT1, UTM, Institut de Mathématique de Toulouse, F-31062 Toulouse, France CNRS, Institut de Mathématique de Toulouse UMR 5219, F-31062 Toulouse, France (email: rebelo@math.univ-toulouse.fr)
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Abstract

This note is a companion paper to ‘Monodromie et classification topologique des germes de feuilletages holomorphes’ (in French) by Marin and Mattei. In their paper the authors develop a very complete theory about topologically conjugate singularities of foliations. However, their techniques require the topological conjugacies to be transversely holomorphic. The purpose of the present note is to show that this assumption can be made without restricting the generality of their results.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 3 , June 2011 , pp. 935 - 950
Copyright
Copyright © Cambridge University Press 2010

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References

[1]Belliart, M.. Sur certains pseudogroupes de biholomorphismes locaux de (ℂn,0). Bull. Soc. Math. France 129 (2001), 259–284.Google Scholar
[2]Belliart, M., Liousse, I. and Loray, F.. Sur l’existence de points fixes attractifs pour les sous-groupes de Aut(ℂ,0). C. R. Acad. Sci. Paris, Sér. I, Math. 324 (1997), 443–446.CrossRefGoogle Scholar
[3]Huddai-Verenov, M. O.. A property of the solutions of a differential equation. Mat. Sb. 56(98) (1962), 301–308 (in Russian).Google Scholar
[4]Ghys, E.. Sur les groupes engendrés par des difféomorphismes proches de l’identité. Bol. Soc. Bras. Mat. Nova Sér. 24 (1993), 137–178.Google Scholar
[5]Loray, F. and Rebelo, J. C.. Minimal, rigid foliations by curves on ℂℙ(n). J. Eur. Math. Soc. 5(2) (2003), 147–201.CrossRefGoogle Scholar
[6]Marin, D. and Mattei, J.-F.. Incompressibilité des feuilles de germes de feuilletages holomorphes singuliers. Ann. Sci. École Norm. Sup. (4) 41 (2008), 855–903.Google Scholar
[7]Marin, D. and Mattei, J.-F.. Monodromie et classification topologique des germes de feuilletages holomorphes. Preprint.Google Scholar
[8]Mattei, J.-F. and Moussu, R.. Holonomie et intégrales premières. Ann. Sci. École Norm. Sup. 16 (1983), 469–523.Google Scholar
[9]Martinet, J. and Ramis, J.-P.. Problèmes de modules pour des équations différentielles non linéaires du premier ordre. Publ. Math. Inst Hautes Études Sci. 55 (1982), 63–164.Google Scholar
[10]Martinet, J. and Ramis, J.-P.. Classification analytique des équations différentielles non linéaires résonnantes du premier ordre. Ann. Sci. École Norm. Sup. (4) 16(4) (1983), 571–621.Google Scholar
[11]Montgomery, D. and Zippin, L.. Topological Transformation Groups, 3rd edn. (Tracts in Pure and Applied Mathematics, 1). Interscience Publishers, New York, 1965.Google Scholar
[12]Nakai, I.. Separatrices for non-solvable dynamics on (ℂ,0). Ann. Inst. Fourier 44 (1994), 569–599.CrossRefGoogle Scholar
[13]Rebelo, J. C.. Ergodicity and rigidity for certain subgroups of Diff ω(S 1). Ann. Sci. École Norm. Sup. (4) 32 (1999), 433–453.Google Scholar
[14]Reis, H.. Equivalence and semi-completude of foliations. Nonlinear Anal. 64(8) (2006), 1654–1665.Google Scholar
[15]Seidenberg, A.. Reduction of singularities of the differentiable equation Ady=Bdx. Amer. J. Math. 90 (1968), 248–269.CrossRefGoogle Scholar
[16]Shcherbakov, A. A.. On the density of an orbit of a pseudogroup of conformal mappings and a generalization of the Hudai–Verenov theorem. Vestn. Mosk. Univ. 31(Ser. I) (1982), 10–15.Google Scholar
[17]Shcherbakov, A. A.. Topological and analytic conjugation of non-commutative groups of conformal mappings. Tr. Semin. Petrovsk 10 (1984), 170–192.Google Scholar