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Mosc. Math. J., 2010, Volume 10, Number 2, Pages 415–468 (Mi mmj387)  

This article is cited in 3 scientific papers (total in 3 papers)

Relations of formal diffeomorphisms and the center problem

Isao Nakaiab, Kana Yanaia

a Ochanomizu University, Dept. of Mathematics, Faculty of Science, Tokyo (Japan)
b Kyoto University, Research Institute of Mathematical Science

Abstract: A word of germs of holomorphic diffeomorphisms of $(\mathbb C,0)$ is a composite of some time-1 maps of formal vector fields fixing 0, in other words, a noncommutative integral of a piecewise constant time depending formal vector field. We calculate its formal-vector-field-valued logarithm applying the Campbell–Hausdorff type formula of the Lie integral due to Chacon and Fomenko to the time depending formal vector field. For words of two time 1-maps we define Cayley diagrams in the plane spanned by the generating two vector fields in the Lie algebra of formal vector fields, and we show that some principal parts in the Taylor coefficients of the logarithm are given in terms of the higher moments of the Cayley diagrams. Solving the so-called center problem, the vanishing of the Lie integral, we show the various results on the existence and non-existence of relations of non-commuting formal diffeomorphisms in terms of the characteristic curves associated to the Cayley diagram.

Key words and phrases: holomorphic diffeomorphism, relation, free group, Campbell–Hausdorff.

DOI: https://doi.org/10.17323/1609-4514-2010-10-2-415-468

Full text: http://www.ams.org/.../abst10-2-2010.html
References: PDF file   HTML file

Bibliographic databases:

MSC: 37F75, 53C12, 81Q70
Received: July 29, 2008; in revised form November 17, 2009
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Citation: Isao Nakai, Kana Yanai, “Relations of formal diffeomorphisms and the center problem”, Mosc. Math. J., 10:2 (2010), 415–468

Citation in format AMSBIB
\Bibitem{NakYan10}
\by Isao~Nakai, Kana~Yanai
\paper Relations of formal diffeomorphisms and the center problem
\jour Mosc. Math.~J.
\yr 2010
\vol 10
\issue 2
\pages 415--468
\mathnet{http://mi.mathnet.ru/mmj387}
\crossref{https://doi.org/10.17323/1609-4514-2010-10-2-415-468}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2722805}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000279342400008}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Mattei J.-F., Rebelo J.C., Reis H., “Generic Pseudogroups on (C, 0) and the Topology of Leaves”, Compos. Math., 149:8 (2013), 1401–1430  crossref  mathscinet  zmath  isi  scopus
    2. Giat Sh., Shelah Y., Shikhelman C., Yomdin Y., “Algebraic Geometry of Abel Differential Equation”, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat., 108:1, SI (2014), 193–210  crossref  mathscinet  zmath  isi  elib  scopus
    3. Briskin M., Pakovich F., Yomdin Y., “Algebraic Geometry of the Center-Focus Problem For Abel Differential Equations”, Ergod. Theory Dyn. Syst., 36:3 (2016), 714–744  crossref  mathscinet  zmath  isi
  • Moscow Mathematical Journal
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