RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mosc. Math. J.: Year: Volume: Issue: Page: Find

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

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
Language:

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} 

• http://mi.mathnet.ru/eng/mmj387
• http://mi.mathnet.ru/eng/mmj/v10/i2/p415

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
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
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