no. 2
Family of intersecting totally real manifolds of (ℂn,0) and germs of holomorphic diffeomorphisms
[Famille d'intersection de variétés totalement réelles de (ℂn,0) et singularités CR]
Bulletin de la Société Mathématique de France, Tome 143 (2015) no. 2, pp. 247-263

We prove the existence (and give a characterization) of a germ of complex analytic set left invariant by an abelian group of germs of holomorphic diffeomorphisms at a common fixed point.We also give condition that ensure that such a group can be linearized holomorphically near the fixed point. It rests on a “small divisors condition” of the family of linear parts.

The second part of this article is devoted to the study families of totally real intersecting n-submanifolds of (ℂn,0). We give some conditions which allow to straighten holomorphically the family. If it is not possible to do this formally, we construct a germ of complex analytic set at the origin which intersection with the family can be holomorphically straightened. The second part is an application of the first.

@article{BSMF_2015__143_2_247_0,
     author = {Stolovitch, Laurent},
     title = {Family of intersecting totally real manifolds of~$({\mathbb {C}}^n,0)$ and germs of holomorphic diffeomorphisms},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {247--263},
     year = {2015},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {143},
     number = {2},
     doi = {10.24033/bsmf.2685},
     mrnumber = {3351178},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/bsmf.2685/}
}
TY  - JOUR
AU  - Stolovitch, Laurent
TI  - Family of intersecting totally real manifolds of $({\mathbb {C}}^n,0)$ and germs of holomorphic diffeomorphisms
JO  - Bulletin de la Société Mathématique de France
PY  - 2015
SP  - 247
EP  - 263
VL  - 143
IS  - 2
PB  - Société mathématique de France
UR  - https://www.numdam.org/articles/10.24033/bsmf.2685/
DO  - 10.24033/bsmf.2685
LA  - en
ID  - BSMF_2015__143_2_247_0
ER  - 
%0 Journal Article
%A Stolovitch, Laurent
%T Family of intersecting totally real manifolds of $({\mathbb {C}}^n,0)$ and germs of holomorphic diffeomorphisms
%J Bulletin de la Société Mathématique de France
%D 2015
%P 247-263
%V 143
%N 2
%I Société mathématique de France
%U https://www.numdam.org/articles/10.24033/bsmf.2685/
%R 10.24033/bsmf.2685
%G en
%F BSMF_2015__143_2_247_0
Stolovitch, Laurent. Family of intersecting totally real manifolds of $({\mathbb {C}}^n,0)$ and germs of holomorphic diffeomorphisms. Bulletin de la Société Mathématique de France, Tome 143 (2015) no. 2, pp. 247-263. doi: 10.24033/bsmf.2685

Ahern, Patrick; Gong, Xianghong A complete classification for pairs of real analytic curves in the complex plane with tangential intersection, J. Dyn. Control Syst., Volume 11 (2005), pp. 1-71 (ISSN: 1079-2724) | MR | Zbl | DOI

Cairns, Grant; Ghys, Étienne The local linearization problem for smooth SL (n)-actions, Enseign. Math., Volume 43 (1997), pp. 133-171 (ISSN: 0013-8584) | MR | Zbl

Chaperon, Marc Géométrie différentielle et singularités de systèmes dynamiques, Astérisque, Volume 138-139 (1986) (ISSN: 0303-1179) | MR | Zbl | Numdam

DeLatte, David; Gramchev, Todor Biholomorphic maps with linear parts having Jordan blocks: linearization and resonance type phenomena, Math. Phys. Electron. J., Volume 8 (2002) (ISSN: 1086-6655) | MR | Zbl

Gong, Xianghong; Stolovitch, Laurent Real submanifolds of maximum complex tangent space at a CR singular point (preprint arXiv:1406.1294, submitted for publication) | MR

Gramchev, Todor; Yoshino, Masafumi Rapidly convergent iteration method for simultaneous normal forms of commuting maps, Math. Z., Volume 231 (1999), pp. 745-770 (ISSN: 0025-5874) | MR | Zbl | DOI

Guillemin, Victor W.; Sternberg, Shlomo Remarks on a paper of Hermann, Trans. Amer. Math. Soc., Volume 130 (1968), pp. 110-116 (ISSN: 0002-9947) | MR | Zbl | DOI

Kasner, E. Conformal geometry, Proc. Fifth Intern. Math. Congr. (1913), pp. 81-87 | JFM

Kushnirenko, A. G. Linear-equivalent action of a semi-simple lie group in the neighbourhood of a stationary point, Funct. Anal. Appl., Volume 1 (1967), pp. 89-90 | DOI

Moser, Jürgen On commuting circle mappings and simultaneous Diophantine approximations, Math. Z., Volume 205 (1990), pp. 105-121 (ISSN: 0025-5874) | DOI | MR

Nakai, Isao The classification of curvilinear angles in the complex plane and the groups of ± holomorphic diffeomorphisms, Ann. Fac. Sci. Toulouse Math., Volume 7 (1998), pp. 313-334 (ISSN: 0240-2963) | Numdam | MR | Zbl

Pfeiffer, G. A. On the Conformal Geometry of Analytic Arcs, Amer. J. Math., Volume 37 (1915), pp. 395-430 (ISSN: 0002-9327) | MR | JFM | DOI

Pöschel, Jürgen On invariant manifolds of complex analytic mappings near fixed points, Exposition. Math., Volume 4 (1986), pp. 97-109 (ISSN: 0723-0869) | MR | Zbl

Rüssmann, Helmut On the convergence of power series transformations of analytic mappings near a fixed point into a normal form (1977) (preprint IHÉS M/77/178)

Rüssmann, Helmut Stability of elliptic fixed points of analytic area-preserving mappings under the Bruno condition, Ergodic Theory Dynam. Systems, Volume 22 (2002), pp. 1551-1573 (ISSN: 0143-3857) | MR | Zbl | DOI

Stolovitch, Laurent Sur un théorème de Dulac, Ann. Inst. Fourier (Grenoble), Volume 44 (1994), pp. 1397-1433 (ISSN: 0373-0956) | Numdam | MR | Zbl

Stolovitch, Laurent Singular complete integrability, Publ. Math. IHÉS, Volume 91 (2000), p. 133-210 (2001) (ISSN: 0073-8301) | Numdam | MR | Zbl | DOI

Stolovitch, Laurent, Hamiltonian dynamical systems and applications (NATO Sci. Peace Secur. Ser. B Phys. Biophys.), Springer, Dordrecht, 2008, pp. 249-284 | DOI | MR

Stolovitch, Laurent Family of intersecting totally real manifolds of ( ℂ n , 0 ) and CR-singularities (preprint arXiv:math/0506052 )

Trépreau, Jean-Marie Discrimination analytique des difféomorphismes résonnants de (ℂ,0) et réflexion de Schwarz, Astérisque, Volume 284 (2003), pp. 271-319 (ISSN: 0303-1179) | MR | Numdam | Zbl

Webster, S. M. Pairs of intersecting real manifolds in complex space, Asian J. Math., Volume 7 (2003), pp. 449-462 (ISSN: 1093-6106) | MR

Yoccoz, J.-C. Petits diviseurs en dimension 1, Astérisque, 231, 1995 | Zbl | Numdam

Cité par Sources :