Complex analysis and geometry, Dynamical systems
A quick proof of the regularity of the flow of analytic vector fields
Comptes Rendus. Mathématique, Volume 359 (2021) no. 9, pp. 1155-1159.

We offer a new and elementary proof of the convergence of the Lie series giving the flow of an analytic vector field as well as a natural deduction of such series.

Received:
Accepted:
Published online:
DOI: 10.5802/crmath.271
Classification: 32M25
Carrillo, Sergio A. 1

1 Programa de matemáticas, Universidad Sergio Arboleda, Calle 74 # 14-14, Bogotá, Colombia.
@article{CRMATH_2021__359_9_1155_0,
     author = {Carrillo, Sergio A.},
     title = {A quick proof of the regularity of the flow of analytic vector fields},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1155--1159},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {9},
     year = {2021},
     doi = {10.5802/crmath.271},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/crmath.271/}
}
TY  - JOUR
AU  - Carrillo, Sergio A.
TI  - A quick proof of the regularity of the flow of analytic vector fields
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 1155
EP  - 1159
VL  - 359
IS  - 9
PB  - Académie des sciences, Paris
UR  - http://www.numdam.org/articles/10.5802/crmath.271/
DO  - 10.5802/crmath.271
LA  - en
ID  - CRMATH_2021__359_9_1155_0
ER  - 
%0 Journal Article
%A Carrillo, Sergio A.
%T A quick proof of the regularity of the flow of analytic vector fields
%J Comptes Rendus. Mathématique
%D 2021
%P 1155-1159
%V 359
%N 9
%I Académie des sciences, Paris
%U http://www.numdam.org/articles/10.5802/crmath.271/
%R 10.5802/crmath.271
%G en
%F CRMATH_2021__359_9_1155_0
Carrillo, Sergio A. A quick proof of the regularity of the flow of analytic vector fields. Comptes Rendus. Mathématique, Volume 359 (2021) no. 9, pp. 1155-1159. doi : 10.5802/crmath.271. http://www.numdam.org/articles/10.5802/crmath.271/

[1] Ahern, Patrick; Rosay, Jean-Pierre Entire functions, in the classification of differentiable germs tangent to the identity, in one or two variables, Trans. Am. Math. Soc., Volume 347 (1995) no. 2, pp. 543-572 | DOI | MR | Zbl

[2] Canalis-Durand, Mireille; Ramis, Jean-Pierre; Schäfke, Reinhard; Sibuya, Yasutaka Gevrey solutions of singularly perturbed differential equations, J. Reine Angew. Math., Volume 518 (2000) no. 2, pp. 95-129 | DOI | MR | Zbl

[3] Cartan, Henri Elementary theory of analytic functions of one or several complex variables, Hermann, 1973

[4] Gröbner, Wolfgang Die Lie-Reihen und ihre Anwendungen, VEB Deutscher Verlag der Wissenschaften, 1960 | MR | Zbl

[5] Ilyashenko, Yulij; Yakovenko, Sergei Lectures on analytic differential equations, Graduate Studies in Mathematics, 86, American Mathematical Society, 2008 | MR | Zbl

[6] Martinet, Jean; Ramis, Jean-Pierre Classification analytique des équations différentielles non linéaires résonnantes du premier ordre, Ann. Sci. Éc. Norm. Supér., Volume 16 (1983) no. 4, pp. 571-621 | DOI | Numdam | Zbl

[7] Nagumo, Mitio Über das Anfangswertproblem partieller Differentialgleichunge, Jap. J. Math., Volume 18 (1942), pp. 41-47 | DOI | MR | Zbl

[8] Steinberg, Stanly Lie series and nonlinear ordinary differential equations, J. Math. Anal. Appl., Volume 101 (1984), pp. 39-63 | DOI | MR | Zbl

[9] Winkel, Rudolf An exponential formula for polynomial vector fields: II. Lie series, exponential substituition, and rooted trees, Adv. Math., Volume 147 (1999), pp. 260-303 | DOI | MR | Zbl

Cited by Sources: