Geodesic
Curves in a~Foliated Plane
Informatics and Automation, Analysis and singularities. Part 2, Tome 259 (2007), pp. 291-303.

Voir la notice de l'article provenant de la source Math-Net.Ru

The paper is devoted to the classification of nonsingular and singular plane curve germs with respect to the group of local diffeomorphisms preserving the foliation of the plane by the phase curves of a fixed vector field, either nonsingular or singular. We define the multiplicity of a pair consisting of a plane curve and a vector field and prove an analog of the Tougeron theorem on finite determinacy. It leads, almost immediately, to a number of classification results; a part of them is contained in the work. 
@article{TRSPY_2007_259_a16,
     author = {M. Ya. Zhitomirskii},
     title = {Curves in {a~Foliated} {Plane}},
     journal = {Informatics and Automation},
     pages = {291--303},
     publisher = {mathdoc},
     volume = {259},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2007_259_a16/}
}
TY  - JOUR
AU  - M. Ya. Zhitomirskii
TI  - Curves in a~Foliated Plane
JO  - Informatics and Automation
PY  - 2007
SP  - 291
EP  - 303
VL  - 259
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2007_259_a16/
LA  - en
ID  - TRSPY_2007_259_a16
ER  - 
%0 Journal Article
%A M. Ya. Zhitomirskii
%T Curves in a~Foliated Plane
%J Informatics and Automation
%D 2007
%P 291-303
%V 259
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TRSPY_2007_259_a16/
%G en
%F TRSPY_2007_259_a16
M. Ya. Zhitomirskii. Curves in a~Foliated Plane. Informatics and Automation, Analysis and singularities. Part 2, Tome 259 (2007), pp. 291-303. http://geodesic.mathdoc.fr/item/TRSPY_2007_259_a16/

[1] Arnold V. I., “Mathematical problems in classical physics”, Trends and perspectives in applied mathematics, Appl. Math. Sci., 100, eds. F. John, J. E. Marsden, L. Sirovich, Springer, New York, 1994, 1–20 | MR | Zbl

[2] Arnold V. I., Ilyashenko Yu. S., “Obyknovennye differentsialnye uravneniya”, Dinamicheskie sistemy–1, Itogi nauki i tekhniki. Sovr. probl. matematiki. Fund. napr., 1, VINITI, M., 1985, 7–149 | MR

[3] Arnold V. I., Varchenko A. N., Gusein-Zade S. M., Osobennosti differentsiruemykh otobrazhenii: Klassifikatsiya kriticheskikh tochek, kaustik i volnovykh frontov, Nauka, M., 1982 | MR

[4] Bogdanov R. I., “Singularities of vector fields on the plane with pointed direction”, Invent. math., 54 (1979), 247–259 | DOI | MR | Zbl

[5] Davydov A. A., “Normalnaya forma differentsialnogo uravneniya, ne razreshennogo otnositelno proizvodnoi, v okrestnosti ego osoboi tochki”, Funkts. analiz i ego pril., 19:2 (1985), 1–10 | MR | Zbl

[6] Moussu R., “Sur l'existence d'intégrales premières pour un germe de forme de Pfaff”, Ann. Inst. Fourier, 26:2 (1976), 171–220 | MR