Abstract
The classical Lyapunov–Poincaré center theorem assures the existence of a first integral for an analytic 1-form near a center singularity in dimension two, provided that the first jet of the 1-form is nondegenerate. The basic point is the existence of an analytic first integral for the given 1-form. In this paper, we consider generalizations for two main frameworks: (1) real analytic foliations of codimension one in higher dimension and (2) singular holomorphic foliations in dimension two. All this is related to the problem of finding criteria assuring the existence of analytic first integrals for a given codimension one germ with a suitable first jet. Our approach consists in giving an interpretation of the center theorem in terms of holomorphic foliations and, following an idea of Moussu, apply the holomorphic foliations arsenal to obtain the required first integral. As a consequence we are able to revisit some of Reeb’s classical results on integrable perturbations of exact homogeneous 1-forms, and prove versions of these in the framework of non-isolated (perturbations of transversely Morse type) singularities.
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References
Baouendi, M.S., Ebenfelt, P., Rothschild, L.P.: Real Submanifolds in Complex Space and Their Mappings. Princeton Mathematical Series, vol. 47. Princeton University Press, Princeton (1999)
Camacho, C.: On the local structure of conformal maps and holomorphic vector fields in \({{\mathbb{C}}}^2\), Jounées Singuliéres de Dijon (Univ. Dijon, Dijon: 3, 83–94). Astérisque 59–60, 1978 (1978)
Cerveau, D., Lins Neto, A.: Local Levi-flat hypersurfaces invariants by a codimension one holomorphic foliation. Am. J. Math. 133(3), 677–716 (2011)
Cerveau, D., Loray, F.: Un théorème de Frobenius singulier via l’arithmétique élémentaire. J. Number Theory 68(2), 217–228 (1998)
Cerveau, D., Scardua, B.: Integrable deformations of foliations: a generalization of Ilyashenko’s result. Moscow Mathematical Jornal, Volume 21, Issue 2, April-June 2021, pp.271–286.
Camacho, C., Lins Neto, A.: Geometric Theory of Foliations, Translated from the Portuguese by Sue E. Goodman. Birkhauser Boston Inc, Boston (1985)
Leau, L.: Étude sur les equations fonctionnelles à une ou à plusieurs variables. Ann. Fac. Sci. Toulouse Math. Sér. I(11), 1–110 (1897)
Mattei, J.F., Moussu, R.: Holonomie et intégrales premières. Ann. Sci. École Norm. Sup. (4) 13, 469–523 (1980)
Moussu, R.: Une démonstration géométrique d’un théorème de Lyapunov–Poincaré. Astérisque 98–99, 216–223 (1982)
Reeb, G.: Sur Certaines Propriétés Topologiques des Variétés Feuilletées. Actualités Sci. Ind, Hermann (1952)
Lyapunov, A.: Etude d’un cas particulier du problème de la stabilité du mouvement. Mat. Sbornik (Russe) 17, 252–333 (1893)
Poincaré, H.: Mémoire sur les courbes définies par une équation différentielle (I). J. Math. Pures Appl 3 Sér. 7, 375–422 (1881)
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The authors are very much indebted to the referee, for the several constructive comments, careful reading, valuable suggestions and various hints, that have greatly improved this article.
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León, V., Scárdua, B. On a Theorem of Lyapunov–Poincaré in Higher Dimensions. Arnold Math J. 7, 561–571 (2021). https://doi.org/10.1007/s40598-021-00183-x
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DOI: https://doi.org/10.1007/s40598-021-00183-x