Abstract
Let S be an orientable compact connected surface without boundary and let X be a C 2 vector field such that its flow is area preserving on S. Assume that X has finitely many singular points, satisfies a Lojasiewicz condition at all its singular points and has singular points if S is the 2-dimensional torus. Then every singular point is either a center or it has a neighbourhood which splits into a finite union of an even number of hyperbolic sectors and all orbits of X are closed except finitely many of them which are either singular points or separatrices joining singular points. The same result is true for measure preserving flows on S. In this case it is not necessary that S be orientable.
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© 1999 Springer-Verlag
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Cima, A., Jibin, L., Llibre, J. (1999). Area preserving flows on compact connected surfaces. In: Leach, P.G.L., Bouquet, S.E., Rouet, JL., Fijalkow, E. (eds) Dynamical Systems, Plasmas and Gravitation. Lecture Notes in Physics, vol 518. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0105936
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DOI: https://doi.org/10.1007/BFb0105936
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