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Poincaré-Bendixson theory for leaves of codimension one


Authors: John Cantwell and Lawrence Conlon
Journal: Trans. Amer. Math. Soc. 265 (1981), 181-209
MSC: Primary 57R30; Secondary 58F18
DOI: https://doi.org/10.1090/S0002-9947-1981-0607116-8
MathSciNet review: 607116
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Abstract: The level of a local minimal set of a $ {C^2}$ codimension-one foliation of a compact manifold is a nonnegative integer defined inductively, level zero corresponding to the minimal sets in the usual sense. Each leaf of a local minimal set at level $ k$ is at level $ k$. The authors develop a theory of local minimal sets, level, and how leaves at level $ k$ asymptotically approach leaves at lower level. This last generalizes the classical Poincaré-Bendixson theorem and provides information relating growth, topological type, and level, e.g. if $ L$ is a totally proper leaf at level $ k$ then $ L$ has exactly polynomial growth of degree $ k$ and topological type $ k - 1$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0607116-8
Keywords: Poincaré-Bendixson theorem, local minimal set, level, infinite level, totally proper leaf, limit set, growth, end, pseudogroup, biregular cover
Article copyright: © Copyright 1981 American Mathematical Society

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