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On $C^r-$closing for flows on 2-manifolds

Carlos Gutierrez
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Abstract

For some full measure subset B of the set of iet's (i.e. interval exchange transformations) the following is satisfied: Let X be a $C^r$, $1\le r\le \infty$, vector field, with finitely many singularities, on a compact orientable surface M. Given a nontrivial recurrent point $p\in M$ of X, the holonomy map around p is semi-conjugate to an iet $E :[0,1) \to [0,1).$ If $E\in B$ then there exists a $C^r$ vector field Y, arbitrarily close to X, in the $C^r-$topology, such that Y has a closed trajectory passing through p.

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Carlos Gutierrez. 1999-11-23. On $C^r-$closing for flows on 2-manifolds. https://doi.org/10.1088/0951-7715%2F13%2F6%2F301

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