Abstract
LetT n be then-torus. We show that strengthened versions of theC r-closing lemma (r≥1) take place for several classes of dynamical systems on tori; namely, 1) for Herman actions of the groupZ k onT 1; 2) for foliations without compact leaves onT 3; 3) for diffeomorphisms ofT 1 with wandering chain recurrent points; 4) for flows onT 2 with wandering chain recurrent trajectories and without fixed points. We also prove a version of theC r-closing lemma for generalized interval exchange transformations onT 1 under the assumption that a nontrivially recurrent point has symbolic expansions sufficiently large, and as a corollary we get a version of theC r Closing lemma under similar assumption in terms of symbolic coding forC r vector fields with finitely many singularities of saddle type on an orientable surface of genus≥2.
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Aranson, S., Malkin, M., Medvedev, V. et al. Versions of the closing lemma for certain dynamical systems on tori. Qual. Th. Dyn. Syst. 4, 1–16 (2003). https://doi.org/10.1007/BF02972818
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DOI: https://doi.org/10.1007/BF02972818