The C 1 closing lemma for generic C 1 endomorphisms
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 6, pp. 1461-1469.

Given a compact m-dimensional manifold M and 1⩽r⩽∞, consider the space C r (M) of self mappings of M. We prove here that for every map f in a residual subset of C 1 (M), the C 1 closing lemma holds. In particular, it follows that the set of periodic points is dense in the nonwandering set of a generic C 1 map. The proof is based on a geometric result asserting that for generic C r maps the future orbit of every point in M visits the critical set at most m times.

DOI : 10.1016/j.anihpc.2010.09.003
Classification : 37Cxx, 58K05
Mots-clés : Closing lemma, Critical points, Transversality 
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     title = {The $ {C}^{1}$ closing lemma for generic $ {C}^{1}$ endomorphisms},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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Rovella, Alvaro; Sambarino, Martín. The $ {C}^{1}$ closing lemma for generic $ {C}^{1}$ endomorphisms. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 6, pp. 1461-1469. doi : 10.1016/j.anihpc.2010.09.003. http://www.numdam.org/articles/10.1016/j.anihpc.2010.09.003/

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