The $ {C}^{1}$ closing lemma for generic $ {C}^{1}$ endomorphisms

Rovella, Alvaro; Sambarino, Martín

doi:10.1016/j.anihpc.2010.09.003

The

C^{1}

closing lemma for generic

C^{1}

endomorphisms

Rovella, Alvaro ; Sambarino, Martín

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 6, pp. 1461-1469

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Résumé

Given a compact m-dimensional manifold M and $1 ⩽ r ⩽ \infty$ , 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.

MR Zbl

DOI : 10.1016/j.anihpc.2010.09.003

Classification : 37Cxx, 58K05
Keywords: Closing lemma, Critical points, Transversality

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     author = {Rovella, Alvaro and Sambarino, Mart{\'\i}n},
     title = {The $ {C}^{1}$ closing lemma for generic $ {C}^{1}$ endomorphisms},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1461--1469},
     publisher = {Elsevier},
     volume = {27},
     number = {6},
     year = {2010},
     doi = {10.1016/j.anihpc.2010.09.003},
     mrnumber = {2738328},
     zbl = {1214.37009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2010.09.003/}
}

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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

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