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 $1⩽r⩽\infty$, consider the space ${C}^{r}\left(M\right)$ of self mappings of M. We prove here that for every map f in a residual subset of ${C}^{1}\left(M\right)$, 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 : https://doi.org/10.1016/j.anihpc.2010.09.003
Classification : 37Cxx,  58K05
Mots clés : 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},
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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://archive.numdam.org/articles/10.1016/j.anihpc.2010.09.003/

[1] M. Golubitsky, V. Guillemin, Stable Mappings and Their Singularities, Grad. Texts in Math. vol. 14, Springer, New York (1973) | MR 341518 | Zbl 0294.58004

[2] D. Kahn, Introduction to Global Analysis, Academic Press (1980) | MR 578917 | Zbl 0443.58001

[3] M. Hirsch, Differential Topology, Grad. Texts in Math. vol. 33, Springer (1991) | MR 1336822

[4] C. Pugh, The closing lemma, Amer. J. Math. 89 (1967), 956-1009 | MR 226669 | Zbl 0167.21803

[5] C. Pugh, C. Robinson, The closing lemma, including Hamiltonians, Ergodic Theory Dynam. Systems 3 no. 2 (1983), 261-313 | MR 742228 | Zbl 0548.58012

[6] M. Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math. 91 (1969), 175-199 | MR 240824 | Zbl 0201.56305

[7] L. Wen, The ${C}^{1}$ closing lemma for non-singular endomorphisms, Ergodic Theory Dynam. Systems 11 (1991), 393-412 | MR 1116648 | Zbl 0712.58037

[8] L. Wen, The ${C}^{1}$ closing lemma for endomorphisms with finitely many singularities, Proc. Amer. Math. Soc. 114 (1992), 217-223 | MR 1087474 | Zbl 0746.58017

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