The ${C}^{1}$ closing lemma for generic ${C}^{1}$ endomorphisms
Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 6, p. 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
Keywords: Closing lemma, Critical points, Transversality
@article{AIHPC_2010__27_6_1461_0,
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},
publisher = {Elsevier},
volume = {27},
number = {6},
year = {2010},
pages = {1461-1469},
doi = {10.1016/j.anihpc.2010.09.003},
zbl = {1214.37009},
mrnumber = {2738328},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2010__27_6_1461_0}
}

Rovella, Alvaro; Sambarino, Martín. The ${C}^{1}$ closing lemma for generic ${C}^{1}$ endomorphisms. Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 6, pp. 1461-1469. doi : 10.1016/j.anihpc.2010.09.003. http://www.numdam.org/item/AIHPC_2010__27_6_1461_0/

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