Given a compact m-dimensional manifold M and , consider the space of self mappings of M. We prove here that for every map f in a residual subset of , the closing lemma holds. In particular, it follows that the set of periodic points is dense in the nonwandering set of a generic map. The proof is based on a geometric result asserting that for generic maps the future orbit of every point in M visits the critical set at most m times.
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}, 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://archive.numdam.org/articles/10.1016/j.anihpc.2010.09.003/} }
TY - JOUR AU - Rovella, Alvaro AU - Sambarino, Martín TI - The $ {C}^{1}$ closing lemma for generic $ {C}^{1}$ endomorphisms JO - Annales de l'I.H.P. Analyse non linéaire PY - 2010 SP - 1461 EP - 1469 VL - 27 IS - 6 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.anihpc.2010.09.003/ DO - 10.1016/j.anihpc.2010.09.003 LA - en ID - AIHPC_2010__27_6_1461_0 ER -
%0 Journal Article %A Rovella, Alvaro %A Sambarino, Martín %T The $ {C}^{1}$ closing lemma for generic $ {C}^{1}$ endomorphisms %J Annales de l'I.H.P. Analyse non linéaire %D 2010 %P 1461-1469 %V 27 %N 6 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.anihpc.2010.09.003/ %R 10.1016/j.anihpc.2010.09.003 %G en %F 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://archive.numdam.org/articles/10.1016/j.anihpc.2010.09.003/
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