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.

Given a compact m-dimensional manifold M and 1r, 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
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},
     zbl = {1214.37009},
     mrnumber = {2738328},
     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
DA  - 2010///
SP  - 1461
EP  - 1469
VL  - 27
IS  - 6
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2010.09.003/
UR  - https://zbmath.org/?q=an%3A1214.37009
UR  - https://www.ams.org/mathscinet-getitem?mr=2738328
UR  - https://doi.org/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 https://doi.org/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/

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

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

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

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

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

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

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

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

Cited by Sources: