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.

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 | 1 citation dans Numdam

DOI : 10.1016/j.anihpc.2010.09.003

Classification : 37Cxx, 58K05
Mots-clés : Closing lemma, Critical points, Transversality

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     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},
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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/

Bibliographie
Cité par

[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

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