Local density of diffeomorphisms with large centralizers
[Densité locale des difféomorphismes ayant un gros centralisateur]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 6, pp. 925-954.

Pour toute variété M compacte, de dimension quelconque, nous construisons une partie 𝒪⊂ Diff 1 (M) non vide, ouverte dans l’espace Diff 1 (M) des C 1 -difféomorphismes de M, et un sous-ensemble 𝒟⊂𝒪 dense en 𝒪, constitué de difféomorphismes dont le centralisateur est non dénombrable, donc non trivial.

Given any compact manifold M, we construct a non-empty open subset 𝒪 of the space Diff 1 (M) of C 1 -diffeomorphisms and a dense subset 𝒟⊂𝒪 such that the centralizer of every diffeomorphism in 𝒟 is uncountable, hence non-trivial.

DOI : 10.24033/asens.2085
Classification : 37C85, 37C80, 37D45, 37D15, 37E30
Keywords: trivial centralizer, trivial symmetries, Mather invariant
Mot clés : centralisateur trivial, symétries triviales, invariant de Mather 
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     title = {Local density of diffeomorphisms with large centralizers},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
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     publisher = {Soci\'et\'e math\'ematique de France},
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Bonatti, Christian; Crovisier, Sylvain; Vago, Gioia M.; Wilkinson, Amie. Local density of diffeomorphisms with large centralizers. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 6, pp. 925-954. doi : 10.24033/asens.2085. http://archive.numdam.org/articles/10.24033/asens.2085/

[1] V. S. Afraimovich & T. Young, Mather invariants and smooth conjugacy on S 2 , J. Dynam. & Control Systems 6 (2000), 341-352. | MR | Zbl

[2] M.-C. Arnaud, C. Bonatti & S. Crovisier, Dynamiques symplectiques génériques, Ergodic Theory & Dynam. Systems 25 (2005), 1401-1436. | MR | Zbl

[3] A. Banyaga, On the structure of the group of equivariant diffeomorphisms, Topology 16 (1977), 279-283. | MR | Zbl

[4] C. Bonatti & S. Crovisier, Récurrence et généricité, Invent. Math. 158 (2004), 33-104. | MR | Zbl

[5] C. Bonatti, S. Crovisier & A. Wilkinson, The centralizer of a C 1 generic diffeomorphism is trivial, Electron. Res. Announc. Math. Sci. 15 (2008), 33-43. | MR | Zbl

[6] C. Bonatti, S. Crovisier & A. Wilkinson, C 1 -generic conservative diffeomorphisms have trivial centralizer, J. Mod. Dyn. 2 (2008), 359-373. | MR | Zbl

[7] C. Bonatti, S. Crovisier & A. Wilkinson, The C 1 generic diffeomorphism has trivial centralizer, preprint arXiv:0804.1416, to appear in Publ. Math. I.H.É.S. | Numdam | Zbl

[8] C. Bonatti & L. Díaz, On maximal transitive sets of generic diffeomorphisms, Publ. Math. Inst. Hautes Études Sci. 96 (2002), 171-197. | Numdam | MR | Zbl

[9] C. Bonatti, L. Díaz & E. R. Pujals, A C 1 -generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math. 158 (2003), 355-418. | MR | Zbl

[10] C. Bonatti, L. Díaz & M. Viana, Dynamics beyond uniform hyperbolicity, Encyclopaedia of Math. Sci. 102, Springer, 2005. | MR | Zbl

[11] B. Farb & J. Franks, Groups of homeomorphisms of one-manifolds. III. Nilpotent subgroups, Ergodic Theory & Dynam. Systems 23 (2003), 1467-1484. | MR | Zbl

[12] N. Kopell, Commuting diffeomorphisms, in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., 1970, 165-184. | MR | Zbl

[13] J. N. Mather, Commutators of diffeomorphisms, Comment. Math. Helv. 49 (1974), 512-528. | MR | Zbl

[14] J. Palis, Vector fields generate few diffeomorphisms, Bull. Amer. Math. Soc. 80 (1974), 503-505. | MR | Zbl

[15] S. Smale, Dynamics retrospective: great problems, attempts that failed, Phys. D 51 (1991), 267-273. | MR | Zbl

[16] S. Smale, Mathematical problems for the next century, Math. Intelligencer 20 (1998), 7-15. | MR | Zbl

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