The C 1 generic diffeomorphism has trivial centralizer
Publications Mathématiques de l'IHÉS, Tome 109 (2009), pp. 185-244

Answering a question of Smale, we prove that the space of C 1 diffeomorphisms of a compact manifold contains a residual subset of diffeomorphisms whose centralizers are trivial.

@article{PMIHES_2009__109__185_0,
     author = {Bonatti, Christian and Crovisier, Sylvain and Wilkinson, Amie},
     title = {The {C} 1 generic diffeomorphism has trivial centralizer},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {185--244},
     year = {2009},
     publisher = {Springer-Verlag},
     address = {Berlin/Heidelberg},
     volume = {109},
     doi = {10.1007/s10240-009-0021-z},
     mrnumber = {2511588},
     zbl = {1177.37025},
     language = {en},
     url = {https://www.numdam.org/articles/10.1007/s10240-009-0021-z/}
}
TY  - JOUR
AU  - Bonatti, Christian
AU  - Crovisier, Sylvain
AU  - Wilkinson, Amie
TI  - The C 1 generic diffeomorphism has trivial centralizer
JO  - Publications Mathématiques de l'IHÉS
PY  - 2009
SP  - 185
EP  - 244
VL  - 109
PB  - Springer-Verlag
PP  - Berlin/Heidelberg
UR  - https://www.numdam.org/articles/10.1007/s10240-009-0021-z/
DO  - 10.1007/s10240-009-0021-z
LA  - en
ID  - PMIHES_2009__109__185_0
ER  - 
%0 Journal Article
%A Bonatti, Christian
%A Crovisier, Sylvain
%A Wilkinson, Amie
%T The C 1 generic diffeomorphism has trivial centralizer
%J Publications Mathématiques de l'IHÉS
%D 2009
%P 185-244
%V 109
%I Springer-Verlag
%C Berlin/Heidelberg
%U https://www.numdam.org/articles/10.1007/s10240-009-0021-z/
%R 10.1007/s10240-009-0021-z
%G en
%F PMIHES_2009__109__185_0
Bonatti, Christian; Crovisier, Sylvain; Wilkinson, Amie. The C 1 generic diffeomorphism has trivial centralizer. Publications Mathématiques de l'IHÉS, Tome 109 (2009), pp. 185-244. doi: 10.1007/s10240-009-0021-z

[BC] Ch. Bonatti, S. Crovisier, Récurrence et généricité, Invent. Math. 158 (2004), p. 33-104 | Zbl | MR

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

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

[BCW1] Ch. Bonatti, S. Crovisier, A. Wilkinson, C 1-generic conservative diffeomorphisms have trivial centralizer, J. Mod. Dyn. 2 (2008), p. 359-373 | Zbl | MR

[BCW2] Ch. Bonatti, S. Crovisier, and A. Wilkinson, The centralizer of a C 1 generic diffeomorphism is trivial. Preprint arXiv:0705.0225 , 2007. | Zbl | MR

[BCVW] Ch. Bonatti, S. Crovisier, G. Vago, and A. Wilkinson, Local density of diffeomorphisms with large centralizers. Preprint arXiv:0709.4319 , 2007. | Zbl | MR | Numdam

[Bu1] L. Burslem, Centralizers of partially hyperbolic diffeomorphisms, Ergod. Theory Dyn. Sys. 24 (2004), p. 55-87 | Zbl | MR

[Bu2] L. Burslem, Centralizers of area preserving diffeomorphisms on S 2 , Proc. Am. Math. Soc. 133 (2005), p. 1101-1108 | Zbl | MR

[Fi] T. Fisher, Trivial centralizers for Axiom A diffeomorphisms. Preprint. | Zbl | MR

[FRW] M. Foreman, D. Rudolph, L. Weiss, On the conjugacy relation in ergodic theory, C. R. Math. Acad. Sci. Paris 343 (2006), p. 653-656 | Zbl | MR

[G] É. Ghys, Groups acting on the circle, L'Enseign. Math. 47 (2001), p. 329-407 | Zbl | MR

[Ko] N. Kopell, Commuting diffeomorphisms, in: Global Analysis, Proc. Sympos. Pure Math. XIV (1970), AMS, Providence | Zbl | MR

[N] A. Navas, Three remarks on one dimensional bi-Lipschitz conjugacies. Preprint arXiv:0705.0034 , 2007.

[PY1] J. Palis, J.-C. Yoccoz, Rigidity of centralizers of diffeomorphisms, Ann. Sci. École Norm. Sup. 22 (1989), p. 81-98 | Zbl | MR | Numdam

[PY2] J. Palis, J.-C. Yoccoz, Centralizers of Anosov diffeomorphisms on tori, Ann. Sci. École Norm. Sup. 22 (1989), p. 99-108 | Zbl | MR | Numdam

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

[R] M. Rees, A minimal positive entropy homeomorphism of the 2-torus, J. Lond. Math. Soc. 23 (1981), p. 537-550 | Zbl | MR

[Sm1] S. Smale, Dynamics retrospective: great problems, attempts that failed. Nonlinear science: the next decade, Los Alamos, NM, 1990, Physica D 51 (1991), p. 267-273 | Zbl | MR

[Sm2] S. Smale, Mathematical problems for the next century, Math. Intell. 20 (1998), p. 7-15 | Zbl | MR

[To1] Y. Togawa, Generic Morse-Smale diffeomorphisms have only trivial symmetries, Proc. Am. Math. Soc. 65 (1977), p. 145-149 | Zbl | MR

[To2] Y. Togawa, Centralizers of C 1-diffeomorphisms, Proc. Am. Math. Soc. 71 (1978), p. 289-293 | Zbl | MR

Cité par Sources :