The C 1 generic diffeomorphism has trivial centralizer

Christian Bonatti; Sylvain Crovisier; Amie Wilkinson

Publications Mathématiques de l'IHÉS (2009)

  • Volume: 109, page 185-244
  • ISSN: 0073-8301

Abstract

top
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.

How to cite

top

Bonatti, Christian, Crovisier, Sylvain, and Wilkinson, Amie. "The C 1 generic diffeomorphism has trivial centralizer." Publications Mathématiques de l'IHÉS 109 (2009): 185-244. <http://eudml.org/doc/273598>.

@article{Bonatti2009,
abstract = {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.},
author = {Bonatti, Christian, Crovisier, Sylvain, Wilkinson, Amie},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {diffeomorphism; centralizer; manifold},
language = {eng},
pages = {185-244},
publisher = {Springer-Verlag},
title = {The C 1 generic diffeomorphism has trivial centralizer},
url = {http://eudml.org/doc/273598},
volume = {109},
year = {2009},
}

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
PB - Springer-Verlag
VL - 109
SP - 185
EP - 244
AB - 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.
LA - eng
KW - diffeomorphism; centralizer; manifold
UR - http://eudml.org/doc/273598
ER -

References

top
  1. [BC] Ch. Bonatti, S. Crovisier, Récurrence et généricité, Invent. Math.158 (2004), p. 33-104 Zbl1071.37015MR2090361
  2. [BD] Ch. Bonatti, L. Díaz, On maximal transitive sets of generic diffeomorphisms, Publ. Math. Inst. Hautes Études Sci.96 (2002), p. 171-197 Zbl1032.37011MR1985032
  3. [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 Zbl1049.37011MR2018925
  4. [BCW1] Ch. Bonatti, S. Crovisier, A. Wilkinson, C 1-generic conservative diffeomorphisms have trivial centralizer, J. Mod. Dyn.2 (2008), p. 359-373 Zbl1149.37017MR2383272
  5. [BCW2] Ch. Bonatti, S. Crovisier, and A. Wilkinson, The centralizer of a C 1 generic diffeomorphism is trivial. Preprint arXiv:0705.0225 , 2007. Zbl1149.37012MR2407535
  6. [BCVW] Ch. Bonatti, S. Crovisier, G. Vago, and A. Wilkinson, Local density of diffeomorphisms with large centralizers. Preprint arXiv:0709.4319 , 2007. Zbl1163.58003MR2504109
  7. [Bu1] L. Burslem, Centralizers of partially hyperbolic diffeomorphisms, Ergod. Theory Dyn. Sys.24 (2004), p. 55-87 Zbl1115.37021MR2041261
  8. [Bu2] L. Burslem, Centralizers of area preserving diffeomorphisms on S 2 , Proc. Am. Math. Soc.133 (2005), p. 1101-1108 Zbl1058.37029MR2117211
  9. [Fi] T. Fisher, Trivial centralizers for Axiom A diffeomorphisms. Preprint. Zbl1173.37013MR2448228
  10. [FRW] M. Foreman, D. Rudolph, L. Weiss, On the conjugacy relation in ergodic theory, C. R. Math. Acad. Sci. Paris343 (2006), p. 653-656 Zbl1115.28002MR2271741
  11. [G] É. Ghys, Groups acting on the circle, L’Enseign. Math.47 (2001), p. 329-407 Zbl1044.37033MR1876932
  12. [Ko] N. Kopell, Commuting diffeomorphisms, in: Global Analysis, Proc. Sympos. Pure Math. XIV (1970), AMS, Providence Zbl0225.57020MR270396
  13. [N] A. Navas, Three remarks on one dimensional bi-Lipschitz conjugacies. Preprint arXiv:0705.0034 , 2007. 
  14. [PY1] J. Palis, J.-C. Yoccoz, Rigidity of centralizers of diffeomorphisms, Ann. Sci. École Norm. Sup.22 (1989), p. 81-98 Zbl0709.58022MR985855
  15. [PY2] J. Palis, J.-C. Yoccoz, Centralizers of Anosov diffeomorphisms on tori, Ann. Sci. École Norm. Sup.22 (1989), p. 99-108 Zbl0675.58029MR985856
  16. [Pu] C. Pugh, The closing lemma, Am. J. Math.89 (1967), p. 956-1009 Zbl0167.21803MR226669
  17. [R] M. Rees, A minimal positive entropy homeomorphism of the 2-torus, J. Lond. Math. Soc.23 (1981), p. 537-550 Zbl0451.58022MR616561
  18. [Sm1] S. Smale, Dynamics retrospective: great problems, attempts that failed. Nonlinear science: the next decade, Los Alamos, NM, 1990, Physica D51 (1991), p. 267-273 Zbl0745.58018MR1128817
  19. [Sm2] S. Smale, Mathematical problems for the next century, Math. Intell.20 (1998), p. 7-15 Zbl0947.01011MR1631413
  20. [To1] Y. Togawa, Generic Morse-Smale diffeomorphisms have only trivial symmetries, Proc. Am. Math. Soc.65 (1977), p. 145-149 Zbl0367.58008MR448449
  21. [To2] Y. Togawa, Centralizers of C 1-diffeomorphisms, Proc. Am. Math. Soc.71 (1978), p. 289-293 Zbl0403.58012MR494312

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.