Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

Simultaneous linearization for commuting quasiperiodically forced circle diffeomorphisms


Authors: Jing Wang and Qi Zhou
Journal: Proc. Amer. Math. Soc. 141 (2013), 625-636
MSC (2010): Primary 37C15; Secondary 37C05
DOI: https://doi.org/10.1090/S0002-9939-2012-11357-7
Published electronically: June 28, 2012
MathSciNet review: 2996967
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For commuting smooth quasiperiodically forced circle diffeomorphisms, we show that if the base frequencies and the fibred rotation numbers jointly satisfy some simultaneous Diophantine condition and if the diffeomorphisms are in some $ C^{\infty }$ neighborhood of the corresponding rotations, then they are simultaneously $ C^{\infty }$-linearizable.


References [Enhancements On Off] (What's this?)

  • 1.
    A. Avila and S. Jitomirskaya,
    The Ten Martini Problem,
    Ann. of Math. (2) 170 (2009) 303-342. MR 2521117 (2011a:47081)
  • 2.
    A. Avila and R. Krikorian,
    Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles,
    Ann. of Math. (2) 164 (2006) 911-940. MR 2259248 (2008h:81044)
  • 3.
    D. Damjanovic, B. Fayad and R. Krikorian,
    Rigidity results for commuting $ SL(2,\mathbb{R})$ cocycles above circular rotations,
    Preprint.
  • 4.
    M. Ding, C. Grebogi, and E. Ott,
    Evolution of attractors in quasiperiodically forced systems: From quasiperiodic to strange nonchaotic to chaotic,
    Physical Review A 39(5) (1989) 2593-2598.
  • 5.
    B. Fayad and R. Krikorian,
    Rigidity results for quasiperiodic $ SL(2,\mathbb{R})$-cocycles,
    Journal of Modern Dynamics 3(4) (2009) 497-510. MR 2587083 (2011f:37046)
  • 6.
    B. Fayad and K. Khanin,
    Smooth linearization of commuting circle diffeomorphisms,
    Ann. of Math. (2) 170 (2009) 961-980. MR 2552115 (2011c:37088)
  • 7.
    A. Haro and J. Puig,
    Strange nonchaotic attractors in Harper maps,
    Chaos 16 (2006), no. 3. MR 2266838 (2007g:37025)
  • 8.
    M. R. Herman,
    Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations,
    Inst. Hautes Etudes Sci. Publ. Math. 49 (1979) 5-233. MR 538680 (81h:58039)
  • 9.
    M. R. Herman,
    Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d'un théorème d'Arnol'd et de Moser sur le tore de dimension $ 2$,
    Comment. Math. Helv. 58 (1983) 453-502. MR 727713 (85g:58057)
  • 10.
    L. Hernández Encinas and J. Muñoz Masqué,
    A short proof of the generalized Faà di Bruno's formula,
    Applied Mathematics Letters 16 (2003) 975-979. MR 2005276 (2004k:26018)
  • 11.
    T. Jäger,
    Linearisation of conservative toral homeomorphisms,
    Invent. Math. 176(3) (2009) 601-616. MR 2501297 (2011e:37095)
  • 12.
    T. Jäger and J. Stark,
    Towards a classification for quasiperiodically forced circle homeomorphisms,
    Journal of the LMS 73(3) (2006) 727-744. MR 2241977 (2008f:37087)
  • 13.
    R. Krikorian,
    Réductibilité des Systèmes Produits-Croisés à Valeurs dans des Groupes Compacts,
    Astérisque 259 (1999). MR 1732061 (2001f:37030)
  • 14.
    J. Moser,
    On commuting circle mappings and simultaneous Diophantine approximations,
    Math. Z. 205 (1990) 105-121. MR 1069487 (92e:58120)
  • 15.
    J. C. Yoccoz,
    Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne,
    Ann. Sci. Ecole Norm. Sup. 17 (1984) 333-359. MR 777374 (86j:58086)
  • 16.
    J. C. Yoccoz,
    Analytic linearization of circle diffeomorphisms,
    in Dynamical Systems and Small Divisors (Cetraro, 1998), Lecture Notes in Math. 1784, Springer-Verlag, New York, (2002) 125-173. MR 1924912 (2004c:37073)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37C15, 37C05

Retrieve articles in all journals with MSC (2010): 37C15, 37C05


Additional Information

Jing Wang
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
Email: jingwang018@gmail.com

Qi Zhou
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
Email: qizhou628@gmail.com

DOI: https://doi.org/10.1090/S0002-9939-2012-11357-7
Received by editor(s): November 30, 2010
Received by editor(s) in revised form: July 10, 2011
Published electronically: June 28, 2012
Additional Notes: This work was supported by NNSF of China (Grant 10531050), NNSF of China (Grant 11031003), and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions
Communicated by: Yingfei Yi
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society