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Linearization of Quasiperiodically Forced Circle Flows Beyond Brjuno Condition

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Abstract

We prove that an analytic quasiperiodically forced circle flow with a not super-Liouvillean base frequency and which is close enough to some constant rotation is C ∞ rotations reducible, provided its fibered rotation number is Diophantine with respect to the base frequency. As a corollary, we obtain that among such systems, the linearizable ones and those displaying mode-locking are locally dense for the C ∞-topology.

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References

  1. Arnold V.I.: Small denominators I. On the mapping of a circle into itself. Akad.Nauk.Math. 25, 21–86 (1961)

    Google Scholar 

  2. Avila A.: Global theory of one-frequency Schrödinger operators. Acta Math. 215, 1–54 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Avila, A.: Almost reducibility and absolute continuity. http://w3.impa.br/~avila/

  4. Avila, A.: KAM, Lyapunov exponents and the spectral dichotomy for one-frequency schrödinger operators. In preparation

  5. Avila A., Fayad B., Krikorian R.: A KAM scheme for \({{SL}(2,{\mathbb{R}})}\) cocycles with Liouvillean frequencies. Geom. Funct. Anal. 21, 1001–1019 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Avila A., Jitomirskaya S.: The ten martini problem. Ann. Math. 170, 303–342 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Avila A., Krikorian R.: Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles. Ann. Math. 164, 911–940 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Avila, A., Krikorian, R.: On almost reducibility of circle diffeomorphisms. In preparation

  9. Béllissard J., Simon B.: Cantor spectrum for the almost Mathieu equation. J. Funct. Anal. 48(3), 408–419 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bjerklöv K., Jäger T.: Rotation numbers for quasiperiodically forced circle maps—mode-locking vs strict monotonicity. J. Am. Math. Soc. 22, 353–362 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dinaburg E., Sinai Y.: The one-dimensional Schrödinger equation with a quasi-periodic potential. Funct. Anal. Appl. 9, 279–289 (1975)

    Article  MATH  Google Scholar 

  12. Ding M., Grebogi C., Ott E.: Evolution of attractors in quasiperiodically forced systems: from quasiperiodic to strange nonchaotic to chaotic. Phys. Rev. A 39(5), 2593–2598 (1989)

    Article  ADS  Google Scholar 

  13. Eliasson H.: Floquet solutions for the one-dimensional quasiperiodic Schrödinger equation. Commun. Math. Phys. 146, 447–482 (1992)

    Article  ADS  MATH  Google Scholar 

  14. Fayad B., Krikorian R.: Rigidity results for quasiperiodic \({{SL}(2,{\mathbb{R}})}\) -cocycles. J. Mod. Dyn. 3(4), 479–510 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Herman M.: Sur la conjugaison diffŕentiable des difféomorphismes du cercle à des rotations. Publ.Math. Inst. Hautes Etudes Sci. 49, 5–233 (1979)

    Article  MATH  Google Scholar 

  16. Herman M.: Une méthode pour minorer les exposants de Lyapunov et quelques exemples montrant le caractère local d’un théorème d’Arnold et de Moser sur le tore de dimension 2. Comment. Math. Helv. 58, 453–502 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hou X., You J.: Almost reducibility and non-perturbative reducibility of quasiperiodic linear systems. Invent. Math. 190, 209–260 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Jäger T., Stark J.: Towards a classification for quasiperiodically forced circle homeomorphisms. J. LMS 73(3), 727–744 (2006)

    MathSciNet  MATH  Google Scholar 

  19. Krikorian, R.: Reducibility, differentiable rigidity and Lyapunov exponents for quasiperiodic cocycles on \({{\mathbb{T}} \times SL(2,{\mathbb{R}})}\) . arXiv:math/0402333

  20. Puig J.: Cantor spectrum for the almost Mathieu operator. Commun. Math. Phys. 244, 297–309 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Wang J.: Lower dimensional invariant tori for quasiperiodically forced circle diffeomorphisms. J. Differ. Equ. 253, 1489–1543 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Wang J., Jäger T.: Abundance of mode-locking for quasiperiodically forced circle maps. Commun. Math. Phys. 353(1), 1–36 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Wang, J., Zhou, Q., Jäger, T.: Genericity of mode-locking for quasiperiodically forced circle maps. arXiv:1607.01700

  24. Yoccoz J.C.: Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation verifie une condition diophantienne. Ann. Sci. Ecole Norm. Sup. 17, 333–359 (1984)

    Article  MATH  Google Scholar 

  25. Yoccoz, J.C.: Analytic linearization of circle diffeomorphisms, In: Dynamical Systems and Small divisors(Cetraro, 1998), Lecture Notes in Math, vol. 1784, pp. 125–173. Springer, New York (2002)

  26. You J., Zhou Q.: Embedding of analytic quasi-periodic cocycles into analytic quasi-periodic linear systems and its applications. Commun. Math. Phys. 323, 975–1005 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Zhou Q., Wang J.: Reducibility results for quasiperiodic cocycles with Liouvillean frequency. J. Dyn. Differ. Equ. 24, 61–83 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, CNRS UMR 8088, Université de Cergy-Pontoise, 2, Av. Adolphe Chauvin, 95302, Cergy-Pontoise, France

    Raphaël Krikorian

  2. Department of Mathematics, Nanjing University of Science and Technology, Nanjing, 210094, China

    Jing Wang

  3. Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, 300071, China

    Jiangong You

  4. Department of Mathematics, Nanjing University, Nanjing, 210093, China

    Qi Zhou

Authors
  1. Raphaël Krikorian
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  2. Jing Wang
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  3. Jiangong You
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  4. Qi Zhou
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Correspondence to Raphaël Krikorian.

Additional information

Communicated by J. Marklof

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Krikorian, R., Wang, J., You, J. et al. Linearization of Quasiperiodically Forced Circle Flows Beyond Brjuno Condition. Commun. Math. Phys. 358, 81–100 (2018). https://doi.org/10.1007/s00220-017-3021-8

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  • DOI: https://doi.org/10.1007/s00220-017-3021-8

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