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Large coupling asymptotics for the entropy of quasi-periodic operators

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Abstract

In this paper, we give an asymptotic estimate for the entropy, i.e., the sum of all positive Lyapunov exponents, of the quasi-periodic finite-range operator with a large trigonometric polynomial potential and Diophantine frequency.

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References

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

    Article  MathSciNet  Google Scholar 

  2. Avila A, Jitomirskaya S. Almost localization and almost reducibility. J Eur Math Soc (JEMS), 2010, 12: 93–131

    Article  MathSciNet  Google Scholar 

  3. Avila A, Jitomirskaya S, Sadel C. Complex one-frequency cocycles. J Eur Math Soc (JEMS), 2014, 16: 1915–1935

    Article  MathSciNet  Google Scholar 

  4. Avila A, You J, Zhou Q. Sharp phase transitions for the almost Mathieu operator. Duke Math J, 2017, 166: 2697–2718

    Article  MathSciNet  Google Scholar 

  5. Avron J, Craig W, Simon B. Large coupling behaviour of the Lyapunov exponent for tight binding one-dimensional random systems. J Phys A, 1983, 16: 209–211

    Article  MathSciNet  Google Scholar 

  6. Bourgain J. Green’s Function Estimates for Lattice Schrödinger Operators and Applications. Princeton: Princeton University Press, 2005

    Book  Google Scholar 

  7. Bourgain J. Positivity and continuity of the Lyapounov exponent for shifts on \({T^d}\) with arbitrary frequency vector and real analytic potential. J Anal Math, 2005, 96: 313–355

    Article  MathSciNet  Google Scholar 

  8. Bourgain J, Jitomirskaya S. Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential. J Stat Phys, 2002, 108: 1203–1218

    Article  MathSciNet  Google Scholar 

  9. Cai A, Chavaudret C, You J, et al. Sharp Hölder continuity of the Lyapunov exponent of finitely differentiable quasi-periodic cocycles. Math Z, 2019, 291: 931–958

    Article  MathSciNet  Google Scholar 

  10. Chavaudret C. Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles. Bull Soc Math France, 2011, 141: 47–106

    Article  MathSciNet  Google Scholar 

  11. Duarte P, Klein S. Positive Lyapunov exponents for higher dimensional quasiperiodic cocycles. Comm Math Phys, 2014, 332: 189–219

    Article  MathSciNet  Google Scholar 

  12. Duarte P, Klein S. Continuity, positivity, and simplicity of the Lyapunov exponents for quasi-periodic cocycles. J Eur Math Soc (JEMS), 2019, 21: 2051–2106

    Article  MathSciNet  Google Scholar 

  13. Eliasson L H. Almost reducibility of linear quasi-periodic systems. Proc Sympos Pure Math, 2001, 69: 679–705

    Article  MathSciNet  Google Scholar 

  14. Ge L, You J. Arithmetic version of Anderson localization via reducibility. ArXiv:2003.13946v1, 2020

  15. Ge L, You J, Zhou Q. Exponential dynamical localization: Criterion and applications. ArXiv:1901.04258, 2019

  16. Han R, Marx C. Large coupling asymptotics for the Lyapunov exponent of quasi-periodic Schrödinger operators with analytic potentials. Ann Henri Poincaré, 2018, 19: 249–265

    Article  MathSciNet  Google Scholar 

  17. Haro A, Puig J. A Thouless formula and Aubry duality for long-range Schrödinger skew-products. Nonlinearity, 2013, 26: 1163–1187

    Article  MathSciNet  Google Scholar 

  18. Herman M. Une methode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractere local d’un théorème d’Arnold et de Moser sur le tore de dimension 2. Comment Math Helv, 1983, 58: 453–502

    Article  MathSciNet  Google Scholar 

  19. Hou X, You J. Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems. Invent Math, 2012, 190: 209–260

    Article  MathSciNet  Google Scholar 

  20. Leguil M, You J, Zhao Z, et al. Asymptotics ofspectral gaps of quasi-periodic Schröodinger operators. ArXiv:1712.04700, 2017

  21. Puig J. A nonperturbative Eliasson’s reducibility theorem. Nonlinearity, 2006, 19: 355–376

    Article  MathSciNet  Google Scholar 

  22. Song Y. On the variation of the spectrum of a normal matrix. Linear Algebra Appl, 1996, 246: 215–223

    Article  MathSciNet  Google Scholar 

  23. Sorets E, Spencer T. Positive Lyapunov exponents for Schröodinger operators with quasi-periodic potentials. Comm Math Phys, 1991, 142: 543–566

    Article  MathSciNet  Google Scholar 

  24. Zhang Z. Positive Lyapunov exponents for quasiperiodic Szego cocycles. Nonlinearity, 2012, 25: 1771–1797

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The second author was supported by National Natural Science Foundation of China (Grant No. 11871286).

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Authors and Affiliations

  1. Department of Mathematics, University of California at Irvine, Irvine, CA, 92612, USA

    Lingrui Ge

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

    Jiangong You

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  1. Lingrui Ge
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  2. Jiangong You
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Correspondence to Jiangong You.

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Ge, L., You, J. Large coupling asymptotics for the entropy of quasi-periodic operators. Sci. China Math. 63, 1745–1756 (2020). https://doi.org/10.1007/s11425-019-1662-8

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  • DOI: https://doi.org/10.1007/s11425-019-1662-8

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