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## Lyapunov exponents of discrete quasi-periodic gevrey schrödinger equations

 College of Sciences, Hohai University, No.1 Xikang Road, Nanjing, Jiangsu, 210098, China

* Corresponding author: Kai Tao

Received  October 2019 Revised  March 2020 Published  July 2020

Fund Project: The second author was supported by the Fundamental Research Funds for the Central Universities(Grant B200202004) and China Postdoctoral Science Foundation (Grant 2019M650094)

In the study of the continuity of the Lyapunov exponent for the discrete quasi-periodic Schrödinger operators, there is a pioneering result by Wang-You [21] that the authors constructed examples whose Lyapunov exponent is discontinuous in the potential with the $C^0$ norm for non-analytic potentials. In this paper, we consider this operators for some Gevrey potential, which is an analytic function having a Gevrey small perturbation, with Diophantine frequency. We prove that in the large coupling regions, the Lyapunov exponent is positive and jointly continuous in all parameters, such as the energy, the frequency and the potential. Note that all analytic functions are also Gevrey ones. Therefore, we also obtain that all of the large analytic potentials are the non-perturbative weak Hölder continuous points of the Lyapunov exponent in the Gevrey topology with $C^0$ norm. It is the first result about the continuity in non-analytic potential with this norm and is complementary to Wang-You's result.

Citation: Wenmeng Geng, Kai Tao. Lyapunov exponents of discrete quasi-periodic gevrey schrödinger equations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020216
##### References:
 [1] A. Avila, S. Jitomirskaya and C. A. Marx, Spectral theory of extended Harper's model and a question by Erdös and Szekeres, Inv. Math., 210 (2017), 283-339.  doi: 10.1007/s00222-017-0729-1.  Google Scholar [2] J. Bochi, Genericity of zero Lyapunov exponents, Ergod. Theory Dyn. Syst., 22 (2002), 1667-1696.  doi: 10.1017/S0143385702001165.  Google Scholar [3] J. Bochi and M. Viana, The Lyapunov exponents of generic volume perserving and symplectic maps, Ann. of Math., 161 (2005), 1423-1485.  doi: 10.4007/annals.2005.161.1423.  Google Scholar [4] J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications, Annals of Mathematics Studies, 158. Princeton University Press, Princeton, NJ, 2005. doi: 10.1515/9781400837144.  Google Scholar [5] J. Bourgain, Positivity and continuity of the Lyapunov exponent for shifts on $\mathbb{T}^d$ with arbitrary frequency vector and real analytic potential, J. Anal. Math., 96 (2005), 313-355.  doi: 10.1007/BF02787834.  Google Scholar [6] J. Bourgain and M. Goldstein, On nonperturbative localization with quasi-periodic potential, Ann. of Math., 152 (2000), 835-879.  doi: 10.2307/2661356.  Google Scholar [7] J. Bourgain and S. Jitomirskaya, Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, J. Stat. Phys., 108 (2002), 1028-1218.  doi: 10.1023/A:1019751801035.  Google Scholar [8] A. Furman, On the multiplicative ergodic theorem for uniquely ergodic systems, Ann. Inst. H. Poincaré Probab. Statist., 33 (1997), 797-815.  doi: 10.1016/S0246-0203(97)80113-6.  Google Scholar [9] L. Ge and Y. Wang,, work in progress. Google Scholar [10] M. Geng and K. Tao, Large deviation theorems for Dirichlet determinants of analytic quasi-periodic Jacobi operators with Brjuno-Rüssmann frequency, preprint, arXiv: 1906.11136. Google Scholar [11] M. Goldstein and W. Schlag, Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions, Ann. of Math., 154 (2001), 155-203.  doi: 10.2307/3062114.  Google Scholar [12] M. Goldstein and W. Schlag, Fine properties of the integrated density of states and a quantitative separation property of the Dirichlet eigenvalues, Geom. Funct. Analysis, 18 (2008), 755-869.  doi: 10.1007/s00039-008-0670-y.  Google Scholar [13] R. Han and S. Zhang, Optimal large deviation estimates and Hölder Regularity of the Lyapunov exponents for quasi-periodic Schrödinger cocycles, preprint, arXiv: 1803.02035v1. Google Scholar [14] S. Jitomirskaya, D. A. Koslover and M. S. Schulteis, Continuity of the Lyapunov exponent for analytic quasiperiodic cocycles, Ergod. Theory Dyn. Syst., 29 (2009), 1881-1905.  doi: 10.1017/S0143385709000704.  Google Scholar [15] S. Jitomirskaya and C. A. Marx, Continuity of the Lyapunov Exponent for analytic quasi-perodic cocycles with singularities, Journal of Fixed Point Theory and Applications, 10 (2011), 129-146.  doi: 10.1007/s11784-011-0055-y.  Google Scholar [16] S. Klein, Anderson localization for the discrete one-dimensional quasi-periodic Schrödinger operator with potential defined by a Gevrey-class function, J. Funct. Anal., 218 (2005), 255-292.  doi: 10.1016/j.jfa.2004.04.009.  Google Scholar [17] J. Liang, Y. Wang and J. You, Hölder continuity of Lyapunov exponent for a family of smooth Schrödinger cocycles, preprint, arXiv: 1806.03284. Google Scholar [18] S. Łojasiewicz, Sur le problème de la division, Studia Math., 18 (1959), 87-136.  doi: 10.4064/sm-18-1-87-136.  Google Scholar [19] K. Tao, Continuity of Lyapunov exponent for analytic quasi-periodic cocycles on higher-dimensional torus, Front. Math. China, 7 (2012), 521-542.  doi: 10.1007/s11464-012-0201-x.  Google Scholar [20] K. Tao, Strong Birkhoff Ergodic Theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles, preprint, arXiv: 1805.00431. Google Scholar [21] Y. Wang and J. You, Examples of discontinuity of Lyapunov exponent in smooth quasi-periodic cocycles, Duke Math., 162 (2013), 2363-2412.  doi: 10.1215/00127094-2371528.  Google Scholar [22] Y. Wang and J. You, The set of smooth quasi-periodic Schrödinger cocycles with positive Lyapunov exponent is not open, Commun. Math. Phys., 362 (2018), 801-826.  doi: 10.1007/s00220-018-3223-8.  Google Scholar [23] Y. Wang and Z. Zhang, Uniform positivity and continuity of Lyapunov exponents for a class of $C^2$ quasiperiodic Schrödinger cocycles, J. Funct. Anal., 268 (2015), 2525-2585.  doi: 10.1016/j.jfa.2015.01.003.  Google Scholar [24] J. Xu, L. Ge and Y. Wang, work in progess. Google Scholar [25] L.-S. Young, Lyapunov exponents for some quasi-periodic cocycles, Ergod. Theory Dyn. Syst., 17 (1997), 483-504.  doi: 10.1017/S0143385797079170.  Google Scholar [26] J. You and S. Zhang, Hölder continuity of the Lyapunov exponent for analytic quasiperiodic Schrödinger cocycles with week Liouville frequency, Ergod. Theory Dyn. Syst., 34 (2014), 1395-1408.  doi: 10.1017/etds.2013.4.  Google Scholar

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##### References:
 [1] A. Avila, S. Jitomirskaya and C. A. Marx, Spectral theory of extended Harper's model and a question by Erdös and Szekeres, Inv. Math., 210 (2017), 283-339.  doi: 10.1007/s00222-017-0729-1.  Google Scholar [2] J. Bochi, Genericity of zero Lyapunov exponents, Ergod. Theory Dyn. Syst., 22 (2002), 1667-1696.  doi: 10.1017/S0143385702001165.  Google Scholar [3] J. Bochi and M. Viana, The Lyapunov exponents of generic volume perserving and symplectic maps, Ann. of Math., 161 (2005), 1423-1485.  doi: 10.4007/annals.2005.161.1423.  Google Scholar [4] J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications, Annals of Mathematics Studies, 158. Princeton University Press, Princeton, NJ, 2005. doi: 10.1515/9781400837144.  Google Scholar [5] J. Bourgain, Positivity and continuity of the Lyapunov exponent for shifts on $\mathbb{T}^d$ with arbitrary frequency vector and real analytic potential, J. Anal. Math., 96 (2005), 313-355.  doi: 10.1007/BF02787834.  Google Scholar [6] J. Bourgain and M. Goldstein, On nonperturbative localization with quasi-periodic potential, Ann. of Math., 152 (2000), 835-879.  doi: 10.2307/2661356.  Google Scholar [7] J. Bourgain and S. Jitomirskaya, Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, J. Stat. Phys., 108 (2002), 1028-1218.  doi: 10.1023/A:1019751801035.  Google Scholar [8] A. Furman, On the multiplicative ergodic theorem for uniquely ergodic systems, Ann. Inst. H. Poincaré Probab. Statist., 33 (1997), 797-815.  doi: 10.1016/S0246-0203(97)80113-6.  Google Scholar [9] L. Ge and Y. Wang,, work in progress. Google Scholar [10] M. Geng and K. Tao, Large deviation theorems for Dirichlet determinants of analytic quasi-periodic Jacobi operators with Brjuno-Rüssmann frequency, preprint, arXiv: 1906.11136. Google Scholar [11] M. Goldstein and W. Schlag, Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions, Ann. of Math., 154 (2001), 155-203.  doi: 10.2307/3062114.  Google Scholar [12] M. Goldstein and W. Schlag, Fine properties of the integrated density of states and a quantitative separation property of the Dirichlet eigenvalues, Geom. Funct. Analysis, 18 (2008), 755-869.  doi: 10.1007/s00039-008-0670-y.  Google Scholar [13] R. Han and S. Zhang, Optimal large deviation estimates and Hölder Regularity of the Lyapunov exponents for quasi-periodic Schrödinger cocycles, preprint, arXiv: 1803.02035v1. Google Scholar [14] S. Jitomirskaya, D. A. Koslover and M. S. Schulteis, Continuity of the Lyapunov exponent for analytic quasiperiodic cocycles, Ergod. Theory Dyn. Syst., 29 (2009), 1881-1905.  doi: 10.1017/S0143385709000704.  Google Scholar [15] S. Jitomirskaya and C. A. Marx, Continuity of the Lyapunov Exponent for analytic quasi-perodic cocycles with singularities, Journal of Fixed Point Theory and Applications, 10 (2011), 129-146.  doi: 10.1007/s11784-011-0055-y.  Google Scholar [16] S. Klein, Anderson localization for the discrete one-dimensional quasi-periodic Schrödinger operator with potential defined by a Gevrey-class function, J. Funct. Anal., 218 (2005), 255-292.  doi: 10.1016/j.jfa.2004.04.009.  Google Scholar [17] J. Liang, Y. Wang and J. You, Hölder continuity of Lyapunov exponent for a family of smooth Schrödinger cocycles, preprint, arXiv: 1806.03284. Google Scholar [18] S. Łojasiewicz, Sur le problème de la division, Studia Math., 18 (1959), 87-136.  doi: 10.4064/sm-18-1-87-136.  Google Scholar [19] K. Tao, Continuity of Lyapunov exponent for analytic quasi-periodic cocycles on higher-dimensional torus, Front. Math. China, 7 (2012), 521-542.  doi: 10.1007/s11464-012-0201-x.  Google Scholar [20] K. Tao, Strong Birkhoff Ergodic Theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles, preprint, arXiv: 1805.00431. Google Scholar [21] Y. Wang and J. You, Examples of discontinuity of Lyapunov exponent in smooth quasi-periodic cocycles, Duke Math., 162 (2013), 2363-2412.  doi: 10.1215/00127094-2371528.  Google Scholar [22] Y. Wang and J. You, The set of smooth quasi-periodic Schrödinger cocycles with positive Lyapunov exponent is not open, Commun. Math. Phys., 362 (2018), 801-826.  doi: 10.1007/s00220-018-3223-8.  Google Scholar [23] Y. Wang and Z. Zhang, Uniform positivity and continuity of Lyapunov exponents for a class of $C^2$ quasiperiodic Schrödinger cocycles, J. Funct. Anal., 268 (2015), 2525-2585.  doi: 10.1016/j.jfa.2015.01.003.  Google Scholar [24] J. Xu, L. Ge and Y. Wang, work in progess. Google Scholar [25] L.-S. Young, Lyapunov exponents for some quasi-periodic cocycles, Ergod. Theory Dyn. Syst., 17 (1997), 483-504.  doi: 10.1017/S0143385797079170.  Google Scholar [26] J. You and S. Zhang, Hölder continuity of the Lyapunov exponent for analytic quasiperiodic Schrödinger cocycles with week Liouville frequency, Ergod. Theory Dyn. Syst., 34 (2014), 1395-1408.  doi: 10.1017/etds.2013.4.  Google Scholar
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