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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 |
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 [
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. |
[2] |
J. Bochi,
Genericity of zero Lyapunov exponents, Ergod. Theory Dyn. Syst., 22 (2002), 1667-1696.
doi: 10.1017/S0143385702001165. |
[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. |
[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. |
[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. |
[6] |
J. Bourgain and M. Goldstein,
On nonperturbative localization with quasi-periodic potential, Ann. of Math., 152 (2000), 835-879.
doi: 10.2307/2661356. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
show all references
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. |
[2] |
J. Bochi,
Genericity of zero Lyapunov exponents, Ergod. Theory Dyn. Syst., 22 (2002), 1667-1696.
doi: 10.1017/S0143385702001165. |
[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. |
[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. |
[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. |
[6] |
J. Bourgain and M. Goldstein,
On nonperturbative localization with quasi-periodic potential, Ann. of Math., 152 (2000), 835-879.
doi: 10.2307/2661356. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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