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A BDF2-approach for the non-linear Fokker-Planck equation
May
2019, 39(5): 2915-2931.
doi: 10.3934/dcds.2019121
A new proof of continuity of Lyapunov exponents for a class of $ C^2 $ quasiperiodic Schrödinger cocycles without LDT
Department of Mathematics, Nanjing University, Nanjing 210093, China |
In this paper, we reconsider the continuity of the Lyapunov exponents for a class of Schrödinger cocycles with a $ C^2 $ cos-type potential and a Diophantine frequency. We propose a new method to prove the Lyapunov exponent is continuous in energies. In particular, a large deviation theorem is not needed in the proof.
Keywords:
Schrödinger cocycle,
Lyapunov exponent,
C2 cos-type potential,
Diophantine frequency,
large deviation theorem.
Mathematics Subject Classification:
Primary: 37A30.
Citation:
Linlin Fu, Jiahao Xu. A new proof of continuity of Lyapunov exponents for a class of $ C^2 $ quasiperiodic Schrödinger cocycles without LDT. Discrete and Continuous Dynamical Systems,
2019, 39
(5)
: 2915-2931.
doi: 10.3934/dcds.2019121
![]()
References:
| [1] |
A. Avila,
Density of positive Lyapunov exponents for SL(2, $\mathbb{R}$)-cocycles, J. Amer. Math. Soc., 24 (2011), 999-1014.
doi: 10.1090/S0894-0347-2011-00702-9. |
| [2] |
A. Avila, D. Damanik and Z. Zhang,
Singular density of states measure for subshift and quasi-periodic Schrödinger operators, Comm. Math. Phys., 330 (2014), 469-498.
doi: 10.1007/s00220-014-1968-2. |
| [3] |
A. Avila and S. Jitomirskaya,
The ten Martini problem, Ann. of Math, 170 (2009), 303-342.
doi: 10.4007/annals.2009.170.303. |
| [4] |
K. Bjerklöv,
The dynamics of a class of quasi-periodic Schrödinger cocycles, Annales Henri Poincaré, 16 (2015), 961-1031.
doi: 10.1007/s00023-014-0330-8. |
| [5] |
J. Bochi, Discontinuity of the Lyapunov exponent for non-hyperbolic cocycles, 1999, unpublished. |
| [6] |
J. Bochi,
Genericity of zero Lyapunov exponents, Ergodic Theory Dynam. Systems, 22 (2002), 1667-1696.
doi: 10.1017/S0143385702001165. |
| [7] |
J. Bourgain and M. Goldstein,
On nonperturbative localization with quasi-periodic potential, Ann. of Math, 152 (2000), 835-879.
doi: 10.2307/2661356. |
| [8] |
J. Bourgain and S. Jitomirskaya,
Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, J. Stat. Phys., 108 (2002), 1203-1218.
doi: 10.1023/A:1019751801035. |
| [9] |
J. Fröhlich, T. Spencer and P. Wittwer,
Localization for a class of one-dimensional quasi-periodic Schrödinger operators, Comm. Math. Phys., 132 (1990), 5-25.
doi: 10.1007/BF02277997. |
| [10] |
A. Furman,
On the multiplicative ergodic theorem for the uniquely ergodic systems, Ann. Inst. Henri Poincaré, 33 (1997), 797-815.
doi: 10.1016/S0246-0203(97)80113-6. |
| [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] |
R. Johnson,
Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients, J. Differential Equations, 61 (1986), 54-78.
doi: 10.1016/0022-0396(86)90125-7. |
| [13] |
O. Knill, The upper Lyapunov exponent of $SL(2, \mathbb{R})$ cocycles: Discontinuity and the problem of positivity, in Lyapunov Exponents, Springer Berlin Heidelberg, 1486 (1991), 86–97. |
| [14] |
J. Liang and P. J. Kung,
Uniform positivity of Lyapunov exponent for a class of smooth Schrödinger cocycles with weak Liouville frequencies, Frontiers of Mathematics in China, 12 (2017), 607-639.
doi: 10.1007/s11464-017-0619-2. |
| [15] |
J. Liang, Y. Wang and J. You, Hölder continuity of Lyapunov exponent for a family of smooth Schrödinger cocyles, preprint, arXiv: 1806.03284. |
| [16] |
R. Mañé, Oseledec's theorem from the generic viewpoint, in Proc. of Int. Congress of Math, (1984), 1269–1276. |
| [17] |
R. Mañé, The Lyapunov exponents of engeric area preserving diffeomorphisms, in Pitman Res. Notes Math. Ser., 362 (1996), 110-119. |
| [18] |
Ya. G. Sinai,
Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential, J. Stat. Phys., 46 (1987), 861-909.
doi: 10.1007/BF01011146. |
| [19] |
J. Thouvenot,
An example of discontinuity in the computation of the Lyapunov exponents, Proc. Steklov Inst. Math, 216 (1997), 366-369.
|
| [20] |
Y. Wang and J. You,
Examples of discontinuity of Lyapunov exponent in smooth quasi-periodic cocycles, Duke Math. J., 162 (2013), 2363-2412.
doi: 10.1215/00127094-2371528. |
| [21] |
Y. Wang and Z. Zhang,
Uniform positivity and continuity of Lyapunov exponents for a class of C2 quasi-periodic Schrödinger cocycles, J. Funct. Anal., 268 (2015), 2525-2585.
doi: 10.1016/j.jfa.2015.01.003. |
| [22] |
Y. Wang and Z. Zhang,
Cantor spectrum for a class of C2 quasiperiodic Schrödinger operators, Int. Math. Res. Not., 8 (2017), 2300-2336.
doi: 10.1093/imrn/rnw079. |
| [23] |
L. Young,
Lyapunov exponents for some quasi-periodic cocycles, Ergodic Theory Dynam. Systems, 17 (1997), 483-504.
doi: 10.1017/S0143385797079170. |
| [24] |
Z. Zhang,
Positive Lyapunov exponents for quasiperiodic Szegő cocycles, Nonlinearity, 25 (2012), 1771-1797.
doi: 10.1088/0951-7715/25/6/1771. |
| [25] |
Z. Zhang, Resolvent set of Schrödinger operators and uniform hyperbolicity, Mathematics, 2013. |
show all references
References:
| [1] |
A. Avila,
Density of positive Lyapunov exponents for SL(2, $\mathbb{R}$)-cocycles, J. Amer. Math. Soc., 24 (2011), 999-1014.
doi: 10.1090/S0894-0347-2011-00702-9. |
| [2] |
A. Avila, D. Damanik and Z. Zhang,
Singular density of states measure for subshift and quasi-periodic Schrödinger operators, Comm. Math. Phys., 330 (2014), 469-498.
doi: 10.1007/s00220-014-1968-2. |
| [3] |
A. Avila and S. Jitomirskaya,
The ten Martini problem, Ann. of Math, 170 (2009), 303-342.
doi: 10.4007/annals.2009.170.303. |
| [4] |
K. Bjerklöv,
The dynamics of a class of quasi-periodic Schrödinger cocycles, Annales Henri Poincaré, 16 (2015), 961-1031.
doi: 10.1007/s00023-014-0330-8. |
| [5] |
J. Bochi, Discontinuity of the Lyapunov exponent for non-hyperbolic cocycles, 1999, unpublished. |
| [6] |
J. Bochi,
Genericity of zero Lyapunov exponents, Ergodic Theory Dynam. Systems, 22 (2002), 1667-1696.
doi: 10.1017/S0143385702001165. |
| [7] |
J. Bourgain and M. Goldstein,
On nonperturbative localization with quasi-periodic potential, Ann. of Math, 152 (2000), 835-879.
doi: 10.2307/2661356. |
| [8] |
J. Bourgain and S. Jitomirskaya,
Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, J. Stat. Phys., 108 (2002), 1203-1218.
doi: 10.1023/A:1019751801035. |
| [9] |
J. Fröhlich, T. Spencer and P. Wittwer,
Localization for a class of one-dimensional quasi-periodic Schrödinger operators, Comm. Math. Phys., 132 (1990), 5-25.
doi: 10.1007/BF02277997. |
| [10] |
A. Furman,
On the multiplicative ergodic theorem for the uniquely ergodic systems, Ann. Inst. Henri Poincaré, 33 (1997), 797-815.
doi: 10.1016/S0246-0203(97)80113-6. |
| [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] |
R. Johnson,
Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients, J. Differential Equations, 61 (1986), 54-78.
doi: 10.1016/0022-0396(86)90125-7. |
| [13] |
O. Knill, The upper Lyapunov exponent of $SL(2, \mathbb{R})$ cocycles: Discontinuity and the problem of positivity, in Lyapunov Exponents, Springer Berlin Heidelberg, 1486 (1991), 86–97. |
| [14] |
J. Liang and P. J. Kung,
Uniform positivity of Lyapunov exponent for a class of smooth Schrödinger cocycles with weak Liouville frequencies, Frontiers of Mathematics in China, 12 (2017), 607-639.
doi: 10.1007/s11464-017-0619-2. |
| [15] |
J. Liang, Y. Wang and J. You, Hölder continuity of Lyapunov exponent for a family of smooth Schrödinger cocyles, preprint, arXiv: 1806.03284. |
| [16] |
R. Mañé, Oseledec's theorem from the generic viewpoint, in Proc. of Int. Congress of Math, (1984), 1269–1276. |
| [17] |
R. Mañé, The Lyapunov exponents of engeric area preserving diffeomorphisms, in Pitman Res. Notes Math. Ser., 362 (1996), 110-119. |
| [18] |
Ya. G. Sinai,
Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential, J. Stat. Phys., 46 (1987), 861-909.
doi: 10.1007/BF01011146. |
| [19] |
J. Thouvenot,
An example of discontinuity in the computation of the Lyapunov exponents, Proc. Steklov Inst. Math, 216 (1997), 366-369.
|
| [20] |
Y. Wang and J. You,
Examples of discontinuity of Lyapunov exponent in smooth quasi-periodic cocycles, Duke Math. J., 162 (2013), 2363-2412.
doi: 10.1215/00127094-2371528. |
| [21] |
Y. Wang and Z. Zhang,
Uniform positivity and continuity of Lyapunov exponents for a class of C2 quasi-periodic Schrödinger cocycles, J. Funct. Anal., 268 (2015), 2525-2585.
doi: 10.1016/j.jfa.2015.01.003. |
| [22] |
Y. Wang and Z. Zhang,
Cantor spectrum for a class of C2 quasiperiodic Schrödinger operators, Int. Math. Res. Not., 8 (2017), 2300-2336.
doi: 10.1093/imrn/rnw079. |
| [23] |
L. Young,
Lyapunov exponents for some quasi-periodic cocycles, Ergodic Theory Dynam. Systems, 17 (1997), 483-504.
doi: 10.1017/S0143385797079170. |
| [24] |
Z. Zhang,
Positive Lyapunov exponents for quasiperiodic Szegő cocycles, Nonlinearity, 25 (2012), 1771-1797.
doi: 10.1088/0951-7715/25/6/1771. |
| [25] |
Z. Zhang, Resolvent set of Schrödinger operators and uniform hyperbolicity, Mathematics, 2013. |
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