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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

Linlin Fu , and  Jiahao Xu

Department of Mathematics, Nanjing University, Nanjing 210093, China

* Corresponding author: Linlin Fu

Received  August 2018 Revised  October 2018 Published  January 2019

Fund Project: The first author was supported by NNSF of China (Grants 11771205). The second author was supported by Qing Lan Project.

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

A. Avila and S. Jitomirskaya, The ten Martini problem, Ann. of Math, 170 (2009), 303-342.  doi: 10.4007/annals.2009.170.303.  Google Scholar

[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.  Google Scholar

[5]

J. Bochi, Discontinuity of the Lyapunov exponent for non-hyperbolic cocycles, 1999, unpublished. Google Scholar

[6]

J. Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory Dynam. Systems, 22 (2002), 1667-1696.  doi: 10.1017/S0143385702001165.  Google Scholar

[7]

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

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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]

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.  Google Scholar

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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. Google Scholar

[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.  Google Scholar

[15]

J. Liang, Y. Wang and J. You, Hölder continuity of Lyapunov exponent for a family of smooth Schrödinger cocyles, preprint, arXiv: 1806.03284. Google Scholar

[16]

R. Mañé, Oseledec's theorem from the generic viewpoint, in Proc. of Int. Congress of Math, (1984), 1269–1276.  Google Scholar

[17]

R. Mañé, The Lyapunov exponents of engeric area preserving diffeomorphisms, in Pitman Res. Notes Math. Ser., 362 (1996), 110-119.  Google Scholar

[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.  Google Scholar

[19]

J. Thouvenot, An example of discontinuity in the computation of the Lyapunov exponents, Proc. Steklov Inst. Math, 216 (1997), 366-369.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

L. Young, Lyapunov exponents for some quasi-periodic cocycles, Ergodic Theory Dynam. Systems, 17 (1997), 483-504.  doi: 10.1017/S0143385797079170.  Google Scholar

[24]

Z. Zhang, Positive Lyapunov exponents for quasiperiodic Szegő cocycles, Nonlinearity, 25 (2012), 1771-1797.  doi: 10.1088/0951-7715/25/6/1771.  Google Scholar

[25]

Z. Zhang, Resolvent set of Schrödinger operators and uniform hyperbolicity, Mathematics, 2013. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

A. Avila and S. Jitomirskaya, The ten Martini problem, Ann. of Math, 170 (2009), 303-342.  doi: 10.4007/annals.2009.170.303.  Google Scholar

[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.  Google Scholar

[5]

J. Bochi, Discontinuity of the Lyapunov exponent for non-hyperbolic cocycles, 1999, unpublished. Google Scholar

[6]

J. Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory Dynam. Systems, 22 (2002), 1667-1696.  doi: 10.1017/S0143385702001165.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  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]

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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[15]

J. Liang, Y. Wang and J. You, Hölder continuity of Lyapunov exponent for a family of smooth Schrödinger cocyles, preprint, arXiv: 1806.03284. Google Scholar

[16]

R. Mañé, Oseledec's theorem from the generic viewpoint, in Proc. of Int. Congress of Math, (1984), 1269–1276.  Google Scholar

[17]

R. Mañé, The Lyapunov exponents of engeric area preserving diffeomorphisms, in Pitman Res. Notes Math. Ser., 362 (1996), 110-119.  Google Scholar

[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.  Google Scholar

[19]

J. Thouvenot, An example of discontinuity in the computation of the Lyapunov exponents, Proc. Steklov Inst. Math, 216 (1997), 366-369.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

L. Young, Lyapunov exponents for some quasi-periodic cocycles, Ergodic Theory Dynam. Systems, 17 (1997), 483-504.  doi: 10.1017/S0143385797079170.  Google Scholar

[24]

Z. Zhang, Positive Lyapunov exponents for quasiperiodic Szegő cocycles, Nonlinearity, 25 (2012), 1771-1797.  doi: 10.1088/0951-7715/25/6/1771.  Google Scholar

[25]

Z. Zhang, Resolvent set of Schrödinger operators and uniform hyperbolicity, Mathematics, 2013. Google Scholar

Figure 1.  $ f_{I_n, m}(x) $ is of the first type
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