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March  2020, 40(3): 1361-1387. doi: 10.3934/dcds.2020080

Positive Lyapunov exponent for a class of quasi-periodic cocycles

Jinhao Liang

Department of Mathematics, Southeast University, Nanjing 211189, China

Received  January 2019 Revised  October 2019 Published  December 2019

Young [17] proved the positivity of Lyapunov exponent in a large set of the energies for some quasi-periodic cocycles. Her result is also proved to be applicable for some quasi-periodic Schrödinger cocycles by Zhang [18]. However, her result cannot be applied to the Schrödinger cocycles with the potential $ v = \cos(4\pi x)+w( x) $, where $ w\in C^2(\mathbb R/\mathbb Z,\mathbb R) $ is a small perturbation. In this paper, we will improve her result such that it can be applied to more cocycles.

Keywords: Quasi-periodic cocycle, Schrödinger cocycle, Lyapunov exponent.
Mathematics Subject Classification: Primary: 37A30.
Citation: Jinhao Liang. Positive Lyapunov exponent for a class of quasi-periodic cocycles. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1361-1387. doi: 10.3934/dcds.2020080
References:
[1]

A. Avila, Global theory of one-frequency Schrödinger operators, Acta Math., 215 (2015), 1-54.  doi: 10.1007/s11511-015-0128-7.

[2]

M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. of Math. (2), 133 (1991), 73-169.  doi: 10.2307/2944326.

[3]

K. Bjerklöv, The dynamics of a class of quasi-periodic Schrödinger cocycles, Ann. Henri Poincaré, 16 (2015), 961-1031.  doi: 10.1007/s00023-014-0330-8.

[4]

J. Bourgain, Positivity and continuity of the Lyapounov 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.

[5]

J. Bourgain and M. Goldstein, On nonperturbative localization with quasi-periodic potential, Ann. of Math. (2), 152 (2000), 835-879.  doi: 10.2307/2661356.

[6]

J. Chan, Method of variations of potential of quasi-periodic Schrödinger equations, Geom. Funct. Anal., 17 (2008), 1416-1478.  doi: 10.1007/s00039-007-0633-8.

[7]

L. H. Eliasson, Discrete one-dimensional quasi-periodic Schrödinger operators with pure point spectrum, Acta Math., 179 (1997), 153-196.  doi: 10.1007/BF02392742.

[8]

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.

[9]

M. Herman, Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d'un théorème d'Arnold et de Moser sur le tore de dimension 2, Comment. Math. Helv., 58 (1983), 453-502. 

[10]

K. Ishii, Localization of eigenstates and transport phenomena in one-dimensional disordered systems, Progr. Theoret. Phys. Suppl., 53 (1973), 77-138.  doi: 10.1143/PTPS.53.77.

[11]

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.

[12]

J. Liang and P. Kung, Uniform positivity of Lyapunov exponent for a class of smooth Schrödinger cocycles with weak Liouville frequencies, Front. Math. China, 12 (2017), 607-639.  doi: 10.1007/s11464-017-0619-2.

[13]

L. Pastur, Spectral properties of disordered systems in the one-body approximation, Comm. Math. Phys., 75 (1980), 179-196.  doi: 10.1007/BF01222516.

[14]

Ya. G. Sinai, Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential, J. Statist. Phys., 46 (1987), 861-909.  doi: 10.1007/BF01011146.

[15]

E. Sorets and T. Spencer, Positive Lyapunov exponents for Schrödinger operators with quasi-periodic potentials, Comm. Math. Phys., 142 (1991), 543-566.  doi: 10.1007/BF02099100.

[16]

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.

[17]

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

[18]

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

show all references

References:
[1]

A. Avila, Global theory of one-frequency Schrödinger operators, Acta Math., 215 (2015), 1-54.  doi: 10.1007/s11511-015-0128-7.

[2]

M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. of Math. (2), 133 (1991), 73-169.  doi: 10.2307/2944326.

[3]

K. Bjerklöv, The dynamics of a class of quasi-periodic Schrödinger cocycles, Ann. Henri Poincaré, 16 (2015), 961-1031.  doi: 10.1007/s00023-014-0330-8.

[4]

J. Bourgain, Positivity and continuity of the Lyapounov 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.

[5]

J. Bourgain and M. Goldstein, On nonperturbative localization with quasi-periodic potential, Ann. of Math. (2), 152 (2000), 835-879.  doi: 10.2307/2661356.

[6]

J. Chan, Method of variations of potential of quasi-periodic Schrödinger equations, Geom. Funct. Anal., 17 (2008), 1416-1478.  doi: 10.1007/s00039-007-0633-8.

[7]

L. H. Eliasson, Discrete one-dimensional quasi-periodic Schrödinger operators with pure point spectrum, Acta Math., 179 (1997), 153-196.  doi: 10.1007/BF02392742.

[8]

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.

[9]

M. Herman, Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d'un théorème d'Arnold et de Moser sur le tore de dimension 2, Comment. Math. Helv., 58 (1983), 453-502. 

[10]

K. Ishii, Localization of eigenstates and transport phenomena in one-dimensional disordered systems, Progr. Theoret. Phys. Suppl., 53 (1973), 77-138.  doi: 10.1143/PTPS.53.77.

[11]

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.

[12]

J. Liang and P. Kung, Uniform positivity of Lyapunov exponent for a class of smooth Schrödinger cocycles with weak Liouville frequencies, Front. Math. China, 12 (2017), 607-639.  doi: 10.1007/s11464-017-0619-2.

[13]

L. Pastur, Spectral properties of disordered systems in the one-body approximation, Comm. Math. Phys., 75 (1980), 179-196.  doi: 10.1007/BF01222516.

[14]

Ya. G. Sinai, Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential, J. Statist. Phys., 46 (1987), 861-909.  doi: 10.1007/BF01011146.

[15]

E. Sorets and T. Spencer, Positive Lyapunov exponents for Schrödinger operators with quasi-periodic potentials, Comm. Math. Phys., 142 (1991), 543-566.  doi: 10.1007/BF02099100.

[16]

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.

[17]

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

[18]

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

Figure 1.  graph of the function in $ \mathcal F $
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Figure 2.  Graphs of the angle functions
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Figure 3.  Bifurcation of Type Ⅲ functions with $ f'_1(c_1)f'_2(c_2)<0 $
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