American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Existence of periodically invariant tori on resonant surfaces for twist mappings
  • DCDS Home
  • This Issue
  • Next Article
    Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps
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 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Graphs of the angle functions
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Bifurcation of Type Ⅲ functions with $ f'_1(c_1)f'_2(c_2)<0 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Wenmeng Geng, Kai Tao. Lyapunov exponents of discrete quasi-periodic gevrey Schrödinger equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2977-2996. doi: 10.3934/dcdsb.2020216
[2]

Pedro Duarte, Silvius Klein, Manuel Santos. A random cocycle with non Hölder Lyapunov exponent. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4841-4861. doi: 10.3934/dcds.2019197
[3]

Jean Bourgain. On quasi-periodic lattice Schrödinger operators. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 75-88. doi: 10.3934/dcds.2004.10.75
[4]

Lei Jiao, Yiqian Wang. The construction of quasi-periodic solutions of quasi-periodic forced Schrödinger equation. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1585-1606. doi: 10.3934/cpaa.2009.8.1585
[5]

Meina Gao, Jianjun Liu. Quasi-periodic solutions for derivative nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2101-2123. doi: 10.3934/dcds.2012.32.2101
[6]

Yingte Sun. Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4531-4543. doi: 10.3934/dcds.2021047
[7]

Sijia Zhang, Shengfan Zhou. Random uniform exponential attractors for Schrödinger lattice systems with quasi-periodic forces and multiplicative white noise. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022056
[8]

William A. Veech. The Forni Cocycle. Journal of Modern Dynamics, 2008, 2 (3) : 375-395. doi: 10.3934/jmd.2008.2.375
[9]

Moulay-Tahar Benameur, Alan L. Carey. On the analyticity of the bivariant JLO cocycle. Electronic Research Announcements, 2009, 16: 37-43. doi: 10.3934/era.2009.16.37
[10]

Claudia Valls. On the quasi-periodic solutions of generalized Kaup systems. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 467-482. doi: 10.3934/dcds.2015.35.467
[11]

Peng Huang, Xiong Li, Bin Liu. Invariant curves of smooth quasi-periodic mappings. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 131-154. doi: 10.3934/dcds.2018006
[12]

Danijela Damjanović, Anatole Katok. Periodic cycle functions and cocycle rigidity for certain partially hyperbolic $\mathbb R^k$ actions. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 985-1005. doi: 10.3934/dcds.2005.13.985
[13]

Qihuai Liu, Dingbian Qian, Zhiguo Wang. Quasi-periodic solutions of the Lotka-Volterra competition systems with quasi-periodic perturbations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1537-1550. doi: 10.3934/dcdsb.2012.17.1537
[14]

Yanling Shi, Junxiang Xu, Xindong Xu. Quasi-periodic solutions of generalized Boussinesq equation with quasi-periodic forcing. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2501-2519. doi: 10.3934/dcdsb.2017104
[15]

Jordi-Lluís Figueras, Thomas Ohlson Timoudas. Sharp $ \frac12 $-Hölder continuity of the Lyapunov exponent at the bottom of the spectrum for a class of Schrödinger cocycles. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4519-4531. doi: 10.3934/dcds.2020189
[16]

Alessandro Fonda, Antonio J. Ureña. Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 169-192. doi: 10.3934/dcds.2011.29.169
[17]

Xavier Blanc, Claude Le Bris. Improving on computation of homogenized coefficients in the periodic and quasi-periodic settings. Networks and Heterogeneous Media, 2010, 5 (1) : 1-29. doi: 10.3934/nhm.2010.5.1
[18]

Danijela Damjanović, James Tanis. Cocycle rigidity and splitting for some discrete parabolic actions. Discrete and Continuous Dynamical Systems, 2014, 34 (12) : 5211-5227. doi: 10.3934/dcds.2014.34.5211
[19]

James Tanis, Zhenqi Jenny Wang. Cohomological equation and cocycle rigidity of discrete parabolic actions. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3969-4000. doi: 10.3934/dcds.2019160
[20]

Hongyong Cui, Mirelson M. Freitas, José A. Langa. On random cocycle attractors with autonomous attraction universes. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3379-3407. doi: 10.3934/dcdsb.2017142

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (325)
  • HTML views (128)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy