• 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

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.

Citation: Jinhao Liang. Positive Lyapunov exponent for a class of quasi-periodic cocycles. Discrete & Continuous Dynamical Systems - A, 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

J. FröhlichT. 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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

J. FröhlichT. 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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

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

Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240

[2]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[3]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[4]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121

[5]

Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020294

[6]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020298

[7]

Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316

[8]

Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, , () : -. doi: 10.3934/era.2021002

[9]

Lingyu Li, Jianfu Yang, Jinge Yang. Solutions to Chern-Simons-Schrödinger systems with external potential. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021008

[10]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[11]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[12]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[13]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[14]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456

[15]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[16]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[17]

Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260

[18]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259

[19]

Jose Anderson Cardoso, Patricio Cerda, Denilson Pereira, Pedro Ubilla. Schrödinger Equations with vanishing potentials involving Brezis-Kamin type problems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020392

[20]

Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020284

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (185)
  • HTML views (125)
  • Cited by (0)

Other articles
by authors

[Back to Top]