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Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps
Positive Lyapunov exponent for a class of quasi-periodic cocycles
Department of Mathematics, Southeast University, Nanjing 211189, China |
Young [
References:
[1] |
A. Avila,
Global theory of one-frequency Schrödinger operators, Acta Math., 215 (2015), 1-54.
doi: 10.1007/s11511-015-0128-7. |
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M. Benedicks and L. Carleson,
The dynamics of the Hénon map, Ann. of Math. (2), 133 (1991), 73-169.
doi: 10.2307/2944326. |
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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. |



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