July  2020, 40(7): 4519-4531. doi: 10.3934/dcds.2020189

Sharp $ \frac12 $-Hölder continuity of the Lyapunov exponent at the bottom of the spectrum for a class of Schrödinger cocycles

1. 

Department of Mathematics, Uppsala University, Box 480, 75106 Uppsala, Sweden

2. 

Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden

* Corresponding author: Jordi-Lluís Figueras

Received  October 2019 Published  April 2020

We consider the setting for the disappearance of uniform hyperbolicity as in Bjerklöv and Saprykina (2008 Nonlinearity 21), where it was proved that the minimum distance between invariant stable and unstable bundles has a linear power law dependence on parameters. In this scenario we prove that the Lyapunov exponent is sharp $ \frac12 $-Hölder continuous.

In particular, we show that the Lyapunov exponent of Schrödinger cocycles with a potential having a unique non-degenerate minimum is sharp $ \frac12 $-Hölder continuous below the lowest energy of the spectrum, in the large coupling regime.

Citation: 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
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]

A. Avila and S. Jitomirskaya, Almost localization and almost reducibility, J. Eur. Math. Soc. (JEMS), 12 (2010), 93-131.  doi: 10.4171/JEMS/191.

[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]

K. Bjerklöv and M. Saprykina, Universal asymptotics in hyperbolicity breakdown, Nonlinearity, 21 (2008), 557-586.  doi: 10.1088/0951-7715/21/3/010.

[5]

J. Bourgain, Hölder regularity of integrated density of states for the almost Mathieu operator in a perturbative regime, Lett. Math. Phys., 51 (2000), 83-118.  doi: 10.1023/A:1007641323456.

[6]

J. Bourgain, Green's function estimates for lattice Schrödinger operators and applications, Annals of Mathematics Studies, Vol. 158, Princeton University Press, Princeton, NJ, 2005. doi: 10.1515/9781400837144.

[7]

J. Bourgain and S. Jitomirskaya, Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, J. Statist. Phys., 108 (2002), 1203-1218.  doi: 10.1023/A:1019751801035.

[8]

R. Calleja and J.-L. Figueras, Collision of invariant bundles of quasi-periodic attractors in the dissipative standard map, Chaos, 22 (2012), 033114, 10 pp. doi: 10.1063/1.4737205.

[9]

J.-L. Figueras and A. Haro, Different scenarios for hyperbolicity breakdown in quasiperiodic area preserving twist maps, Chaos, 25 (2015), 123119, 16 pp. doi: 10.1063/1.4938185.

[10]

J.-L. Figueras and A. Haro, A note on the fractalization of saddle invariant curves in quasiperiodic systems, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1095-1107.  doi: 10.3934/dcdss.2016043.

[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.

[12]

M. Goldstein and W. Schlag, Fine properties of the integrated density of states and a quantitative separation property of the Dirichlet eigenvalues, Geom. Funct. Anal., 18 (2008), 755-869.  doi: 10.1007/s00039-008-0670-y.

[13]

S. Hadj Amor, Hölder continuity of the rotation number for quasi-periodic co-cycles in $ {\rm{SL}} (2,\mathbb{R})$, Comm. Math. Phys., 287 (2009), 565-588.  doi: 10.1007/s00220-008-0688-x.

[14]

A. Haro and R. de la Llave, Manifolds on the verge of a hyperbolicity breakdown, Chaos, 16 (2006), 013120, 8 pp. doi: 10.1063/1.2150947.

[15]

M.-R. Herman, Une méthode pour minorer les exposants de lyapounov et quelques exemples montrant le caractére local d'un théorème d'arnold et de moser sur le tore de dimension $2$, Comment. Math. Helv., 58 (1983), 453-502. 

[16]

S. JitomirskayaD. A. Koslover and M. S. Schulteis, Continuity of the Lyapunov exponent for analytic quasiperiodic cocycles, Ergodic Theory Dynam. Systems, 29 (2009), 1881-1905.  doi: 10.1017/S0143385709000704.

[17]

R. A. Johnson, Lyapounov numbers for the almost periodic Schrödinger equation, Illinois J. Math., 28 (1984), 397-419.  doi: 10.1215/ijm/1256046068.

[18]

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.

[19]

O. Knill, The upper Lyapunov exponent of $ \rm{SL} $(2,R) cocycles: Discontinuity and the problem of positivity, in Lyapunov Exponents (Oberwolfach, 1990), Lecture Notes in Math., Vol. 1486, Springer, Berlin, 1991, 86–97. doi: 10.1007/BFb0086660.

[20]

C. A. Marx and S. Jitomirskaya, Dynamics and spectral theory of quasi-periodic Schrödinger-type operators, Ergodic Theory Dynam. Systems, 37 (2017), 2353-2393.  doi: 10.1017/etds.2016.16.

[21]

T. Ohlson Timoudas, Power law asymptotics in the creation of strange attractors in the quasi-periodically forced quadratic family, Nonlinearity, 30 (2017), 4483-4522.  doi: 10.1088/1361-6544/aa8c9e.

[22]

T. Ohlson Timoudas, Asymptotic laws for a class of quasi-periodic Schrödinger cocycles at the lowest energy of the spectrum, preprint, arXiv: 1809.05418, 2018.

[23]

Y. Wang and J. You, Examples of discontinuity of Lyapunov exponent in smooth quasiperiodic cocycles, Duke Math. J., 162 (2013), 2363-2412.  doi: 10.1215/00127094-2371528.

[24]

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.

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]

A. Avila and S. Jitomirskaya, Almost localization and almost reducibility, J. Eur. Math. Soc. (JEMS), 12 (2010), 93-131.  doi: 10.4171/JEMS/191.

[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]

K. Bjerklöv and M. Saprykina, Universal asymptotics in hyperbolicity breakdown, Nonlinearity, 21 (2008), 557-586.  doi: 10.1088/0951-7715/21/3/010.

[5]

J. Bourgain, Hölder regularity of integrated density of states for the almost Mathieu operator in a perturbative regime, Lett. Math. Phys., 51 (2000), 83-118.  doi: 10.1023/A:1007641323456.

[6]

J. Bourgain, Green's function estimates for lattice Schrödinger operators and applications, Annals of Mathematics Studies, Vol. 158, Princeton University Press, Princeton, NJ, 2005. doi: 10.1515/9781400837144.

[7]

J. Bourgain and S. Jitomirskaya, Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, J. Statist. Phys., 108 (2002), 1203-1218.  doi: 10.1023/A:1019751801035.

[8]

R. Calleja and J.-L. Figueras, Collision of invariant bundles of quasi-periodic attractors in the dissipative standard map, Chaos, 22 (2012), 033114, 10 pp. doi: 10.1063/1.4737205.

[9]

J.-L. Figueras and A. Haro, Different scenarios for hyperbolicity breakdown in quasiperiodic area preserving twist maps, Chaos, 25 (2015), 123119, 16 pp. doi: 10.1063/1.4938185.

[10]

J.-L. Figueras and A. Haro, A note on the fractalization of saddle invariant curves in quasiperiodic systems, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1095-1107.  doi: 10.3934/dcdss.2016043.

[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.

[12]

M. Goldstein and W. Schlag, Fine properties of the integrated density of states and a quantitative separation property of the Dirichlet eigenvalues, Geom. Funct. Anal., 18 (2008), 755-869.  doi: 10.1007/s00039-008-0670-y.

[13]

S. Hadj Amor, Hölder continuity of the rotation number for quasi-periodic co-cycles in $ {\rm{SL}} (2,\mathbb{R})$, Comm. Math. Phys., 287 (2009), 565-588.  doi: 10.1007/s00220-008-0688-x.

[14]

A. Haro and R. de la Llave, Manifolds on the verge of a hyperbolicity breakdown, Chaos, 16 (2006), 013120, 8 pp. doi: 10.1063/1.2150947.

[15]

M.-R. Herman, Une méthode pour minorer les exposants de lyapounov et quelques exemples montrant le caractére local d'un théorème d'arnold et de moser sur le tore de dimension $2$, Comment. Math. Helv., 58 (1983), 453-502. 

[16]

S. JitomirskayaD. A. Koslover and M. S. Schulteis, Continuity of the Lyapunov exponent for analytic quasiperiodic cocycles, Ergodic Theory Dynam. Systems, 29 (2009), 1881-1905.  doi: 10.1017/S0143385709000704.

[17]

R. A. Johnson, Lyapounov numbers for the almost periodic Schrödinger equation, Illinois J. Math., 28 (1984), 397-419.  doi: 10.1215/ijm/1256046068.

[18]

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.

[19]

O. Knill, The upper Lyapunov exponent of $ \rm{SL} $(2,R) cocycles: Discontinuity and the problem of positivity, in Lyapunov Exponents (Oberwolfach, 1990), Lecture Notes in Math., Vol. 1486, Springer, Berlin, 1991, 86–97. doi: 10.1007/BFb0086660.

[20]

C. A. Marx and S. Jitomirskaya, Dynamics and spectral theory of quasi-periodic Schrödinger-type operators, Ergodic Theory Dynam. Systems, 37 (2017), 2353-2393.  doi: 10.1017/etds.2016.16.

[21]

T. Ohlson Timoudas, Power law asymptotics in the creation of strange attractors in the quasi-periodically forced quadratic family, Nonlinearity, 30 (2017), 4483-4522.  doi: 10.1088/1361-6544/aa8c9e.

[22]

T. Ohlson Timoudas, Asymptotic laws for a class of quasi-periodic Schrödinger cocycles at the lowest energy of the spectrum, preprint, arXiv: 1809.05418, 2018.

[23]

Y. Wang and J. You, Examples of discontinuity of Lyapunov exponent in smooth quasiperiodic cocycles, Duke Math. J., 162 (2013), 2363-2412.  doi: 10.1215/00127094-2371528.

[24]

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.

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