Article Contents

2023, Volume 22, Issue 10: 3068-3082. Doi: 10.3934/cpaa.2023099

This issue Previous Article Spectral stability and instability of solitary waves of the Dirac equation with concentrated nonlinearity Next Article Existence and nonexistence of global solutions for time-dependent damped NLS equations

Non-perturbative positivity and weak Hölder continuity of Lyapunov exponent for some discrete multivariable Jacobi operators

Wenwen Chen and
Kai Tao^,

College of Sciences, Hohai University, 1 Xikang Road Nanjing, Jiangsu 210098, P. R. China

^*Corresponding author: Kai Tao
^*Corresponding author: Kai Tao

Received: March 2023

Revised: June 2023

Early access: September 2023

Published: October 2023

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

In this paper, we construct a class of special Jacobi cocycles, some element of which is a measurable function defined on the high dimensional torus. We prove that no matter the underlying dynamics is the shift or the skew-shift, the non-perturbative positive Lyapunov exponent holds for any irrational frequency, when the coupling number is large. What's more, if the frequency becomes to be the Diophantine number, then the Lyapunov exponent is weak continuity in the energy.

Keywords:

Mathematics Subject Classification: 37C55, 37F10.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	A. Avila, S. Jitomirskaya and C. Sadel, Complex one-frequency cocycles, J. Euro. Math. Soc., 9 (2013), 1915-1935. doi: 10.4171/JEMS/479.
[2]	J. Bochi and M. Viana, The Lyapunov exponents of generic volume perserving and symplectic maps, Ann. Math., 161 (2005), 1-63. doi: 10.4007/annals.2005.161.1423.
[3]	J. Bourgain, Green 's Function Estimates for Lattice Schrödinger Operators and Applications, Ann. Math. Stud. Princeton, 158, NJ: Princeton University Press, 2004. doi: 10.1515/9781400837144.
[4]	J. Bourgain and M. Goldstein, On nonperturbative localization with quasi-periodic potential, Ann. Math., 152 (2000), 835-879. doi: 10.2307/2661356.
[5]	J. Bourgain, M. Goldstein and W. Schlag, Anderson localization for Schrödinger operators on $\mathbb{Z}^2$ with potentials given by the skew-shift, Commun. Math. Phys., 220 (2001), 583-621. doi: 10.1007/PL00005570.
[6]	J. Bourgain and S. Jitomirskaya, Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, J. Stat. Phys., 108 (2002), 1028-1218. doi: 10.1023/A:1019751801035.
[7]	J. Garnett and D. Marshall, Harmonic Measure, Cambridge University Press, 2005. doi: 10.1017/CBO9780511546617.
[8]	L. Ge, Y. Wang and J. You et al., Transition space for the contuinuity of the Lyapunov exponent of quasiperiodic Schrödinger Cocycles, arXiv: 2102.05175.
[9]	M. Goldstein and W. Schlag, Hölder continuity of the integrated density of states for quasiperiodic Schrödinger equations and averages of shifts of subharmonic functions, Ann. Math., 154 (2001), 155-203. doi: 10.2307/3062114.
[10]	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.
[11]	N. M. Korobov, Exponential Sums and Their Applications (Mathematics and Its Applications, Soviet Series, volume 80, Kluwer Academic Publishers, 1992. doi: 10.1007/978-94-015-8032-8.
[12]	E. Stein, Harmonic analysis, Princeton Mathematical Series 43, Princeton, NJ: Princeton University Press, 1993.
[13]	K. Tao, Strong Birkhoff Ergodic Theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles, Discrete Contin. Dyn. Syst., 42 (2022), 1495-1533. doi: 10.3934/dcds.2021162.
[14]	Y. Wang and J. You, Examples of discontinuity of Lyapunov exponent in smooth quasi-periodic cocycles, Duke Math. J., 162 (2013), 2363-2412. doi: 10.1215/00127094-2371528.
[15]	Y. Wang and J. You, The Set of Smooth Quasi-periodic Schrödinger Cocycles with Positive Lyapunov Exponent is Not Open, Commun. Math. Phys., 362 (2018), 801-826. doi: 10.1007/s00220-018-3223-8.
[16]	L. Young, Lyapunov exponents for some quasi-periodic cocycles, Ergod. Theory Dyn. Syst., 17 (1997), 483-504. doi: 10.1017/S0143385797079170.