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Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency
College of Sciences, Hohai University, No.1 Xikang Road, Nanjing, Jiangsu, 210098, China |
In this paper, we first study the strong Birkhoff Ergodic Theorem for subharmonic functions with the Brjuno-Rüssmann shift on the Torus. Then, we apply it to prove the large deviation theorems for the finite scale Dirichlet determinants of quasi-periodic analytic Jacobi operators with this frequency. It shows that the Brjuno-Rüssmann function, which reflects the irrationality of the frequency, plays the key role in these theorems via the smallest deviation. At last, as an application, we obtain a distribution of the eigenvalues of the Jacobi operators with Dirichlet boundary conditions, which also depends on the smallest deviation, essentially on the irrationality of the frequency.
References:
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A. Avila and S. Jitomirskaya,
The Ten Martini Problem, Ann. Math., 170 (2009), 303-342.
doi: 10.4007/annals.2009.170.303. |
[2] |
A. Avila, S. Jitomirskaya and C. A. Marx,
Spectral theory of extended Harper's model and a question by Erdös and Szekeres, Inv. Math., 210 (2017), 1-57.
doi: 10.1016/j.aim.2017.08.026. |
[3] |
A. Avila, Y. Last, M. Shamis and Q. Zhou, On the abominable properties of the Almost Mathieu operator with well approximated frequencies, In preparation. Google Scholar |
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A. Avila, J. You and Z. Zhou,
Sharp Phase transitions for the almost Mathieu operator, Duke Math., 166 (2017), 2697-2718.
doi: 10.1215/00127094-2017-0013. |
[5] |
J. Bourgain and S. Jitomirskaya,
Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, J. Stat. Phys., 108 (2002), 1203-1218.
doi: 10.1023/A:1019751801035. |
[6] |
J. Bourgain and M. Goldstein,
On nonperturbative localization with quasi-periodic potential, Ann. Math., 152 (2000), 835-879.
doi: 10.2307/2661356. |
[7] |
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. |
[8] |
I. Binder and M. Voda,
An estimate on the number of eigenvalues of a quasiperiodic Jacobi matrix of size n contained in an interval of size $n^{-C}$, J. Spectr. Theory, 3 (2013), 1-45.
doi: 10.4171/JST/36. |
[9] |
I. Binder and M. Voda, On optimal separation of eigenvalues for a quasiperiodic Jacobi matrix, Commun. Math. Phys., 325 (2014), 1063-1106.
doi: 10.1007/s00220-013-1836-5. |
[10] |
M. Goldstein, D. Damanik, W. Schlag and M. Voda,
Homogeneity of the spectrum for quasi-perioidic Schrödinger operators, J. Eur. Math. Soc., 20 (2018), 3073-3111.
doi: 10.4171/JEMS/829. |
[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. Math., 2 (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] |
M. Goldstein and W. Schlag,
On resonances and the formation of gaps in the spectrum of quasi-periodic Schrödinger equations, Ann. Math., 173 (2011), 337-475.
doi: 10.4007/annals.2011.173.1.9. |
[14] |
R. Han,
Dry Ten Martini problem for the non-self-dual extended Harper's model, Trans. Am. Math. Soc., 370 (2018), 197-217.
doi: 10.1090/tran/6989. |
[15] |
R. Han and S. Zhang, Optimal Large Deviation Estimates and Hölder Regularity of the Lyapunov Exponents for Quasi-periodic Schrödinger Cocycles, arXiv: 1803.02035 Google Scholar |
[16] |
S. Jitomirskaya, D. A. Koslover and M. S. Schulteis,
Continuity of the Lyapunov exponent for analytic quasiperiodic cocycles, Ergod. Theor. Dyn. Syst., 29 (2009), 1881-1905.
doi: 10.1017/S0143385709000704. |
[17] |
S. Jitomirskaya, D. A. Koslover and M. S. Schulteis,
Localization for a family of one-dimensional quasiperiodic operators of magnetic origin, Ann. Henri. Poincar., 6 (2005), 103-125.
doi: 10.1007/s00023-005-0200-5. |
[18] |
S. Jitomirskaya and C. A. Marx,
Continuity of the Lyapunov Exponent for analytic quasi-perodic cocycles with singularities, J. Fix. Point Theory A., 10 (2011), 129-146.
doi: 10.1007/s11784-011-0055-y. |
[19] |
Ya. B. Levin, Lectures on Entire Functions, AMS, Providence, RI, 1996. |
[20] |
K. Tao,
Hölder continuity of Lyapunov exponent for quasi-periodic Jacobi operators, Bulletin de la SMF, 142 (2014), 635-671.
doi: 10.24033/bsmf.2675. |
[21] |
K. Tao, Strong Birkhoff ergodic theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles, arXiv: 1805.00431. Google Scholar |
[22] |
J. You and S. Zhang,
Hölder continuity of the Lyapunov exponent for analytic quasiperiodic Schrödinger cocycles with week Liouville frequency, Ergod. Theor. Dyn. Syst., 34 (2014), 1395-1408.
doi: 10.1017/etds.2013.4. |
show all references
References:
[1] |
A. Avila and S. Jitomirskaya,
The Ten Martini Problem, Ann. Math., 170 (2009), 303-342.
doi: 10.4007/annals.2009.170.303. |
[2] |
A. Avila, S. Jitomirskaya and C. A. Marx,
Spectral theory of extended Harper's model and a question by Erdös and Szekeres, Inv. Math., 210 (2017), 1-57.
doi: 10.1016/j.aim.2017.08.026. |
[3] |
A. Avila, Y. Last, M. Shamis and Q. Zhou, On the abominable properties of the Almost Mathieu operator with well approximated frequencies, In preparation. Google Scholar |
[4] |
A. Avila, J. You and Z. Zhou,
Sharp Phase transitions for the almost Mathieu operator, Duke Math., 166 (2017), 2697-2718.
doi: 10.1215/00127094-2017-0013. |
[5] |
J. Bourgain and S. Jitomirskaya,
Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, J. Stat. Phys., 108 (2002), 1203-1218.
doi: 10.1023/A:1019751801035. |
[6] |
J. Bourgain and M. Goldstein,
On nonperturbative localization with quasi-periodic potential, Ann. Math., 152 (2000), 835-879.
doi: 10.2307/2661356. |
[7] |
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. |
[8] |
I. Binder and M. Voda,
An estimate on the number of eigenvalues of a quasiperiodic Jacobi matrix of size n contained in an interval of size $n^{-C}$, J. Spectr. Theory, 3 (2013), 1-45.
doi: 10.4171/JST/36. |
[9] |
I. Binder and M. Voda, On optimal separation of eigenvalues for a quasiperiodic Jacobi matrix, Commun. Math. Phys., 325 (2014), 1063-1106.
doi: 10.1007/s00220-013-1836-5. |
[10] |
M. Goldstein, D. Damanik, W. Schlag and M. Voda,
Homogeneity of the spectrum for quasi-perioidic Schrödinger operators, J. Eur. Math. Soc., 20 (2018), 3073-3111.
doi: 10.4171/JEMS/829. |
[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. Math., 2 (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] |
M. Goldstein and W. Schlag,
On resonances and the formation of gaps in the spectrum of quasi-periodic Schrödinger equations, Ann. Math., 173 (2011), 337-475.
doi: 10.4007/annals.2011.173.1.9. |
[14] |
R. Han,
Dry Ten Martini problem for the non-self-dual extended Harper's model, Trans. Am. Math. Soc., 370 (2018), 197-217.
doi: 10.1090/tran/6989. |
[15] |
R. Han and S. Zhang, Optimal Large Deviation Estimates and Hölder Regularity of the Lyapunov Exponents for Quasi-periodic Schrödinger Cocycles, arXiv: 1803.02035 Google Scholar |
[16] |
S. Jitomirskaya, D. A. Koslover and M. S. Schulteis,
Continuity of the Lyapunov exponent for analytic quasiperiodic cocycles, Ergod. Theor. Dyn. Syst., 29 (2009), 1881-1905.
doi: 10.1017/S0143385709000704. |
[17] |
S. Jitomirskaya, D. A. Koslover and M. S. Schulteis,
Localization for a family of one-dimensional quasiperiodic operators of magnetic origin, Ann. Henri. Poincar., 6 (2005), 103-125.
doi: 10.1007/s00023-005-0200-5. |
[18] |
S. Jitomirskaya and C. A. Marx,
Continuity of the Lyapunov Exponent for analytic quasi-perodic cocycles with singularities, J. Fix. Point Theory A., 10 (2011), 129-146.
doi: 10.1007/s11784-011-0055-y. |
[19] |
Ya. B. Levin, Lectures on Entire Functions, AMS, Providence, RI, 1996. |
[20] |
K. Tao,
Hölder continuity of Lyapunov exponent for quasi-periodic Jacobi operators, Bulletin de la SMF, 142 (2014), 635-671.
doi: 10.24033/bsmf.2675. |
[21] |
K. Tao, Strong Birkhoff ergodic theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles, arXiv: 1805.00431. Google Scholar |
[22] |
J. You and S. Zhang,
Hölder continuity of the Lyapunov exponent for analytic quasiperiodic Schrödinger cocycles with week Liouville frequency, Ergod. Theor. Dyn. Syst., 34 (2014), 1395-1408.
doi: 10.1017/etds.2013.4. |
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