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March  2022, 42(3): 1495-1533. doi: 10.3934/dcds.2021162

Strong Birkhoff ergodic theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles

College of Sciences, Hohai University, No.1 Xikang Road, Nanjing, Jiangsu 210098, China

* Corresponding author: Kai Tao

Received  February 2020 Revised  August 2020 Published  March 2022 Early access  November 2021

Fund Project: The author is supported by The author is supported by the Fundamental Research Funds for the Central Universities (Grant B200202004) and China Postdoctoral Science Foundation (Grant 2019M650094)

In this paper, we first prove the strong Birkhoff Ergodic Theorem for subharmonic functions with the irrational shift on the Torus. Then, we apply it to the analytic quasi-periodic Jacobi cocycles and show that for suitable frequency and coupling number, if the Lyapunov exponent of these cocycles is positive at one point, then it is positive on an interval centered at this point and Hölder continuous in $ E $ on this interval. What's more, if the coupling number of the potential is large, then the Lyapunov exponent is always positive for all irrational frequencies and Hölder continuous in $ E $ for all finite Liouville frequencies. For the Schrödinger cocycles, a special case of the Jacobi ones, its Lyapunov exponent is also Hölder continuous in the frequency and the lengths of the intervals where the Hölder condition of the Lyapunov exponent holds only depend on the coupling number.

Citation: Kai Tao. Strong Birkhoff ergodic theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1495-1533. doi: 10.3934/dcds.2021162
References:
[1]

P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev., 109 (1958), 1492-1501.  doi: 10.1103/PhysRev.109.1492.

[2]

A. AvilaS. 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: 10.1007/s00222-017-0729-1.

[3]

A. AvilaS. Jitomirskaya and C. Sadel, Complex one-frequency cocycles, J. of Euro. Math. Soc., 9 (2013), 1915-1935.  doi: 10.4171/JEMS/479.

[4]

A. Avila, Y. Last, M. Shamis and Q. Zhou, On the abominable properties of the Almost Mathieu operator with well approximated frequencies, In preparation.

[5] J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications, Princeton University Press, 2005.  doi: 10.1515/9781400837144.
[6]

J. Bourgain and M. Goldstein, On nonperturbative localization with quasi-periodic potential, Ann. of Math., 152 (2000), 835-879.  doi: 10.2307/2661356.

[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]

A. Furman, On the multiplicative ergodic theorem for uniquely ergodic systems, Ann. Inst. Henri Poincare Probab. Statist., 33 (1997), 797-815.  doi: 10.1016/S0246-0203(97)80113-6.

[9] J. B. Garnett and D. E. Marshall, Harmonic Measure, Cambridge University Press, 2005.  doi: 10.1017/CBO9780511546617.
[10]

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., 2 (2001), 155-203.  doi: 10.2307/3062114.

[11]

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

[12]

R. Han and S. Zhang, Optimal large deviation estimates and Hölder regularity of the Lyapunov exponents for quasi-periodic Schrödinger cocycles, arXiv: 1803.02035v1.

[13]

P. G. Harper, Single band motion of conduction electrons in a uniform magnetic field, Proc. Phys. Soc. London, 68 (1955), 874-892.  doi: 10.1088/0370-1298/68/10/304.

[14]

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.

[15]

S. Jitomirskaya and C. A. Marx, Continuity of the Lyapunov exponent for analytic quasi-perodic cocycles with singularities, Journal of Fixed Point Theory and Applications, 10 (2011), 129-146.  doi: 10.1007/s11784-011-0055-y.

[16]

S. Jitomirskaya and C. A. Marx, Analytic quasi-perodic cocycles with singularities and the Lyapunov exponent of extended Harper's model, Commun. Math. Phys., 316 (2012), 237-267.  doi: 10.1007/s00220-012-1465-4.

[17]

S. Jitomirskaya and C. A. Marx, Erratum to: Analytic quasi-perodic cocycles with singularities and the Lyapunov exponent of extended Harper's model, Commun. Math. Phys., 317 (2012), 269-271.  doi: 10.1007/s00220-012-1637-2.

[18]

W. Schlag, Regularity and convergence rates for the Lyapunov exponents of linear cocycles, J. Mod. Dyn., 7 (2013), 619-637.  doi: 10.3934/jmd.2013.7.619.

[19]

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.

[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]

J. You and S. Zhang, Hölder continuity of the Lyapunov exponent for analytic quasiperiodic Schrödinger cocycles with week Liouville frequency, Ergod. Th. Dynam. Sys., 34 (2014), 1395-1408.  doi: 10.1017/etds.2013.4.

show all references

References:
[1]

P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev., 109 (1958), 1492-1501.  doi: 10.1103/PhysRev.109.1492.

[2]

A. AvilaS. 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: 10.1007/s00222-017-0729-1.

[3]

A. AvilaS. Jitomirskaya and C. Sadel, Complex one-frequency cocycles, J. of Euro. Math. Soc., 9 (2013), 1915-1935.  doi: 10.4171/JEMS/479.

[4]

A. Avila, Y. Last, M. Shamis and Q. Zhou, On the abominable properties of the Almost Mathieu operator with well approximated frequencies, In preparation.

[5] J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications, Princeton University Press, 2005.  doi: 10.1515/9781400837144.
[6]

J. Bourgain and M. Goldstein, On nonperturbative localization with quasi-periodic potential, Ann. of Math., 152 (2000), 835-879.  doi: 10.2307/2661356.

[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]

A. Furman, On the multiplicative ergodic theorem for uniquely ergodic systems, Ann. Inst. Henri Poincare Probab. Statist., 33 (1997), 797-815.  doi: 10.1016/S0246-0203(97)80113-6.

[9] J. B. Garnett and D. E. Marshall, Harmonic Measure, Cambridge University Press, 2005.  doi: 10.1017/CBO9780511546617.
[10]

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., 2 (2001), 155-203.  doi: 10.2307/3062114.

[11]

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

[12]

R. Han and S. Zhang, Optimal large deviation estimates and Hölder regularity of the Lyapunov exponents for quasi-periodic Schrödinger cocycles, arXiv: 1803.02035v1.

[13]

P. G. Harper, Single band motion of conduction electrons in a uniform magnetic field, Proc. Phys. Soc. London, 68 (1955), 874-892.  doi: 10.1088/0370-1298/68/10/304.

[14]

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.

[15]

S. Jitomirskaya and C. A. Marx, Continuity of the Lyapunov exponent for analytic quasi-perodic cocycles with singularities, Journal of Fixed Point Theory and Applications, 10 (2011), 129-146.  doi: 10.1007/s11784-011-0055-y.

[16]

S. Jitomirskaya and C. A. Marx, Analytic quasi-perodic cocycles with singularities and the Lyapunov exponent of extended Harper's model, Commun. Math. Phys., 316 (2012), 237-267.  doi: 10.1007/s00220-012-1465-4.

[17]

S. Jitomirskaya and C. A. Marx, Erratum to: Analytic quasi-perodic cocycles with singularities and the Lyapunov exponent of extended Harper's model, Commun. Math. Phys., 317 (2012), 269-271.  doi: 10.1007/s00220-012-1637-2.

[18]

W. Schlag, Regularity and convergence rates for the Lyapunov exponents of linear cocycles, J. Mod. Dyn., 7 (2013), 619-637.  doi: 10.3934/jmd.2013.7.619.

[19]

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.

[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]

J. You and S. Zhang, Hölder continuity of the Lyapunov exponent for analytic quasiperiodic Schrödinger cocycles with week Liouville frequency, Ergod. Th. Dynam. Sys., 34 (2014), 1395-1408.  doi: 10.1017/etds.2013.4.

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