Sharp Hölder continuity of the Lyapunov exponent of finitely differentiable quasi-periodic cocycles

Abstract

We show that if the base frequency is Diophantine, then the Lyapunov exponent of a \(C^{k}\) quasi-periodic \(SL(2,{\mathbb {R}})\) cocycle is 1 / 2-Hölder continuous in the almost reducible regime. As a consequence, we show that if the frequency is Diophantine, and the potential is small, then the integrated density of states of the corresponding quasi-periodic Schrödinger operator is 1 / 2-Hölder continuous.

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References

  1. 1.

    Hadj Amor, S.: Hölder continuity of the rotation number for the quasi-periodic co-cycles in \(SL(2,\mathbb{R})\). Commun. Math. Phys. 287, 565–588 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Avila, A.: The absolutely continuous spectrum of the almost Mathieu operator. Preprint. arXiv:0810.2965

  3. 3.

    Avila, A.: Global theory of one-frequency Schrödinger operators. Acta Math. 215, 1–54 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Avila, A., Jitomirskaya, S.: Almost localization and almost reducibility. J. Eur. Math. Soc. 12, 93–131 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Avila, A., Jitomirskaya, S., Sadel, C.: Complex one-frequency cocycles. J. Eur. Math. Soc. 16(9), 1915–1935 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Avila, A., Krikorian, R.: Monotonic cocycles. Invent. Math. 202, 271–331 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Avila, A., Viana, M., Eskin, A.: Continuity of Lyapunov exponents of random matrix products. In preparation

  8. 8.

    Avila, A., Fayad, B., Krikorian, R.: A KAM scheme for \(SL(2,\mathbb{R})\) cocycles with Liouvillean frequencies. Geom. Funct. Anal. 21, 1001–1019 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Avila, A., You, J., Zhou, Q.: Sharp phase transitions for the almost Mathieu operator. Duke Math. J. 166(14), 2697–2718 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Avila, A., You, J., Zhou, Q.: Dry ten Martini problem in the non-critical case. Preprint

  11. 11.

    Avron, J., Simon, B.: Almost periodic Schrödinger operators. II: The integrated density of states. Duke Math. J. 50, 369–391 (1983)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Backes, L., Brown, A.W., Butler, C.: Continuity of Lyapunov exponents for cocycles with invariant holonomies. Preprint. https://arxiv.org/pdf/1507.08978v2.pdf

  13. 13.

    Berti, M., Biasco, L.: Forced vibrations of wave equations with non-monotone nonlinearities. Ann. I. H. Poincaré-AN. 23(4), 439–474 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Bocker-Neto, C., Viana, M.: Continuity of Lyapunov exponents for random two-dimensional matrices. Ergodic Theory Dyn. Syst. 37, 1413–1442 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Bochi, J.: Discontinuity of the Lyapunov exponent for non-hyperbolic cocycles. Unpublished. http://www.mat.uc.cl/ jairo.bochi/docs/discont.pdf (1999)

  16. 16.

    Bochi, J.: Genericity of zero Lyapunov exponents. Ergodic Theory Dyn. Syst. 22(6), 1667–1696 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Bourgain, J.: Positivity and continuity of the Lyapunov exponent for shifts on \(\mathbb{T}^d\) with arbitrary frequency vector and real analytic potential. J. Anal. Math. 96, 313–355 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Bourgain, J., Jitomirskaya, S.: Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential. J. Stat. Phys. 108, 1203–1218 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

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

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Chavaudret, C.: Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles. Bull. Soc. Math. Fr. 141, 47–106 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Chavaudret, C.: Almost reducibility for finitely differentiable \(SL(2,\mathbb{R})\)-valued quasi-periodic cocycles. Nonlinearity 25, 481–494 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)

    Google Scholar 

  23. 23.

    Eliasson, L.H.: Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation. Commun. Math. Phys. 146, 447–482 (1992)

    Article  MATH  Google Scholar 

  24. 24.

    Fayad, B., Krikorian, R.: Rigidity results for quasiperiodic \(SL(2,\mathbb{R})\)-cocycles. J. Mod. Dyn. 3(4), 479–510 (2009)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Furman, A.: On the multiplicative ergodic theorem for the uniquely ergodic systems. Ann. Inst. H. Poincaré Prob. Stat. 33, 797–815 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Goldstein, M., Schlag, W.: Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions. Ann. Math. 154(1), 155–203 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Goldstein, M., Schlag, W.: Fine properties of the integrated density of states and a quantitative separation property of the Dirichlet eigenvalues. Geom. Funct. Anal. 18(3), 755–869 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Hou, X., You, J.: Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems. Invent. Math. 190, 209–260 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Jitomirskaya, S., Kachkovskiy, I.: All couplings localization for quasiperiodic operators with Lipschitz monotone potentials. To appear in J. Eur. Math. Soc. arxiv:1509.02226

  30. 30.

    Jitomirskaya, S., Koslover, D.A., Schulteis, M.S.: Continuity of the Lyapunov exponent for analytic quasiperiodic cocycles. Ergodic Theory Dyn. Syst. 29, 1881–1905 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Johnson, R., Moser, J.: The rotation number for almost periodic potentials. Commun. Math. Phys. 84, 403–438 (1982)

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Klein, S.: Localization for quasiperiodic Schrödinger operators with multivariable Gevrey potential functions. J. Spectr. Theory 4, 1–53 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Knill, O.: The upper Lyapunov exponent of \(SL(2,\mathbb{R})\) cocycles: discontinuity and the problem of positivity. Lect. Notes Math. 1486, 86–97 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Krikorian, R.: Reducibility of skew-product systems with values in compact groups. Asterisque- Societe Mathematique de France 259(259) (1999)

  35. 35.

    Leguil, M., You, J., Zhao, Z., Zhou, Q.: Asymptotics of spectral gaps of quasi-periodic Schrödinger operators. Preprint. https://arxiv.org/pdf/1712.04700.pdf

  36. 36.

    Ruelle, D.: Analyticity properties of the characteristic exponents of random matrix products. Adv. Math. 32, 68–80 (1979)

    Article  MATH  Google Scholar 

  37. 37.

    Thouvenot, J.: An example of discontinuity in the computation of the Lyapunov exponents. Proc. Steklov Inst. Math. 216, 366–369 (1997)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Viana, M.: Lectures on Lyapunov Exponents. Cambridge Studies in Advanced Mathematics, vol. 145. Cambridge University Press, Cambridge (2014)

    Google Scholar 

  39. 39.

    Wang, Y., You, J.: Examples of discontinuity of Lyapunov exponent in smooth quasi-periodic cocycles. Duke Math. J. 162, 2363–2412 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  40. 40.

    Wang, Y., Zhang, Z.: Uniform positivity and continuity of Lyapunov exponents for a class of \(C^2\) quasiperiodic Schrödinger cocycles. J. Func. Anal. 268, 2525–2585 (2015)

    Article  MATH  Google Scholar 

  41. 41.

    You, J., Zhou, Q.: Embedding of analytic quasi-periodic cocycles into analytic quasi-periodic linear systems and its applications. Commun. Math. Phys. 323, 975–1005 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  42. 42.

    Zehnder, E.: Generalized implicit function theorems with applications to some small divisor problems, I. Commun. Pure Appl. Math. XXVIII, 91–140 (1975)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

C. Chavaudret was supported by the ANR “BEKAM” and the ANR “Dynamics and CR Geometry”. J. You was partially supported by NSFC grant (11471155) and 973 projects of China (2014CB340701). Q. Zhou was partially supported by NSFC grant (11671192), “Deng Feng Scholar Program B” of Nanjing University, Specially-appointed professor programe of Jiangsu province.

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Correspondence to Qi Zhou.

5 Appendix: Proof of Lemma 3.1

5 Appendix: Proof of Lemma 3.1

In this appendix, we prove Lemma 3.1 by the following quantitative Implicit Function Theorem:

Theorem 5.1

[13, 22] Let XYZ be Banach spaces, \(U\subset X\) and \(V\subset Y\) neighborhoods of \(x_0\) and \(y_0\) respectively. Fix \(s,\delta >0\) and define \(B_s(x_0)=\{x\in X\mid ||x-x_0||_ X\leqslant s \}, B_{\delta }(y_0)=\{y\in Y\mid ||y-y_0||_Y \leqslant \delta \}.\) Let \(\Psi \in C^1(U\times V,Z)\) and \(B_s(x_0)\times B_{\delta }(y_0)\subset U\times V\). Suppose also that \(\Psi (x_0,y_0)=0\), and that \(D_y\Psi (x_0,y_0)\in \mathcal {L}(Y,Z)\) is invertible. If

$$\begin{aligned}&\sup _{\begin{array}{c} \overline{B_s(x_0)} \end{array}}||\Psi (x,y_0)||_Z\leqslant \frac{\delta }{2||({D_y\Psi (x_0,y_0)}^{-1})||}, \end{aligned}$$
(5.1)
$$\begin{aligned}&\sup _{\begin{array}{c} \overline{B_s(x_0)}\times \overline{B_{\delta }(y_0)} \end{array}}||Id_Y-(D_y\Psi (x_0,y_0))^{-1}D_y\Psi (x,y) ||_{\mathcal {L}(Y,Y)}\leqslant \frac{1}{2}, \end{aligned}$$
(5.2)

then there exists \(y\in C^1(B_s(x_0),\overline{B_{\delta }(y_0)})\) such that \(\Psi (x,y(x))=0\).

With Theorem 5.1 in hand, now we can prove Lemma 3.1 easily. We construct the nonlinear functional

$$\begin{aligned} \Psi :\mathcal {B}_r^{nre}(\eta ) \times C^{\omega }_r({\mathbb {T}}^d,sl(2,{\mathbb {R}}))\rightarrow \mathcal {B}_r^{nre}(\eta ) \end{aligned}$$

by

$$\begin{aligned} \Psi (Y,g)=\mathbb {P}_{nre} \ln (e^{A^{-1}Y(\theta +\alpha )A}e^{g(\theta )}e^{-Y(\theta )}) \end{aligned}$$

Immediate check shows that

$$\begin{aligned} \Psi (0,0)=0, \ \ ||\Psi (0,g)||\leqslant |g|_r. \end{aligned}$$

and

$$\begin{aligned}&\Psi (Y+Y',g)-\Psi (Y,g)\\&\quad =\mathbb {P}_{nre} \ln (e^{A^{-1}(Y(\theta +\alpha )+Y'(\theta +\alpha ))A}e^{g(\theta )}e^{-(Y(\theta )+Y'(\theta ))})\\&\qquad -\mathbb {P}_{nre} \ln (e^{A^{-1}Y(\theta +\alpha )A}e^{g(\theta )}e^{-Y(\theta )})\\&\quad =\mathbb {P}_{nre} \ln (e^{A^{-1}(Y(\theta +\alpha )+Y'(\theta +\alpha ))A}e^{g(\theta )}e^{-(Y(\theta )+Y'(\theta ))})\\&\qquad -\mathbb {P}_{nre}\ln (e^{A^{-1}(Y(\theta +\alpha )+Y'(\theta +\alpha ))A}e^{g(\theta )}e^{-Y(\theta )})\\&\qquad +\mathbb {P}_{nre}\ln (e^{A^{-1}(Y(\theta +\alpha )+Y'(\theta +\alpha ))A}e^{g(\theta )}e^{-Y(\theta )})\\&\qquad -\mathbb {P}_{nre} \ln (e^{A^{-1}Y(\theta +\alpha )A}e^{g(\theta )}e^{-Y(\theta )}) \end{aligned}$$

To make further computations, we need the fact that if ABC are small \(sl(2,{\mathbb {R}})\) matrices, then there exists \(D,E\in sl(2,{\mathbb {R}})\) such that

$$\begin{aligned} e^{A}e^{B}e^{C}=e^{D+E} \end{aligned}$$

and

$$\begin{aligned} D=A+B+C \end{aligned}$$

where E is a sum of terms at least 2 orders in ABC. Also, the famous Baker-Campbell-Hausdorff Formula shows that

$$\begin{aligned} \ln (e^X e^Y)=X+Y+\frac{1}{2}[X,Y]+\frac{1}{12}([X,[X,Y]+[Y,[Y,X]])+\cdots , \end{aligned}$$
(5.3)

where \([X,Y]=XY-YX\) denotes the Lie Bracket and \(\cdots \) denotes the sum of higher order terms.

Therefore, we can compute that

$$\begin{aligned}&\mathbb {P}_{nre} \ln (e^{A^{-1}(Y(\theta +\alpha )+Y'(\theta +\alpha ))A}e^{g(\theta )}e^{-(Y(\theta )+Y'(\theta ))})\\&\qquad -\mathbb {P}_{nre}\ln (e^{A^{-1}(Y(\theta +\alpha )+Y'(\theta +\alpha ))A}e^{g(\theta )}e^{-Y(\theta )})\\&\quad =\mathbb {P}_{nre} \ln (e^{A^{-1}(Y(\theta +\alpha )+Y'(\theta +\alpha ))A}e^{g(\theta )}e^{-Y(\theta )}e^{-Y''(\theta )})\\&\qquad -\mathbb {P}_{nre}\ln (e^{A^{-1}(Y(\theta +\alpha )+Y'(\theta +\alpha ))A}e^{g(\theta )}e^{-Y(\theta )})\\&\quad =\mathbb {P}_{nre} \ln (e^{D+E}e^{-Y''(\theta )})-\mathbb {P}_{nre}\ln (e^{D+E})\\&\quad =\mathbb {P}_{nre}(D+E-Y''+\frac{1}{2}[D+E,-Y'']+\cdots )-\mathbb {P}_{nre}(D+E)\\&\quad =\mathbb {P}_{nre}(-Y''+\frac{1}{2}[D+E,-Y'']+\cdots ) \end{aligned}$$

where

$$\begin{aligned} Y''(\theta )= & {} Y'(\theta )+\mathcal {O}(Y(\theta ))Y'(\theta ),\\ D(\theta )= & {} A^{-1}(Y(\theta +\alpha )+Y'(\theta +\alpha ))A+g(\theta )-Y(\theta ), \end{aligned}$$

and E is a sum of terms at least 2 orders in \(A^{-1}(Y(\theta +\alpha )+Y'(\theta +\alpha ))A,g(\theta ),-Y(\theta )\).

Similarly, we have

$$\begin{aligned}&\mathbb {P}_{nre}\ln (e^{A^{-1}(Y(\theta +\alpha )+Y'(\theta +\alpha ))A}e^{g(\theta )}e^{-Y(\theta )})\\&\qquad -\mathbb {P}_{nre} \ln (e^{A^{-1}Y(\theta +\alpha )A}e^{g(\theta )}e^{-Y(\theta )})\\&\quad =\mathbb {P}_{nre}\ln (e^{Y'''}e^{A^{-1}Y(\theta +\alpha )A}e^{g(\theta )}e^{-Y(\theta )})\\&\qquad -\mathbb {P}_{nre} \ln (e^{A^{-1}Y(\theta +\alpha )A}e^{g(\theta )}e^{-Y(\theta )})\\&\quad =\mathbb {P}_{nre} \ln (e^{Y'''}e^{F+H})-\mathbb {P}_{nre}\ln (e^{F+H})\\&\quad =\mathbb {P}_{nre}(Y'''+F+H+\frac{1}{2}[Y''',F+H]+\cdots )-\mathbb {P}_{nre}(F+H)\\&\quad =\mathbb {P}_{nre}(Y'''+\frac{1}{2}[Y''',F+H]+\cdots ) \end{aligned}$$

where

$$\begin{aligned} Y'''(\theta +\alpha )= & {} A^{-1}Y'(\theta +\alpha )A+\mathcal {O}(A^{-1}Y(\theta +\alpha )A)\times A^{-1}Y'(\theta +\alpha )A,\\ F(\theta )= & {} A^{-1}Y(\theta +\alpha )A+g(\theta )-Y(\theta ) \end{aligned}$$

and H is a sum of terms at least 2 orders in \(A^{-1}Y(\theta +\alpha )A,g(\theta ),-Y(\theta )\).

By the definition of Fréchet Differential, we only need to consider the linear terms of \(\Psi (Y+Y',g)-\Psi (Y,g)\), thus we have

$$\begin{aligned} D_Y\Psi (Y,g)(Y')&=\mathbb {P}_{nre}(A^{-1}Y'(\theta +\alpha )A+\mathcal {O}(A^{-1}Y(\theta +\alpha )A)\times A^{-1}Y'(\theta +\alpha )A\\&\quad +\frac{1}{2}[Y''',F+H]+\cdots )\\&\quad +\mathbb {P}_{nre}(-Y'(\theta )-\mathcal {O}(Y(\theta ))Y'(\theta )+\frac{1}{2}[F+H',-Y'']+\cdots ), \end{aligned}$$

where \(H'\) is a sum of terms at least 2 orders in \(A^{-1}Y(\theta +\alpha )A,g(\theta ),-Y(\theta )\). Moreover, the first “\(\cdots \)” denotes the sum of terms which are at least 2 orders in \(F+H\) but only 1 order in \(Y'''\). The second “\(\cdots \)” denotes the sum of terms which are at least 2 orders in \(F+H'\) but only 1 order in \(Y''\).

Let \(Y=0\) and \(g=0\), then all the Lie Brackets vanish. So we immediately obtain

$$\begin{aligned} D_Y\Psi (0,0)(Y')&=\mathbb {P}_{nre}(A^{-1}Y'(\theta +\alpha )A)+\mathbb {P}_{nre}(-Y'(\theta ))\\&=A^{-1}Y'(\theta +\alpha )A-Y'(\theta ) \end{aligned}$$

Thus

$$\begin{aligned} ||D_Y\Psi (0,0)(Y')||&\geqslant |A^{-1}Y'(\theta +\alpha )A-Y'(\theta )|_r \\&\geqslant \eta |Y'|_r.\\ \end{aligned}$$

So we have

$$\begin{aligned} ||(D_Y\Psi (0,0))^{-1}||\leqslant \eta ^{-1}. \end{aligned}$$

For our purpose, we set \(s=\epsilon ,\delta =\epsilon ^\frac{1}{2}\) and \(\eta \geqslant 13||A||^2\epsilon ^{\frac{1}{2}}\). Then we have

$$\begin{aligned} 2\times \sup _{\begin{array}{c} \overline{B_{s}(0)} \end{array}}||\Psi (0,g)||\times ||(D_Y\Psi (0,0))^{-1}||\leqslant 2\times \epsilon \times \frac{1}{13||A||^2}\epsilon ^{-\frac{1}{2}}\leqslant \epsilon ^{\frac{1}{2}}=\delta , \end{aligned}$$

then (5.1) is fulfilled.

On the other hand, direct computation shows that

$$\begin{aligned}&D_Y\Psi (Y,g)(Y')-D_Y\Psi (0,0)(Y')\\&\quad =\mathbb {P}_{nre}(\mathcal {O}(A^{-1}Y(\theta +\alpha )A)\times A^{-1}Y'(\theta +\alpha )A+\frac{1}{2}[Y''',F+H]+\cdots )\\&\qquad +\mathbb {P}_{nre}(-\mathcal {O}(Y(\theta ))Y'(\theta )+\frac{1}{2}[F+H',-Y'']+\cdots ). \end{aligned}$$

Therefore, we have

$$\begin{aligned} \sup _{\begin{array}{c} \overline{B_s(0)}\times \overline{B_{\delta }(0)} \end{array}}||D_Y\Psi (Y,g)(Y')-D_Y\Psi (0,0)(Y')||&\leqslant 6(||A||^2|Y|_r+||A||^2|g|_r)|Y'|\\&\leqslant 6(||A||^2\delta +||A||^2s)|Y'|\\&\leqslant 6(||A||^2 \epsilon ^{\frac{1}{2}}+||A||^2\epsilon )|Y'|, \end{aligned}$$

which implies

$$\begin{aligned} \sup _{\begin{array}{c} \overline{B_s(0)}\times \overline{B_{\delta }(0)} \end{array}}||D_Y\Psi (0,0)-D_Y\Psi (Y,g)||\leqslant 6(||A||^2 \epsilon ^{\frac{1}{2}}+||A||^2\epsilon ). \end{aligned}$$

Thus we have

$$\begin{aligned}&\sup _{\begin{array}{c} \overline{B_s(0)}\times \overline{B_{\delta }(0)} \end{array}}||Id_{\mathcal {B}_r^{nre}(\eta )}-(D_Y\Psi (0,0))^{-1}\times D_Y\Psi (Y,g) ||\\&\quad \leqslant \sup _{\begin{array}{c} \overline{B_s(0)}\times \overline{B_{\delta }(0)} \end{array}}||D_Y\Psi (0,0)-D_Y\Psi (Y,g)||\times ||(D_Y\Psi (0,0))^{-1}||\\&\quad \leqslant 6(||A||^2 \epsilon ^{\frac{1}{2}}+||A||^2\epsilon ) \times \frac{1}{13||A||^2}\epsilon ^{-\frac{1}{2}}\\&\quad \leqslant \frac{1}{2}, \end{aligned}$$

which satisfies (5.2). By Theorem 5.1, for \(|g|_r\leqslant \epsilon \) and \(\eta \geqslant 13||A||^2\epsilon ^{\frac{1}{2}}\), there exists \(|Y|_r\leqslant \epsilon ^{\frac{1}{2}}\) such that \(\Psi (Y,g)=0\), i.e.

$$\begin{aligned} e^{A^{-1}Y(\theta +\alpha )A}e^{g(\theta )}e^{Y(\theta )}=e^{g^{re}(\theta )}, \end{aligned}$$

which is equivalent to say

$$\begin{aligned} e^{Y(\theta +\alpha )}Ae^{g(\theta )}e^{Y(\theta )}=Ae^{g^{re}(\theta )} \end{aligned}$$

and it is easy to check \(|g^{re} (\theta )|_r\leqslant 2\epsilon \). This finishes the proof of Lemma 3.1.

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Cai, A., Chavaudret, C., You, J. et al. Sharp Hölder continuity of the Lyapunov exponent of finitely differentiable quasi-periodic cocycles. Math. Z. 291, 931–958 (2019). https://doi.org/10.1007/s00209-018-2147-5

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