October  2013, 7(4): 619-637. doi: 10.3934/jmd.2013.7.619

Regularity and convergence rates for the Lyapunov exponents of linear cocycles

1. 

Department of Mathematics, The University of Chicago, 5734 South University Avenue, Chicago, IL 60615

Received  August 2013 Published  March 2014

We consider cocycles $\tilde A: \mathbb{T}\times K^d \ni (x,v)\mapsto ( x+\omega, A(x,E)v)$ with $\omega$ Diophantine, $K=\mathbb{R}$ or $K=\mathbb{C}$. We assume that $A: \mathbb{T}\times \mathfrak{E} \to GL(d,K)$ is continuous, depends analytically on $x\in\mathbb{T}$ and is Hölder in $E\in \mathfrak{E} $, where $\mathfrak{E}$ is a compact metric space. It is shown that if all Lyapunov exponents are distinct at one point $E_{0}\in\mathfrak{E}$, then they remain distinct near $E$. Moreover, they depend in a Hölder fashion on $E\in B$ for any ball $B\subset \mathfrak{E}$ where they are distinct. Similar results, with a weaker modulus of continuity, hold for higher-dimensional tori $\mathbb{T}^\nu$ with a Diophantine shift. We also derive optimal statements about the rate of convergence of the finite-scale Lyapunov exponents to their infinite-scale counterparts. A key ingredient in our arguments is the Avalanche Principle, a deterministic statement about long finite products of invertible matrices, which goes back to work of Michael Goldstein and the author. We also discuss applications of our techniques to products of random matrices.
Citation: Wilhelm Schlag. Regularity and convergence rates for the Lyapunov exponents of linear cocycles. Journal of Modern Dynamics, 2013, 7 (4) : 619-637. doi: 10.3934/jmd.2013.7.619
References:
[1]

A. Avila, Density of positive Lyapunov exponents for quasiperiodic $SL(2,R)$-cocycles in arbitrary dimension, J. Mod. Dyn., 3 (2009), 631-636. doi: 10.3934/jmd.2009.3.631.

[2]

A. Avila and R. Krikorian, Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles, Ann. of Math. (2), 164 (2006), 911-940. doi: 10.4007/annals.2006.164.911.

[3]

A. Avila and M. Viana, Simplicity of Lyapunov spectra: A sufficient criterion, Port. Math. (N.S.), 64 (2007), 311-376. doi: 10.4171/PM/1789.

[4]

A. Avila and M. Viana, Extremal Lyapunov exponents: An invariance principle and applications, Invent. Math., 181 (2010), 115-189. doi: 10.1007/s00222-010-0243-1.

[5]

L. Barreira and Y. Pesin, Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents, Encyclopedia of Mathematics and its Applications, 115, Cambridge University Press, Cambridge, 2007.

[6]

C. Bonatti and M. Viana, Lyapunov exponents with multiplicity 1 for deterministic products of matrices, Ergodic Theory Dynam. Systems, 24 (2004), 1295-1330. doi: 10.1017/S0143385703000695.

[7]

C. Bonatti, X. Gómez-Mont and M. Viana, Généricité d'exposants de Lyapunov non-nuls pour des produits déterministes de matrices, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 579-624. doi: 10.1016/S0294-1449(02)00019-7.

[8]

P. Bougerol and J. Lacroix, Products of random matrices with applications to Schrödinger operators, Progress in Probability and Statistics, 8, Birkhäuser Boston, Inc., Boston, MA, 1985.

[9]

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.

[10]

J. Bourgain, Green's Function Estimates for Lattice Schr\"odinger Operators and Applications, Annals of Mathematics Studies, 158, Princeton University Press, Princeton, NJ, 2005.

[11]

J. Bourgain and M. Goldstein, On nonperturbative localization with quasi-periodic potential, Ann. of Math. (2), 152 (2000), 835-879. doi: 10.2307/2661356.

[12]

M. Campanino and A. Klein, A supersymmetric transfer matrix and differentiability of the density of states in the one-dimensional Anderson model, Comm. Math. Phys., 104 (1986), 227-241. doi: 10.1007/BF01211591.

[13]

P. Duarte and S. Klein, Continuity of the Lyapunov exponents for quasiperiodic cocycles, arXiv:1305.7504.

[14]

P. Duarte and S. Klein, Positive Lyapunov exponents for higher dimensional quasiperiodic cocycles, arXiv:1211.4002.

[15]

B. Fayad and R. Krikorian, Rigidity results for quasiperiodic $SL(2,R)$-cocycles, J. Mod. Dyn., 3 (2009), 497-510. doi: 10.3934/jmd.2009.3.479.

[16]

H. Fürstenberg, Noncommuting random products, Trans. Amer. Math. Soc., 108 (1963), 377-428. doi: 10.1090/S0002-9947-1963-0163345-0.

[17]

H. Fürstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist., 31 (1960), 457-469. doi: 10.1214/aoms/1177705909.

[18]

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), 154 (2001), 155-203. doi: 10.2307/3062114.

[19]

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.

[20]

M. Goldstein and W. Schlag, On resonances and the formation of gaps in the spectrum of quasi-periodic Schrödinger equations, Ann. of Math. (2), 173 (2011), 337-475. doi: 10.4007/annals.2011.173.1.9.

[21]

M. Herman, Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d'un théorème d'Arnol'd et de Moser sur le tore de dimension $2$, Comment. Math. Helv., 58 (1983), 453-502. doi: 10.1007/BF02564647.

[22]

R. Krikorian, Global density of reducible quasi-periodic cocycles on $\mathbbT\times SU(2)$, Ann. of Math. (2), 154 (2001), 269-326. doi: 10.2307/3062098.

[23]

F. Ledrappier, Quelques propriétés des exposants caractéristiques, École d'été de probabilités de Saint-Flour, XII - 1982, Lecture Notes in Math., 1097, Springer, Berlin, 1984, 305-396. doi: 10.1007/BFb0099434.

[24]

É. Le Page, Théorèmes limites pour les produits de matrices aléatoires, in Probability Measures on Groups (Oberwolfach, 1981), Lecture Notes in Math., 928, Springer, Berlin-New York, 1982, 258-303.

[25]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov exponents of dynamical systems, (Russian) Trudy Moskov. Mat. Obšč. 19 (1968), 179-210.

[26]

M. S. Raghunathan, A proof of Oseledec's multiplicative ergodic theorem, Israel J. Math., 32 (1979), 356-362. doi: 10.1007/BF02760464.

[27]

D. Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 27-58.

[28]

B. Simon and M. Taylor, Harmonic analysis on $\mathbbSL(2,R)$ and smoothness of the density of states in the one-dimensional Anderson model, Comm. Math. Phys., 101 (1985), 1-19. doi: 10.1007/BF01212354.

show all references

References:
[1]

A. Avila, Density of positive Lyapunov exponents for quasiperiodic $SL(2,R)$-cocycles in arbitrary dimension, J. Mod. Dyn., 3 (2009), 631-636. doi: 10.3934/jmd.2009.3.631.

[2]

A. Avila and R. Krikorian, Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles, Ann. of Math. (2), 164 (2006), 911-940. doi: 10.4007/annals.2006.164.911.

[3]

A. Avila and M. Viana, Simplicity of Lyapunov spectra: A sufficient criterion, Port. Math. (N.S.), 64 (2007), 311-376. doi: 10.4171/PM/1789.

[4]

A. Avila and M. Viana, Extremal Lyapunov exponents: An invariance principle and applications, Invent. Math., 181 (2010), 115-189. doi: 10.1007/s00222-010-0243-1.

[5]

L. Barreira and Y. Pesin, Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents, Encyclopedia of Mathematics and its Applications, 115, Cambridge University Press, Cambridge, 2007.

[6]

C. Bonatti and M. Viana, Lyapunov exponents with multiplicity 1 for deterministic products of matrices, Ergodic Theory Dynam. Systems, 24 (2004), 1295-1330. doi: 10.1017/S0143385703000695.

[7]

C. Bonatti, X. Gómez-Mont and M. Viana, Généricité d'exposants de Lyapunov non-nuls pour des produits déterministes de matrices, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 579-624. doi: 10.1016/S0294-1449(02)00019-7.

[8]

P. Bougerol and J. Lacroix, Products of random matrices with applications to Schrödinger operators, Progress in Probability and Statistics, 8, Birkhäuser Boston, Inc., Boston, MA, 1985.

[9]

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.

[10]

J. Bourgain, Green's Function Estimates for Lattice Schr\"odinger Operators and Applications, Annals of Mathematics Studies, 158, Princeton University Press, Princeton, NJ, 2005.

[11]

J. Bourgain and M. Goldstein, On nonperturbative localization with quasi-periodic potential, Ann. of Math. (2), 152 (2000), 835-879. doi: 10.2307/2661356.

[12]

M. Campanino and A. Klein, A supersymmetric transfer matrix and differentiability of the density of states in the one-dimensional Anderson model, Comm. Math. Phys., 104 (1986), 227-241. doi: 10.1007/BF01211591.

[13]

P. Duarte and S. Klein, Continuity of the Lyapunov exponents for quasiperiodic cocycles, arXiv:1305.7504.

[14]

P. Duarte and S. Klein, Positive Lyapunov exponents for higher dimensional quasiperiodic cocycles, arXiv:1211.4002.

[15]

B. Fayad and R. Krikorian, Rigidity results for quasiperiodic $SL(2,R)$-cocycles, J. Mod. Dyn., 3 (2009), 497-510. doi: 10.3934/jmd.2009.3.479.

[16]

H. Fürstenberg, Noncommuting random products, Trans. Amer. Math. Soc., 108 (1963), 377-428. doi: 10.1090/S0002-9947-1963-0163345-0.

[17]

H. Fürstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist., 31 (1960), 457-469. doi: 10.1214/aoms/1177705909.

[18]

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), 154 (2001), 155-203. doi: 10.2307/3062114.

[19]

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.

[20]

M. Goldstein and W. Schlag, On resonances and the formation of gaps in the spectrum of quasi-periodic Schrödinger equations, Ann. of Math. (2), 173 (2011), 337-475. doi: 10.4007/annals.2011.173.1.9.

[21]

M. Herman, Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d'un théorème d'Arnol'd et de Moser sur le tore de dimension $2$, Comment. Math. Helv., 58 (1983), 453-502. doi: 10.1007/BF02564647.

[22]

R. Krikorian, Global density of reducible quasi-periodic cocycles on $\mathbbT\times SU(2)$, Ann. of Math. (2), 154 (2001), 269-326. doi: 10.2307/3062098.

[23]

F. Ledrappier, Quelques propriétés des exposants caractéristiques, École d'été de probabilités de Saint-Flour, XII - 1982, Lecture Notes in Math., 1097, Springer, Berlin, 1984, 305-396. doi: 10.1007/BFb0099434.

[24]

É. Le Page, Théorèmes limites pour les produits de matrices aléatoires, in Probability Measures on Groups (Oberwolfach, 1981), Lecture Notes in Math., 928, Springer, Berlin-New York, 1982, 258-303.

[25]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov exponents of dynamical systems, (Russian) Trudy Moskov. Mat. Obšč. 19 (1968), 179-210.

[26]

M. S. Raghunathan, A proof of Oseledec's multiplicative ergodic theorem, Israel J. Math., 32 (1979), 356-362. doi: 10.1007/BF02760464.

[27]

D. Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 27-58.

[28]

B. Simon and M. Taylor, Harmonic analysis on $\mathbbSL(2,R)$ and smoothness of the density of states in the one-dimensional Anderson model, Comm. Math. Phys., 101 (1985), 1-19. doi: 10.1007/BF01212354.

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