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. doi: 10.3934/jmd.2009.3.631. Google Scholar

[2]

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

[3]

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

[4]

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

[5]

L. Barreira and Y. Pesin, Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents,, Encyclopedia of Mathematics and its Applications, 115 (2007). Google Scholar

[6]

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

[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. doi: 10.1016/S0294-1449(02)00019-7. Google Scholar

[8]

P. Bougerol and J. Lacroix, Products of random matrices with applications to Schrödinger operators,, Progress in Probability and Statistics, (1985). Google Scholar

[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. doi: 10.1023/A:1007641323456. Google Scholar

[10]

J. Bourgain, Green's Function Estimates for Lattice Schr\"odinger Operators and Applications,, Annals of Mathematics Studies, (2005). Google Scholar

[11]

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

[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. doi: 10.1007/BF01211591. Google Scholar

[13]

P. Duarte and S. Klein, Continuity of the Lyapunov exponents for quasiperiodic cocycles,, , (). Google Scholar

[14]

P. Duarte and S. Klein, Positive Lyapunov exponents for higher dimensional quasiperiodic cocycles,, , (). Google Scholar

[15]

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

[16]

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

[17]

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

[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. doi: 10.2307/3062114. Google Scholar

[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. doi: 10.1007/s00039-008-0670-y. Google Scholar

[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. doi: 10.4007/annals.2011.173.1.9. Google Scholar

[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. doi: 10.1007/BF02564647. Google Scholar

[22]

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

[23]

F. Ledrappier, Quelques propriétés des exposants caractéristiques,, École d'été de probabilités de Saint-Flour, (1982), 305. doi: 10.1007/BFb0099434. Google Scholar

[24]

É. Le Page, Théorèmes limites pour les produits de matrices aléatoires,, in Probability Measures on Groups (Oberwolfach, (1981), 258. Google Scholar

[25]

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

[26]

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

[27]

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

[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. doi: 10.1007/BF01212354. Google Scholar

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. doi: 10.3934/jmd.2009.3.631. Google Scholar

[2]

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

[3]

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

[4]

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

[5]

L. Barreira and Y. Pesin, Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents,, Encyclopedia of Mathematics and its Applications, 115 (2007). Google Scholar

[6]

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

[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. doi: 10.1016/S0294-1449(02)00019-7. Google Scholar

[8]

P. Bougerol and J. Lacroix, Products of random matrices with applications to Schrödinger operators,, Progress in Probability and Statistics, (1985). Google Scholar

[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. doi: 10.1023/A:1007641323456. Google Scholar

[10]

J. Bourgain, Green's Function Estimates for Lattice Schr\"odinger Operators and Applications,, Annals of Mathematics Studies, (2005). Google Scholar

[11]

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

[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. doi: 10.1007/BF01211591. Google Scholar

[13]

P. Duarte and S. Klein, Continuity of the Lyapunov exponents for quasiperiodic cocycles,, , (). Google Scholar

[14]

P. Duarte and S. Klein, Positive Lyapunov exponents for higher dimensional quasiperiodic cocycles,, , (). Google Scholar

[15]

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

[16]

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

[17]

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

[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. doi: 10.2307/3062114. Google Scholar

[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. doi: 10.1007/s00039-008-0670-y. Google Scholar

[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. doi: 10.4007/annals.2011.173.1.9. Google Scholar

[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. doi: 10.1007/BF02564647. Google Scholar

[22]

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

[23]

F. Ledrappier, Quelques propriétés des exposants caractéristiques,, École d'été de probabilités de Saint-Flour, (1982), 305. doi: 10.1007/BFb0099434. Google Scholar

[24]

É. Le Page, Théorèmes limites pour les produits de matrices aléatoires,, in Probability Measures on Groups (Oberwolfach, (1981), 258. Google Scholar

[25]

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

[26]

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

[27]

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

[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. doi: 10.1007/BF01212354. Google Scholar

[1]

Cecilia González-Tokman, Anthony Quas. A concise proof of the multiplicative ergodic theorem on Banach spaces. Journal of Modern Dynamics, 2015, 9: 237-255. doi: 10.3934/jmd.2015.9.237

[2]

Alex Blumenthal. A volume-based approach to the multiplicative ergodic theorem on Banach spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2377-2403. doi: 10.3934/dcds.2016.36.2377

[3]

Luciana A. Alves, Luiz A. B. San Martin. Multiplicative ergodic theorem on flag bundles of semi-simple Lie groups. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1247-1273. doi: 10.3934/dcds.2013.33.1247

[4]

Andrey Gogolev, Ali Tahzibi. Center Lyapunov exponents in partially hyperbolic dynamics. Journal of Modern Dynamics, 2014, 8 (3&4) : 549-576. doi: 10.3934/jmd.2014.8.549

[5]

Antonio Siconolfi, Gabriele Terrone. A metric proof of the converse Lyapunov theorem for semicontinuous multivalued dynamics. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4409-4427. doi: 10.3934/dcds.2012.32.4409

[6]

Matthias Rumberger. Lyapunov exponents on the orbit space. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 91-113. doi: 10.3934/dcds.2001.7.91

[7]

Edson de Faria, Pablo Guarino. Real bounds and Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1957-1982. doi: 10.3934/dcds.2016.36.1957

[8]

Andy Hammerlindl. Integrability and Lyapunov exponents. Journal of Modern Dynamics, 2011, 5 (1) : 107-122. doi: 10.3934/jmd.2011.5.107

[9]

Sebastian J. Schreiber. Expansion rates and Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 433-438. doi: 10.3934/dcds.1997.3.433

[10]

James Kingsbery, Alex Levin, Anatoly Preygel, Cesar E. Silva. Dynamics of the $p$-adic shift and applications. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 209-218. doi: 10.3934/dcds.2011.30.209

[11]

Chao Liang, Wenxiang Sun, Jiagang Yang. Some results on perturbations of Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4287-4305. doi: 10.3934/dcds.2012.32.4287

[12]

Shrihari Sridharan, Atma Ram Tiwari. The dependence of Lyapunov exponents of polynomials on their coefficients. Journal of Computational Dynamics, 2019, 6 (1) : 95-109. doi: 10.3934/jcd.2019004

[13]

Nguyen Dinh Cong, Thai Son Doan, Stefan Siegmund. On Lyapunov exponents of difference equations with random delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 861-874. doi: 10.3934/dcdsb.2015.20.861

[14]

Jianyu Chen. On essential coexistence of zero and nonzero Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4149-4170. doi: 10.3934/dcds.2012.32.4149

[15]

Paul L. Salceanu, H. L. Smith. Lyapunov exponents and persistence in discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 187-203. doi: 10.3934/dcdsb.2009.12.187

[16]

Luis Barreira, César Silva. Lyapunov exponents for continuous transformations and dimension theory. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 469-490. doi: 10.3934/dcds.2005.13.469

[17]

Fei Yu, Kang Zuo. Weierstrass filtration on Teichmüller curves and Lyapunov exponents. Journal of Modern Dynamics, 2013, 7 (2) : 209-237. doi: 10.3934/jmd.2013.7.209

[18]

Lucas Backes, Aaron Brown, Clark Butler. Continuity of Lyapunov exponents for cocycles with invariant holonomies. Journal of Modern Dynamics, 2018, 12: 223-260. doi: 10.3934/jmd.2018009

[19]

Alena Erchenko. Flexibility of Lyapunov exponents for expanding circle maps. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2325-2342. doi: 10.3934/dcds.2019098

[20]

Sigurdur Freyr Hafstein. A constructive converse Lyapunov theorem on exponential stability. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 657-678. doi: 10.3934/dcds.2004.10.657

2018 Impact Factor: 0.295

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]