Approximating Lyapunov Exponents and Stationary Measures

  • Alexandre BaravieraEmail author
  • Pedro Duarte


We give a new proof of E. Le Page’s theorem on the Hölder continuity of the first Lyapunov exponent in the class of irreducible Bernoulli cocycles. This suggests an algorithm to approximate the first Lyapunov exponent, as well as the stationary measure, for such random cocycles.


Lyapunov exponent Random cocycle Stationary measure 

Mathematics Subject Classification

37H15 37D25 



The first author was partially supported by CNPq through the Project 312698/2013-5. The second author was partially supported by Fundação para a Ciência e a Tecnologia through the strategic Project PEst-OE/MAT/UI0209/2013.


  1. 1.
    Bougerol, P.: Théorèmes limite pour les systèmes linéaires à coefficients Markoviens. Probab. Theory Rel. Fields 78(2), 193–221 (1988)CrossRefGoogle Scholar
  2. 2.
    Duarte, P., Klein, S.: Lyapunov Exponents of Linear Cocycles; Continuity via Large Deviations. Atlantis Studies in Dynamical Systems, vol. 3. Atlantis Press, Paris (2016)CrossRefGoogle Scholar
  3. 3.
    Furstenberg, H., Kesten, H.: Products of random matrices. Ann. Math. Stat. 31, 457–469 (1960)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Furstenberg, H.: Noncommuting random products. Trans. Am. Math. Soc. 108, 377–428 (1963)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Galatolo, S., Monge, M., Nisoli, I.: Rigorous approximation of stationary measures and convergence to equilibrium for iterated function systems. J. Phys. A 49(27), 274001 (2016). 22MathSciNetCrossRefGoogle Scholar
  6. 6.
    Galatolo, S., Nisoli, I.: An elementary approach to rigorous approximation of invariant measures. SIAM J. Appl. Dyn. Syst. 13(2), 958–985 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hennion, H., Hervé, L.: Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-compactness. Lecture Notes in Mathematics, vol. 1766. Springer, Berlin (2001)zbMATHGoogle Scholar
  8. 8.
    Herman, M.-R.: Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnol\(\prime \)d et de Moser sur le tore de dimension \(2\). Comment. Math. Helv. 58(3), 453–502 (1983)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Le Page, É.: Régularité du plus grand exposant caractéristique des produits de matrices aléatoires indépendantes et applications. Ann. Inst. Henri Poincaré Probab. Stat. 25(2), 109–142 (1989)zbMATHGoogle Scholar
  10. 10.
    Peres, Y.: Analytic dependence of Lyapunov exponents on transition probabilities. In: Arnold, L., Crauel, H., Eckmann, J.P. (eds.) Lyapunov Exponents (Oberwolfach, 1990). Lecture Notes in Mathematics, vol. 1486, pp. 64–80. Springer, Berlin (1991)CrossRefGoogle Scholar
  11. 11.
    Pollicott, M.: Maximal Lyapunov exponents for random matrix products. Invent. Math. 181(1), 209–226 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Riesz, F., Sz.-Nagy, B.: Functional Analysis. Frederick Ungar Publishing Co., New York (1955) (translated by Leo F. Boron)Google Scholar
  13. 13.
    Ruelle, D.: Analycity properties of the characteristic exponents of random matrix products. Adv. Math. 32(1), 68–80 (1979)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Simon, B., Taylor, M.: Harmonic analysis on \({\rm SL}(2,{ R})\) and smoothness of the density of states in the one-dimensional Anderson model. Commun. Math. Phys. 101(1), 1–19 (1985)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Sternberg, S.: Lectures on Differential Geometry. Prentice-Hall Inc., Englewood Cliffs, NJ (1964)zbMATHGoogle Scholar
  16. 16.
    Walters, P.: An Introduction to Ergodic Theory. Graduate Texts in Mathematics. Springer, New York (2000)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Instituto de Matemática e Estatística-UFRGSPorto AlegreBrazil
  2. 2.Centro de Matemática, Aplicações Fundamentais e Investigação Operacional, Faculdade de CiênciasUniversidade de LisboaLisbonPortugal

Personalised recommendations