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Continuity of Lyapunov exponents for random two-dimensional matrices

Published online by Cambridge University Press:  08 March 2016

CARLOS BOCKER-NETO  and
MARCELO VIANA
CARLOS BOCKER-NETO
Affiliation:
Departamento de Matemática, Universidade Federal da Paraíba, Cidade Universitária, 58051-900 João Pessoa, PB, Brazil email carlos@mat.ufpb.br
MARCELO VIANA
Affiliation:
IMPA, Est. D. Castorina 110, Jardim Botânico, 22460-320 Rio de Janeiro, RJ, Brazil email viana@impa.br
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Abstract

The Lyapunov exponents of locally constant $\text{GL}(2,\mathbb{C})$-cocycles over Bernoulli shifts vary continuously with the cocycle and the invariant probability measure.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 5 , August 2017 , pp. 1413 - 1442
Copyright
© Cambridge University Press, 2016 

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References

Arbieto, A. and Bochi, J.. L p -generic cocycles have one-point Lyapunov spectrum. Stoch. Dyn. 3 (2003), 73–81.Google Scholar
Arnold, L. and Cong, N. D.. On the simplicity of the Lyapunov spectrum of products of random matrices. Ergod. Th. & Dynam. Sys. 17 (1997), 1005–1025.Google Scholar
Avila, A. and Viana, M.. Extremal Lyapunov exponents: an invariance principle and applications. Invent. Math. 181(1) (2010), 115–189.CrossRefGoogle Scholar
Bochi, J.. Discontinuity of the Lyapunov exponents for non-hyperbolic cocycles. Preprint, http://www.mat.uc.cl/∼jairo.bochi/docs/discont.pdf.Google Scholar
Bochi, J.. Genericity of zero Lyapunov exponents. Ergod. Th. & Dynam. Sys. 22 (2002), 1667–1696.Google Scholar
Bochi, J.. C 1 -generic symplectic diffeomorphisms: partial hyperbolicity and zero centre Lyapunov exponents. J. Inst. Math. Jussieu 8 (2009), 49–93.Google Scholar
Bochi, J. and Viana, M.. The Lyapunov exponents of generic volume-preserving and symplectic maps. Ann. of Math. (2) 161 (2005), 1423–1485.Google Scholar
Bourgain, J.. Positivity and continuity of the Lyapounov exponent for shifts on T d with arbitrary frequency vector and real analytic potential. J. Anal. Math. 96 (2005), 313–355.CrossRefGoogle Scholar
Bourgain, J. and Jitomirskaya, S.. Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential. J. Stat. Phys. 108 (2002), 1203–1218.CrossRefGoogle Scholar
Castaing, C. and Valadier, M.. Convex Analysis and Measurable Multifunctions (Lecture Notes in Mathematics, 580) . Springer, Heidelberg, 1977.Google Scholar
Furstenberg, H. and Kesten, H.. Products of random matrices. Ann. Math. Statist. 31 (1960), 457–469.Google Scholar
Furstenberg, H. and Kifer, Yu.. Random matrix products and measures in projective spaces. Israel J. Math. 10 (1983), 12–32.Google Scholar
Gol’dsheid, I. Ya. and Margulis, G. A.. Lyapunov indices of a product of random matrices. Uspekhi Mat. Nauk 44 (1989), 13–60.Google Scholar
Guivarc’h, Y. and Raugi, A.. Products of random matrices: convergence theorems. Contemp. Math. 50 (1986), 31–54.Google Scholar
Hennion, H.. Loi des grands nombres et perturbations pour des produits réductibles de matrices aléatoires indépendantes. Z. Wahrsch. Verw. Gebiete 67 (1984), 265–278.Google Scholar
Itō, K.. Introduction to Probability Theory. Cambridge University Press, Cambridge, 1984.Google Scholar
Johnson, R.. Lyapounov numbers for the almost periodic Schrödinger equation. Illinois J. Math. 28 (1984), 397–419.Google Scholar
Kifer, Yu.. Perturbations of random matrix products. Z. Wahrsch. Verw. Gebiete 61 (1982), 83–95.Google Scholar
Kifer, Yu. and Slud, E.. Perturbations of random matrix products in a reducible case. Ergod. Th. & Dynam. Sys. 2 (1983), 367–382, 1982.Google Scholar
Kingman, J.. The ergodic theory of subadditive stochastic processes. J. Roy. Statist. Soc. 30 (1968), 499–510.Google Scholar
Knill, O.. The upper Lyapunov exponent of SL(2, R) cocycles: discontinuity and the problem of positivity. Lyapunov Exponents (Oberwolfach, 1990) (Lecture Notes in Mathematics, 1486) . Springer, Heidelberg, 1991, pp. 86–97.Google Scholar
Knill, O.. Positive Lyapunov exponents for a dense set of bounded measurable SL (2, R)-cocycles. Ergod. Th. & Dynam. Sys. 12 (1992), 319–331.CrossRefGoogle Scholar
Mañé, R.. Oseledec’s theorem from the generic viewpoint. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983). PWN Publishers, Warsaw, 1984, pp. 1269–1276.Google Scholar
Oseledets, V. I.. A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19 (1968), 197–231.Google Scholar
Le Page, É.. Théorèmes limites pour les produits de matrices aléatoires. Probability Measures on Groups (Oberwolfach, 1981) (Lecture Notes in Mathematics, 928) . Springer, Heidelberg, 1982, pp. 258–303.Google Scholar
Le Page, É.. Régularité du plus grand exposant caractéristique des produits de matrices aléatoires indépendantes et applications. Ann. Inst. H. Poincaré Probab. Stat. 25 (1989), 109–142.Google Scholar
Peres, Y.. Analytic dependence of Lyapunov exponents on transition probabilities. Lyapunov Exponents (Oberwolfach, 1990) (Lecture Notes in Mathematics, 1486) . Springer, Heidelberg, 1991, pp. 64–80.Google Scholar
Ruelle, D.. Analyticity properties of the characteristic exponents of random matrix products. Adv. Math. 32 (1979), 68–80.Google Scholar
Simon, B. and Taylor, M.. Harmonic analysis on SL(2, R) and smoothness of the density of states in the one-dimensional Anderson model. Comm. Math. Phys. 101 (1985), 1–19.Google Scholar
Viana, M.. Lyapunov exponents of Teichmüller flows. Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow (Fields Institute Communications, 51) . American Mathematical Society, 2007, pp. 139–201.Google Scholar
Viana, M.. Lectures on Lyapunov Exponents. Cambridge University Press, Cambridge, 2014.CrossRefGoogle Scholar
Viana, M. and Oliveira, K.. Foundations of Ergodic Theory. Cambridge University Press, Cambridge, 2015.CrossRefGoogle Scholar
Young, L.-S.. Random perturbations of matrix cocycles. Ergod. Th. & Dynam. Sys. 6 (1986), 627–637.Google Scholar