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Continuity of Lyapunov exponents for non-uniformly fiber-bunched cocycles

Part of: Random dynamical systems Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  02 December 2020

CATALINA FREIJO  and
KARINA MARIN
CATALINA FREIJO
Affiliation:
Departamento de Matemática, Instituto de Ciências Exatas (ICEx), Universidade Federal de Minas Gerais, Belo Horizonte, Minas Gerais, Brazil (e-mail: catalinafreijo@gmail.com)
KARINA MARIN*
Affiliation:
Departamento de Matemática, Instituto de Ciências Exatas (ICEx), Universidade Federal de Minas Gerais, Belo Horizonte, Minas Gerais, Brazil (e-mail: catalinafreijo@gmail.com)
*
e-mail: kmarin@mat.ufmg.br
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Abstract

We provide conditions that imply the continuity of the Lyapunov exponents for non-uniformly fiber-bunched cocycles in $SL(2,\mathbb {R})$ . The main theorem is an extension of the result of Backes, Brown and Butler and gives a partial answer to a conjecture of Marcelo Viana.

Keywords

Lyapunov exponentslinear cocyclesnon-uniformly fiber-bunched

MSC classification

Primary: 37H15: Multiplicative ergodic theory, Lyapunov exponents
Secondary: 37D30: Partially hyperbolic systems and dominated splittings 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 12 , December 2021 , pp. 3740 - 3767
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Avila, A., Eskin, A. and Viana, M.. Continuity of Lyapunov exponents of random matrix products. In preparation.Google Scholar
Avila, A., Santamaria, J. and Viana, M.. Holonomy invariance: rough regularity and applications to Lyapunov exponents. Astérisque 358 (2013), 13–74.Google Scholar
Avila, A. and Viana, M.. Extremal Lyapunov exponents: an invariance principle and applications. Invent. Math. 181(1) (2010), 115–178.CrossRefGoogle Scholar
Backes, L., Brown, A. and Butler, C.. Continuity of Lyapunov exponents for cocycles with invariant holonomies. J. Mod. Dyn. 12 (2018), 223–260.10.3934/jmd.2018009CrossRefGoogle Scholar
Bochi, J.. Genericity of zero Lyapunov exponents. Ergod. Th. & Dynam. Sys. 22(6) (2002), 1667–1696.CrossRefGoogle Scholar
Bocker-Neto, C. and Viana, M.. Continuity of Lyapunov exponents for random $2d$ matrices. Ergod. Th. & Dynam. Sys. 37(5) (2017), 1413–1442.CrossRefGoogle Scholar
Bonatti, C., Gómez-Mont, X. and Viana, M.. Généricité d’exposants de Lyapunov non-nuls pour des produits déterministes de matrices. Ann. Inst. H. Poincaré Anal. Non Linéaire 20(4) (2003), 579–624.CrossRefGoogle Scholar
Bonatti, C. and Viana, M.. Lyapunov exponents with multiplicity 1 for deterministic products of matrices. Ergod. Th. & Dynam. Sys. 24(5) (2004), 1295–1330.CrossRefGoogle Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lectures Notes in Mathematics, 470). Springer, Berlin, 1975, pp. 90–107.10.1007/BFb0081284CrossRefGoogle Scholar
Butler, C.. Discontinuity of Lyapunov exponents near fiber-bunched cocycles. Ergod. Th. & Dynam. Sys. 38(2) (2018), 523–539.CrossRefGoogle Scholar
Furstenberg, H.. Non-commuting random products. Trans. Amer. Math. Soc. 108 (1963), 377–428.CrossRefGoogle Scholar
Furstenberg, H. and Kesten, H.. Products of random matrices. Ann. Math. 31(2) (1960), 457–469.Google Scholar
Kalinin, B.. Livs̆ic theorem for matrix cocycles. Ann. Math. 173 (2011), 1025–1042.CrossRefGoogle Scholar
Kalinin, B. and Sadovskaya, V.. Holonomy and cohomology for cocycles over partially hyperbolic diffeomorphisms. Discrete Contin. Dyn. Syst. 36 (2016), 245–259.CrossRefGoogle Scholar
Ledrappier, F.. Positivity of the exponent for stationary sequences of matrices. Lyapunov Exponents. Vol. 1186. Springer, Berlin, 1986, pp. 56–73.CrossRefGoogle Scholar
Malheiro, E. C. and Viana, M.. Lyapunov exponents of linear cocycles over Markov shifts. Stoch. Dyn. 15 (2015), 1–27.CrossRefGoogle Scholar
Oseledets, V. I.. A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19 (1968), 197–231.Google Scholar
Pesin, Ya.. Characteristic Lyapunov exponents and smooth ergodic theory. Uspekhi Mat. Nauk 32 (1977), 55–114.Google Scholar
Poletti, M.. Stably positive Lyapunov exponents for symplectic linear cocycles over partially hyperbolic diffeomorphisms. Discrete Contin. Dyn. Syst. 38 (2018), 5163–5188.CrossRefGoogle Scholar
Rokhlin, V. A.. On the fundamental ideas of measure theory. Mat. Sb. 67(1) (1949), 107–150.Google Scholar
Tahzibi, A. and Yang, J.. Invariance Principle and rigidity of high entropy measures. Trans. Amer. Math. Soc. 371 (2019), 1231–1251.CrossRefGoogle Scholar
Tall, E. Y. and Viana, M.. Moduli of continuity for the Lyapunov exponents of random $\textsf{GL}(2)$ -cocycles. Trans. Amer. Math. Soc. 373 (2020), 1343–1383.CrossRefGoogle Scholar
Viana, M.. Almost all cocycles over any hyperbolic system have non-vanishing exponents. Ann. Math. 167 (2008), 643–680.CrossRefGoogle Scholar
Viana, M.. Lectures on Lyapunov Exponents. Vol. 145. Cambridge University Press, Cambridge, 2014.CrossRefGoogle Scholar
Viana, M.. (Dis)continuity of Lyapunov exponents. Ergod. Th. & Dynam. Sys. 40 (2018), 1–35.Google Scholar
Viana, M. and Yang, J.. Continuity of Lyapunov exponents in the ${C}^0$ topology. Israel J. Math. 229(1) (2019), 461–485.CrossRefGoogle Scholar