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A formula with some applications to the theory of Lyapunov exponents

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Abstract

We prove an elementary formula about the average expansion of certain products of 2 by 2 matrices. This permits us to quickly re-obtain an inequality by M. Herman and a theorem by Dedieu and Shub, both concerning Lyapunov exponents. Indeed, we show that equality holds in Herman’s result. Finally, we give a result about the growth of the spectral radius of products.

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References

  1. J. Bochi,Genericity of zero Lyapunov exponents, Preprint 2000, available at www.preprint.impa.br.

  2. K. Burns, C. Pugh, M. Shub and A. Wilkinson,Recent results about stable ergodicity, inThe Seattle Conference of Smooth Ergodic Theory, Proceedings on Symposia in Pure Mathematics, to appear.

  3. J. E. Cohen,Open problems — The spectral radius of a product of random matrices, Contemporary Mathematics50 (1986), 329–330.

    Google Scholar 

  4. J.-P. Dedieu and M. Shub,On random and mean exponents for unitarily invariant probability measures on \(\mathbb{G}\mathbb{L}_n (\mathbb{C})\), Preprint 2001, available at www.research.ibm.com/people/s/shub.

  5. H. Furstenberg and H. Kesten,Products of random matrices, Annals of Mathematical Statistics31 (1960), 457–469.

    MathSciNet  MATH  Google Scholar 

  6. M. R. Herman,Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnold et de Moser sur le tore de dimension 2, Commentarii Mathematici Helvetici58 (1983), 453–502.

    Article  MATH  MathSciNet  Google Scholar 

  7. O. Knill,Positive Lyapunov exponents for a dense set of bounded measurable SL(2,R)-cocycles, Ergodic Theory and Dynamical Systems12 (1992), 319–331.

    Article  MATH  MathSciNet  Google Scholar 

  8. F. Ledrappier,Quelques propriétés des exposants caractéristiques, Lecture Notes in Mathematics1097, Springer-Verlag, Berlin, 1982, pp. 305–396.

    Google Scholar 

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Authors and Affiliations

  1. Collège de France, 3 rue d’Ulm, 75005, Paris, France

    Artur Avila

  2. IMPA, Estr. D. Castorina 110, 22460-320, Rio de Janeiro, Brazil

    Jairo Bochi

Authors
  1. Artur Avila
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  2. Jairo Bochi
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Corresponding author

Correspondence to Artur Avila.

Additional information

Financial support from Pronex-Dynamical Systems, CNPq 001/2000 and from Faperj is gratefully acknowledged.

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Avila, A., Bochi, J. A formula with some applications to the theory of Lyapunov exponents. Isr. J. Math. 131, 125–137 (2002). https://doi.org/10.1007/BF02785853

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  • DOI: https://doi.org/10.1007/BF02785853

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