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The upper Lyapunov exponent of Sl(2,R) cocycles: Discontinuity and the problem of positivity

  • Chapter 1: Linear Random Dynamical Systems
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Lyapunov Exponents

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1486))

Abstract

Let T be an aperiodic automorphism of a standard probability space (X,m). Let P be the subset of A=L ∞(X, Sl(2, R)) where the upper Lyapunov exponent is positive almost everywhere.

We prove that the set P∖int(P) is not empty. So, there are always points in A where the Lyapunov exponents are discontinuous.

We show further that the decision whether a given cocycle is in P is at least as hard as the following cohomology problem: Can a given measurable set Z ⊂ X be represented as YΔT(Y) for a measurable set Y ⊂ X?

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Authors and Affiliations

  1. Mathematikdepartement, ETH Zentrum, CH-8092, Zürich

    Oliver Knill

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  1. Oliver Knill
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Editor information

Ludwig Arnold Hans Crauel Jean-Pierre Eckmann

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© 1991 Springer-Verlag

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Knill, O. (1991). The upper Lyapunov exponent of Sl(2,R) cocycles: Discontinuity and the problem of positivity. In: Arnold, L., Crauel, H., Eckmann, JP. (eds) Lyapunov Exponents. Lecture Notes in Mathematics, vol 1486. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086660

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  • DOI: https://doi.org/10.1007/BFb0086660

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-54662-7

  • Online ISBN: 978-3-540-46431-0

  • eBook Packages: Springer Book Archive

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