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Lyapunov Exponents pp 86–97Cite as

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

Chapter 1: Linear Random Dynamical Systems

Part of the Lecture Notes in Mathematics book series (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?

Keywords

  • Lyapunov Exponent
  • Invariant Measure
  • Exponential Dichotomy
  • Ergodic Measure
  • Positive Lyapunov Exponent

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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

    Oliver Knill

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  1. Oliver Knill
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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

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