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

  • Oliver Knill
Chapter 1: Linear Random Dynamical Systems
Part of the Lecture Notes in Mathematics book series (LNM, volume 1486)


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?


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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Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Oliver Knill
    • 1
  1. 1.MathematikdepartementETH ZentrumZürich

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