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Analysis establishes a dichotomy in matrix valued cocycles over Bernoulli or Markov shifts, indicating finite Lyapunov exponents.
In this paper we establish a Bochi-Ma\~n\'e type dichotomy in the space of two dimensional, nonnegative determinant matrix valued, locally constant linear cocycles over a Bernoulli or Markov shift. Moreover, we prove that Lebesgue almost every such cocycle has finite first Lyapunov exponent, which then implies a break in the regularity of the Lyapunov exponent, from analyticity to discontinuity.
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Duarte et al. (2025) studied this question.
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