Abstract
We consider a random product of two-by-two matrices of determinant one over an abstract dynamical system. When the two Lyapunov exponents are distinct, Oseledets’ theorem asserts that the matrix cocycle is cohomologous to a diagonal matrix cocycle. When they are equal, we show that the cocycle is conjugate to one of three cases: a rotation matrix cocycle, an upper triangular matrix cocycle, or a diagonal matrix cocycle modulo a rotation by π/2.
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Thieullen, P. Ergodic reduction of random products of two-by-two matrices. J. Anal. Math. 73, 19–64 (1997). https://doi.org/10.1007/BF02788137
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DOI: https://doi.org/10.1007/BF02788137