Exact Lyapunov exponent for infinite products of random matrices

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Abstract
Despite significant work since the original paper by H Furstenberg(1963), explicit formulae for Lyapunov exponents of infinite products of random matrices are available only in a very few cases. In this work, we give a rigorous explicit formula for the Lyapunov exponent for some binary infinite products of random 2*2 real matrices. All these products are constructed using only two types of matrices, A and B, which are chosen according to a stochastic process. The matrix A is singular, namely its determinant is zero. This formula is derived by using a particular decomposition for the matrix B, which allows us to write the Lyapunov exponent as a sum of convergent series. The key point is the computation of all the integer powers of B, which is achieved by a suitable change of frame. The computation then follows by looking at each of the special types of B (hyperbolic, parabolic and elliptic). Finally, we show, with an example, that the Lyapunov exponent is a discontinuous function of the given parameter.
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