Computation of Lyapunov exponents of matrix products

For m given square matrices A₀,A₁,…,A_m−1 (m≥2), one of which is assumed to be of rank 1, and for a given sequence (ω_n) in {0,1,…,m−1}^N, the following limit, if it exists, [Formula presented] defines the Lyapunov exponent of the sequence of matrices (A_ωn)_n≥0. It is proven that the Lyapunov exponent L(ω) has a closed-form expression under certain conditions. One special case arises when A_j’s are non-negative and ω is generic with respect to some shift-invariant measure; a second special case occurs when A_j’s (for 1≤j<m) are invertible and ω is a typical point with respect to some shift-ergodic measure. Substitutive sequences and characteristic sequences of B-free integers are considered as examples. An application is presented for the computation of multifractal spectrum of weighted Birkhoff averages.