Random matrix products and applications to cellular automata

Let λ be the upper Lyapunov exponent corresponding to a product of i.i.d. random m × m matrices (X_i)_i=0 ^∞ over C. Assume that the X_i 's are chosen from a finite set {D₀, D₁, ..., D_q-1} ⊆ M _m(ℂ), with P(X_i = D_j) > 0, and that the monoid generated by D₀, D₁,..., D_q-1 contains a matrix of rank 1. We obtain an explicit formula for A as a sum of a convergent series. We also consider the case where the X_i 's are chosen according to a Markov process and thus generalize a result of Lima and Rahibe [22]. Our results on A enable us to provide an approximation for the number N_≠0(F(x)ⁿ,r) of nonzero coefficients in F(x) ⁿ (mod r), where F(x) ∈ ℤ[x] and r ≥ 2. We prove the existence of and supply a formula for a constant α (< 1) such that N_≠0(F(x)ⁿ, r) ≈ n^α for "almost" every n.