Efficient algorithms for deciding the type of growth of products of integer matrices

For a given finite set Σ of matrices with nonnegative integer entries we study the growth with t ofmax{‖A1⋯At‖:Ai∈Σ}.max{‖A1⋯At‖:Ai∈Σ}.We show how to determine in polynomial time whether this growth is bounded, polynomial, or exponential, and we characterize all possible behaviors.