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

Abstract

For a given finite set $Σ$ of matrices with nonnegative integer entries we study the growth of $$ \max_t(Σ) = \max\{\|A_{1}... A_{t}\|: A_i \in Σ\}.$$ We show how to determine in polynomial time whether the growth with $t$ is bounded, polynomial, or exponential, and we characterize precisely all possible behaviors.