Totally expanding multiplicative systems

A single-matrix multiplicative system consists of an N × N nonnegative matrix Q and an N × 1 semi-positive vector x(0). This system is said to be totally expanding if each entry of the sequence {Qnx(0)}n=0,1, … is unbounded. A multiple-matrix multiplicative system replaces Q by a set {Qδ:δ ∈ D} of N × N nonnegative matrices, where D is in “product form,” and is said to be totally expanding if for every δ in D each entry of the sequence {(Qδ)nx(0)}n=0,1, … is unbounded. Each of these systems is shown to be totally expanding if and only if it has no “degenerate” coordinates and a particular set of linear inequalities has a solution. These sets of linear inequalities can also be used to approximate the smallest coordinate-dependent growth rate of the output of the respective system.