Characteristic Points of Recursive Systems

Characteristic points have been a primary tool in the study of a generating function defined by a single recursive equation. We investigate the proper way to adapt this tool when working with multi-equation recursive systems. Given an irreducible non-negative power series system with m equations, let ρ be the radius of convergence of the solution power series and let τ be the values of the solution series evaluated at ρ. The main results of the paper include: (a) the set of characteristic points form an antichain in Rm+1, (b) given a characteristic point (a,b), (i) the spectral radius of the Jacobian of G at (a,b) is > 1, and (ii) it is = 1 iff (a,b) = (ρ,τ ), (c) if (ρ,τ ) is a characteristic point, then (i) ρ is the largest a for (a,b) a characteristic point, and (ii) a characteristic point (a,b) with a = ρ is the extreme point (ρ,τ ).