Characteristic Points of Recursive Systems

Abstract

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 $\rho$ be the radius of convergence of the solution power series and let $\pmb{\tau}$ be the values of the solution series evaluated at $\rho$. The main results of the paper include: (a) the set of characteristic points form an antichain in ${\mathbb R}^{m+1}$, (b) given a characteristic point $(a,\mathbf{b})$, (i) the spectral radius of the Jacobian of $\pmb \gamma$ at $(a, \mathbf{b})$ is $\ge 1$, and (ii) it is $=1$ iff $(a,\mathbf{b}) = (\rho,\pmb{\tau})$, (c) if $(\rho,\pmb{\tau})$ is a characteristic point, then (i) $\rho$ is the largest $a$ for $(a,\mathbf{b})$ a characteristic point, and (ii) a characteristic point $(a,\mathbf{b})$ with $a=\rho$ is the extreme point $(\rho,\pmb{\tau})$.