The growth of iterates of multivariate generating functions

The vector-valued function $m(\theta)$ of a $p$-vector $\theta$ has components $m_1(\theta), m_2(\theta), \dots, m_p(\theta)$. For each $i$, $\exp(m_i(-\theta))$ is the (multivariate) Laplace transform of a discrete measure concentrated on $[0,\infty)^p$ with only a finite number of atoms. The main objective is to give conditions for the functional iterates $m^{(n)}$ of $m$ to grow like $\rho^n$ for a suitable $\rho>1$. The initial stimulus was provided by results of Miller and O'Sullivan (1992) on enumeration issues in `context free languages', results which can be improved using the theory developed here. The theory also allows certain results in Jones (2004) on multitype branching to be proved under significantly weaker conditions.