Dilatation of outer automorphisms of Right-angled Artin Groups

We study the dilatation of outer automorphisms of right-angled Artin groups. Given a right-angled Artin group defined on a simplicial graph: $A(\Gamma) = \langle V | E \rangle$ and an automorphism $\phi \in Out(A(\Gamma))$ there is a natural measure of how fast the length of a word of $A(\Gamma)$ grows after $n$ iterations of $\phi$ as a function of $n$, which we call the dilatation of $w$ under $\phi$. We define the dilatation of $\phi$ as the supremum over dilatations of all $w \in A(\Gamma)$. Assuming that $\phi$ is a pure and square map, we show that if the dilatation of $\phi$ is positive, then either there exists a free abelian special subgroup on which that dilatation is realized; or there exists a strata of either free or free abelian groups on which the dilatation is realized.