  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.
