  We study the dynamics of a family of non cohomologically hyperbolic
automorphisms $f$ of $\mathbb{C}^3$. We construct a compactification $X$ of
$\mathbb{C}^3$ where their extensions are algebraically stable. We finally
construct canonical invariant closed positive $(1,1)$-currents for $f^*$, $f_*$
and we study several of their properties. Moreover, we study the well defined
current $T_f \wedge T_{f^{-1}}$ and the dynamics of $f$ on its support. Then we
construct an invariant positive measure $T_f \wedge T_{f^{-1}}\wedge
\phi_{\infty}$, where $\phi_{\infty}$ is a function defined on the support of
$T_f \wedge T_{f^{-1}}$. We prove that the support of this measure is compact
and pluripolar. We prove also that this measure is canonical, in some sense
that will be precised.
