Remarks on Automorphisms of ${\mathbb{C}}^{\ast }\times {\mathbb{C}}^{\ast }$ and their basins

We study basins of attraction of automorphisms of ${\text{CC}}^2$ tangent to the identity that fix both axes. Our main result is that, if a well known conjecture about automorphisms of ${\text{CC}}^{\ast }\times {\text{CC}}^{\ast }$ holds, then there are no basins of attraction associated to the non-degenerate characteristic directions (in the sense of Hakim), and therefore we cannot find a Fatou-Bieberbach domain that does not intersect both axis with this method.