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We study basins of attraction of automorphisms of $\CC^2$ tangent to theidentity that fix both axes. Our main result is that, if a well knownconjecture about automorphisms of $\CC^*\times\CC^*$ holds, then there are nobasins of attraction associated to the non-degenerate characteristic directions(in the sense of Hakim), and therefore we cannot find a Fatou-Bieberbach domainthat does not intersect both axis with this method.

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