We investigate the following question: Is there a biholomorphic map from C2 into the set {zw ≠ 0}?;In order to answer this question we study the construction of Fatou-Bieberbach Domains for maps tangent to the Identity, and we prove that it is not possible to have a Fatou-Bieberbach domain that avoid both axes as the basin of an automorphisms of C2 along non-degenerate characteristic directions, for a large class of automorphisms of C2 that fixes both axes.;We find new examples of Fatou-Bieberbach domains as basins of attractions of automorphisms tangent to the identity along degenerate characteristic directions. Using a specific map we find a Fatou-Bieberbach domain that avoids one axis and a complex curve Gamma tangent to the other axis to an arbitrarily high degree.
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