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We prove that there exists an automorphism of ℂ2 tangent to the identity with a domain of attraction D to the origin, biholomorphic to ℂ2, along a degenerate characteristic direction. Our automorphism of ℂ2 is conjugate to a translation in D. We also prove the existence of a curve Γ, a biholomorphic copy of ℂ, entirely contained in the boundary of D. In our construction Γ is tangent to the z-axis in a neighborhood of the origin. The automorphisms we construct also fix the w-axis; therefore we obtain D, a Fatou–Bieberbach domain that does not intersect two biholomorphic copies of ℂ locally transversal at the origin.

Fatou–Bieberbach Domains as Basins of Attraction of Automorphisms Tangent to the Identity