Attracting basins for automorphisms of \(\mathbb{C}^2\) (Q1127755)

The paper deals with the interesting question whether an automorphism of \(\mathbb{C}^n\) tangent to the identity can have a domain of attraction, and if so, whether that domain is biholomorphic to \(\mathbb{C}^n\). It is obvious that this cannot happen when \(n=1\), due to the fact that the automorphism group of \(\mathbb{C}\) is the group of affine mappings \(z\mapsto az+b\), \(a\neq 0\). The main result of the paper states that there exist an automorphism \(\mathbb{C}^2\) tangent to the identity with an invariant domain of attraction to the origin, biholomorphic to \(\mathbb{C}^2\), on which the automorphism is biholomorphic conjugate to the map \((x,y)\mapsto(x-1,y)\).