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Attracting basins for automorphisms of ${ℂ}^{2}$. (English) Zbl 0932.37028
The paper deals with the interesting question whether an automorphism of ${ℂ}^{n}$ tangent to the identity can have a domain of attraction, and if so, whether that domain is biholomorphic to ${ℂ}^{n}$. It is obvious that this cannot happen when $n=1$, due to the fact that the automorphism group of $ℂ$ is the group of affine mappings $z↦az+b$, $a\ne 0$. The main result of the paper states that there exist an automorphism ${ℂ}^{2}$ tangent to the identity with an invariant domain of attraction to the origin, biholomorphic to ${ℂ}^{2}$, on which the automorphism is biholomorphic conjugate to the map $\left(x,y\right)↦\left(x-1,y\right)$.
MSC:
 37F10 Polynomials; rational maps; entire and meromorphic functions 37F15 Expanding maps; hyperbolicity; structural stability