Nonwandering, nonrecurrent Fatou components in \({\mathbb{P}}^2\).
Let \(f\) be a holomorphic endomorphism of the projective plane \(\mathbb{P}^2\) and \(\Omega\) be a fixed Fatou component. We assume that \(\Omega\) is nonrecurrent, i.e., the orbit of its points go to the boundary of \(\Omega\). Let \(h\) be a limit of the sequence of the iterates of \(f\) on \(\Omega\) and note \(\Sigma\) the set \(h(\Omega)\). The author shows that \(\Sigma\) is either a fixed-point or a hyperbolic Riemann surface. In the latter case, the restriction of \(f\) on \(\Sigma\) can be lifted to an holomorphic map \(g\) on the unit disc \(D\), and there are two possibilities. Either \(g\) is an irrational rotation of \(D\), or the sequence of iterates \(g^n\) converges to the boundary of \(D\). The author also proves the following independant result : the basins of attraction of holomorphic endomorphisms of \(\mathbb{P}^k\) are taut, i.e., the family of holomorphic discs (into such a basin) is normal.
scientific article; zbMATH DE number 2101117