Classification of recurrent domains for some holomorphic maps
The paper deals with the holomorphic map \(f: \mathbb{P}^2\to \mathbb{P}^2\) of degree \(d\geq 2\). In the first part of the paper the recurrent properties of the Fatou set are investigated. A Fatou component \(U\) is recurrent if for some \(p_0\in U\) the \(\omega\)-limit set of \(p_0\) intersects \(U\) and a Fatou component \(U\) is a Siegel domain if there exists a subsequence \(\{f^{n_i}\}\) that converges uniformly to the identity map \(\text{id}: U\to U\) on compact sets. The authors study three situations which can happen for recurrent Fatou components \(U\): (i) \(U\) contains a fixed attractive point; (ii) \(U\) contains an \(f\)-invariant Riemann surface; (iii) \(U\) is a Siegel domain. In the second part of the paper the recurrent Fatou components for Hénon mappings in \(\mathbb{P}^2\) are studied. The mapping \(f: \mathbb{C}^2\to \mathbb{C}^2\) is a Hénon map if it is a finite composition of maps of the following type: \(f_j (z,w)= (p_j (z)- aw, z)\), where \(p_j\) is a polynomial of degree \(d_j\geq 2\). The analogous properties as (i)--(iii) above are investigated. This interesting paper is well written and organized.
scientific article; zbMATH DE number 759295