Attracting domains of maps tangent to the identity whose only characteristic direction is non-degenerate (Q2863013)

Let \(f\) be a holomorphic map of \(\mathbb{C}^2\) tangent to the identity at the origin, i.e. \(f(0)=0\) and \(df(0)=id\). In local coordinates \((z,w)\) centered at 0, let \((P_k(z,w), Q_k(z,w))\), \(k\geq 2\), be the leading nonlinear term in the homogeneous expansion of \(f\). A direction \([v]=[z:w]\) is called a ``non-degenerate characteristic direction'' for \(f\) if \((P_k(z,w),Q_k(z,w))=\lambda(z,w)\) for some \(\lambda\neq 0\). \textit{M. Hakim} [Duke Math. J. 92, No. 2, 403--428 (1998; Zbl 0952.32012)] defined a ``director'' \(\alpha\) associated to each non-degenerate characteristic direction and showed the existence of attracting domains when \(\mathrm{Re\,}\alpha>0\). NEWLINENEWLINENEWLINENEWLINE In the paper under review, the author studies a special family of maps tangent to the identity, whose only characteristic direction is non-degenerate. For these maps, the director associated to the unique non-degenerate characteristic direction is 0. The author shows the existence of attracting domains for this family. Furthermore, she shows the existence of a Fatou-Bieberbach domain when the special family comes from automorphisms of \(\mathbb{C}^2\).NEWLINENEWLINEIn [Int. J. Math. 25, No. 1, Article ID 1450003, 10 p. (2014; Zbl 1288.32024)], the reviewer defined the ``dicritical order'' \(\nu\) associated to each non-degenerate characteristic direction and showed the existence of attracting domains when \(\nu\geq 1\). We note that the dicritical order associated to the unique non-degenerate characteristic direction of the maps studied by the author is \(k-1\geq 1\).