Periodic cycles of attracting Fatou components of type \(\mathbb{C}\times(\mathbb{C}^*)^{d-1}\) in automorphisms of \(\mathbb{C}^d\) (Q2035973)

The author generalizes the results obtained by \textit{F. Bracci} et al. [J. Eur. Math. Soc. (JEMS) 23, No. 2, 639--666 (2021; Zbl 1469.37036)]. In particular, he constructs automorphisms of \(\mathbb{C}^d\) having an arbitrary finite number of non-recurrent Fatou components, each biholomorphic to \(\mathbb{C}\times(\mathbb{C}^*)^{d-1}\) and such that all orbits in each component converge to a common boundary fixed point. The automorphisms can be chosen such that each Fatou component is invariant or such that the Fatou components are grouped into periodic cycles of any common period. The author further proves that in such attracting Fatou components no orbit can converge tangentially to a complex submanifold, and that every stable orbit near the fixed point is contained either in these attracting components or in one of \(d\) invariant hypersurfaces tangent to each coordinate hyperplane where the automorphism acts as an irrational rotation.