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

Abstract

We generalise a recent example by F. Bracci, J. Raissy and B. Stensønes to construct automorphisms of \({\mathbb {C}}^{d}\) admitting an arbitrary finite number of non-recurrent Fatou components, each biholomorphic to \({\mathbb {C}}\times ({\mathbb {C}}^{*})^{d-1}\) and all attracting to a common boundary fixed point. These automorphisms can be chosen such that each Fatou component is invariant or such that the components are grouped into periodic cycles of any common period. We further show that no orbit in these attracting Fatou components can converge tangent 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 on which the automorphism acts as an irrational rotation.