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

## TL;DR

This paper constructs automorphisms of complex spaces with multiple attracting Fatou components arranged in cycles, each biholomorphic to a product space, and analyzes their dynamical properties near a fixed point.

## Contribution

It generalizes previous examples to create automorphisms with arbitrary finite cycles of Fatou components of a specific biholomorphic type, including invariant and periodic arrangements.

## Key findings

- Automorphisms admit any finite number of non-recurrent Fatou components.
- Each Fatou component is biholomorphic to ^{*})^{d-1} and attracts to a common boundary fixed point.
- No orbit in these components can converge tangent to a complex submanifold.

## Abstract

We generalise a recent example by F. Bracci, J. Raissy and B. Stens{\o}nes 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.