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

Small divisors in discrete local holomorphic dynamics

“…, d , they use Pöschel's theorem 3.4.7 to ensure the existence of these Siegel hypersurfaces. In [Rep21a], we extend their considerations to the case k > 1 and observe that under the same partial Brjuno condition, for any ℓ ∈ Theorem 3.4.16 yields coordinates such that the tail on the parabolic shadow in (3.8.3) is of order O (u ℓ ) on a full neighbourhood of the origin. This allows us to classify the stable orbits on some neighbourhood of the origin: Theorem 3.8.9 (Reppekus [Rep21a]).…”

“…More precisely, we observed in [Rep21a] that each B h is connected, if and only if gcd(α) = 1. Generally, each B h consists of ℓ = gcd(α) components that are cyclically permuted by F and these components correspond to the invariant attracting basins of F •ℓ , which is one-resonant of generator α 0 := α/ℓ (gcd(α 0 ) = 1), weighted order k • ℓ and parabolically attracting.…”

“…Another interesting observation in [Rep21a] is, that in this case, the convergence to 0 of orbits in B h is never tangent to any one complex direction, since the arguments accumulate on the whole central hyperplane from 3.8.7. In fact orbits accumulate at each complex direction outside the coordinate hyperplanes.…”