We show the existence of automorphisms $F$ of $\mathbb{C}^{2}$ with a non-recurrent Fatou component $\Omega $ biholomorphic to $\mathbb{C}\times \mathbb{C}^{*}$ that is the basin of attraction to an invariant entire curve on which $F$ acts as an irrational rotation. We further show that the biholomorphism $\Omega \to \mathbb{C}\times \mathbb{C}^{*}$ can be chosen such that it conjugates $F$ to a translation $(z,w)\mapsto (z+1,w)$, making $\Omega $ a parabolic cylinder as recently defined by L. Boc Thaler, F. Bracci, and H. Peters. $F$ and $\Omega $ are obtained by blowing up a fixed point of an automorphism of $\mathbb{C}^{2}$ with a Fatou component of the same biholomorphic type attracted to that fixed point, established by F. Bracci, J. Raissy, and B. Stensønes. A crucial step is the application of the density property of a suitable Lie algebra to show that the automorphism in their work can be chosen such that it fixes a coordinate axis. We can then remove the proper transform of that axis from the blow-up to obtain an $F$-stable subset of the blow-up that is biholomorphic to $\mathbb{C}^{2}$. Thus, we can interpret $F$ as an automorphism of $\mathbb{C}^{2}$.

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We construct automorphisms of C d admitting an arbitrary (finite) number of non-recurrent Fatou components, each biholomorphic to C × (C * ) d−1 and all attracting to the same fixed point contained in the boundary of each of the components. These automorphisms can be chosen such that each Fatou component is invariant or such that the components are grouped into periodic cycles of any (sensible) common period. Convergence to the fixed point in these attracting Fatou components is not tangent to any one complex direction and the whole family of Fatou components avoids hypersurfaces tangent to each coordinate hyperplane. The construction is a generalisation of a result by F. Bracci, J. Raissy and B. Stensønes in the spirit of a generalised Leau-Fatou flower.

We give an introduction to the study of local dynamics of iterated holomorphic mappings near a fixed point via local conjugations in one and several complex variables. Starting with the systematic construction of formal conjugations to Poincaré–Dulac normal form in general and formal linearisations in particular, we discuss conditions for convergence of the normalising series in terms of the linear part. The convergence is closely related to the size of denominators that show up in the normalising series and depend only on the linear part of the mapping. Hence, we speak of a small-divisor problem. The central result on the linearisation problem is the Brjuno condition, that ensures holomorphic linearisability. The first part of these notes is a survey of the local dynamics in one variable from the viewpoint of local normalisations. In this case, the picture is fairly complete. In the second part we introduce the additional obstacles to both formal and holomorphic normalisations that emerge from interactions of multiple eigenvalues, such as resonances, that preclude the Brjuno condition in particular. For these cases, we proceed with several generalisations of the Brjuno condition, that allow us to find convergent conjugations to at least partial normalisations. The last part reviews a recent application of such a partial normalisation to illuminate the local dynamics near certain one-resonant fixed points.