Automorphisms of $\mathbb{C}^2$ with non-recurrent Siegel cylinders

Luka Boc Thaler, Filippo Bracci, Han Peters

Published 2019-07-17Version 1

A non-recurrent Siegel cylinder is an invariant, non-recurrent Fatou component $\Omega$ of an automorphism $F$ of $\mathbb{C}^2$ satisfying: (1) The closure of the $\omega$-limit set of $F$ on $\Omega$ contains an isolated fixed point, (2) there exists a univalent map $\Phi$ from $\Omega$ into $\mathbb{C}^2$ conjugating $F$ to the translation $(z,w) \rightarrow (z+1, w)$, and (3) every limit map of $\{F^{\circ n}\}$ on $\Omega$ has one-dimensional image. In this paper we prove that the existence of non-recurrent Siegel cylinders. In fact, we provide an explicit class of maps having such Fatou components, and show that examples in this class can be constructed as compositions of shears and overshears.