Automorphisms of $\mathbb C^2$ with parabolic cylinders

TL;DR

This paper investigates the structure of certain invariant regions called parabolic cylinders in complex two-dimensional automorphisms, providing explicit examples and construction methods involving shears and overshears.

Contribution

It proves the existence of parabolic cylinders for a specific class of automorphisms and demonstrates their construction via compositions of shears and overshears.

Findings

Existence of parabolic cylinders in explicit automorphisms

Construction of examples through shears and overshears

Characterization of limit maps on these cylinders

Abstract

A {\sl parabolic 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 $(z,w) \mapsto (z+1, w)$, and (3) every limit map of $\{F^{\circ n}\}$ on $\Omega$ has one-dimensional image. In this paper we prove the existence of parabolic cylinders for an explicit class of maps, and show that examples in this class can be constructed as compositions of shears and overshears.