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

A {\sl parabolic cylinder} is an invariant, non-recurrent Fatou component $\mathrm{\Omega }$ of an automorphism $F$ of ${\mathbb{C}}^2$ satisfying: (1) The closure of the $\omega$-limit set of $F$ on $\mathrm{\Omega }$ contains an isolated fixed point, (2) there exists a univalent map $\mathrm{\Phi }$ from $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ 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.