On the rank-one limit sets for maps in the constant-Jacobian shear group of ${\mathbb{C}}^2$

Let ${\mathcal{S}}_c$ be the subgroup of $Aut({\mathbb{C}}^2)$ generated by affine maps and shear automorphisms with constant Jacobian. We determine the possible conformal types of the rank-one limit set of an invariant recurrent Fatou component when the induced dynamics has infinite order. In particular we show that, under this infinite-order hypothesis, the limit set of a finite composition of Hénon maps (polynomial or transcendental), can only be conformally equivalent to a disk or an annulus. Thus we exclude the remaining three conformal types, i.e. ${\mathbb{D}}^{\ast }$, ${\mathbb{C}}^{\ast }$ and $\mathbb{C}$, from the classification result by Arosio, Benini, Fornæss and Peters. We also prove that the set of fixed points of a finite composition of Hénon maps (polynomial or transcendental) contains no entire curves.