Automorphisms of $\mathbb{C}^2$ with cycles of escaping Fatou components with hyperbolic limit sets

We study the stable dynamics of non-polynomial automorphisms of $\mathbb{C}^2$ of the form $F(z,w)=(e^{-z^m}+ \delta e^{\frac{2 \pi}{m}i}\, w\,,\,z)$, with $m\ge 2$ a natural number and $\mathbb{R}\ni\delta>2$. If $m$ is even, there are $\frac{m}{2}$ cycles of escaping Fatou components, all of period $2m$. If $m$ is odd there are $\frac{m-1}{2}$ cycles of escaping Fatou components of period $2m$ and just one cycle of escaping Fatou components of period $m$. These maps have two distinct limit functions on each cycle, both of which have generic rank 1. Each Fatou component in each cycle has two disjoint and hyperbolic limit sets on the line at infinity, except for the Fatou components that belong to the unique cycle of period $m$: the latter in fact have the same hyperbolic limit set on the line at infinity.