Automorphisms of ℂ² with cycles of escaping Fatou components with hyperbolic limit sets

We study the stable dynamics of non-polynomial automorphisms of [Formula: see text], more precisely we construct a special class of transcendental Hénon maps of the form [Formula: see text] with [Formula: see text] a natural number and [Formula: see text]. One of the points of greatest interest is that we get a combinatorial dynamic behavior, indeed there are cycles of escaping Fatou components where the dynamics vary depending on whether [Formula: see text] is even or odd. Moreover, we have two distinct limit functions on each cycle, both of which have generic rank 1. Additionally, each Fatou component in each cycle has two disjoint and hyperbolic limit sets on the line at infinity, with the exception of the Fatou components belonging to the short cycle of period [Formula: see text] (that occurs if [Formula: see text] is odd) which have the same hyperbolic limit set.