Escaping Fatou components with disjoint hyperbolic limit sets

We construct automorphisms of C2 of constant Jacobian with a cycle of escaping Fatou components, on which there are exactly two limit functions, both of rank 1. On each such Fatou component, the limit sets for these limit functions are two disjoint hyperbolic subsets of the line at infinity. In the literature there are currently very few examples of automorphisms of C2 with rank one limit sets on the boundary of Fatou components. To our knowledge, this is the first example in which such limit sets are hyperbolic, and moreover different limit sets of rank 1 coexist.