Escaping Fatou components with disjoint hyperbolic limit sets

We construct automorphisms of C(2 )of constant Jacobian with a cycle of escaping Fatou components, on which there are exactly two limit functions, both of rank 1. On each suchFatou component, the limit sets for these limit functions are two disjoint hyperbolic subsetsof the line at infinity. In the literature there are currently very few examples of automorphisms of C(2 )with rank one limit sets on the boundary of Fatou components. To our knowledge, thisis the first example in which such limit sets are hyperbolic, and moreover different limit setsof rank 1 coexist